Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
SlideShare a Scribd company logo
Section 4.4
                  Curve Sketching

                V63.0121.002.2010Su, Calculus I

                        New York University


                        June 10, 2010



Announcements
   Homework 4 due Tuesday


                                              .   .   .   .   .   .
Announcements




           Homework 4 due Tuesday




                                                                      .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010       2 / 45
Objectives




           given a function, graph it
           completely, indicating
                   zeroes (if easy)
                   asymptotes if applicable
                   critical points
                   local/global max/min
                   inflection points




                                                                      .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010       3 / 45
Why?




  Graphing functions is like
  dissection




                                                                      .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010       4 / 45

Recommended for you

A basic introduction to learning
A basic introduction to learningA basic introduction to learning
A basic introduction to learning

This document provides an introduction to machine learning concepts including loss functions, empirical risk, and two basic methods of learning - least squared error and nearest neighborhood. It describes how machine learning aims to find an optimal function that minimizes empirical risk under a given loss function. Least squared error learning is discussed as minimizing the squared differences between predictions and labels. Nearest neighborhood is also introduced as an alternative method. The document serves as a high-level overview of fundamental machine learning principles.

machine learningfunction approximationloss functions
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: Antiderivatives

This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. Examples are provided of finding the antiderivative of power functions like x^3 through identifying the power rule relationship between a function and its derivative.

antiderivativecalculusfunction
April 3, 2014
April 3, 2014April 3, 2014
April 3, 2014

The document provides examples and explanations for graphing quadratic functions. It begins with an overview of how the a, b, and c values in the quadratic function y=ax2 + bx + c impact the graph. Examples are then worked through step-by-step to show how to find the axis of symmetry, vertex, y-intercept, and additional points to graph the function. An application example models the height of a basketball shot as a quadratic function to find the maximum height and time to reach it. The document concludes with a check your understanding example modeling the height of a dive.

Why?




  Graphing functions is like
  dissection … or diagramming
  sentences




                                                                      .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010       4 / 45
Why?




  Graphing functions is like
  dissection … or diagramming
  sentences
  You can really know a lot about
  a function when you know all of
  its anatomy.




                                                                      .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010       4 / 45
The Increasing/Decreasing Test

 Theorem (The Increasing/Decreasing Test)
 If f′ > 0 on (a, b), then f is increasing on (a, b). If f′ < 0 on (a, b), then f
 is decreasing on (a, b).

 Example
 Here f(x) = x3 + x2 , and f′ (x) = 3x2 + 2x.

                                                                      f
                                                                      .(x)
                                                           .′ (x)
                                                           f



                                                .


                                                                       .     .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                  June 10, 2010       5 / 45
Testing for Concavity
 Theorem (Concavity Test)
 If f′′ (x) > 0 for all x in (a, b), then the graph of f is concave upward on
 (a, b) If f′′ (x) < 0 for all x in (a, b), then the graph of f is concave
 downward on (a, b).

 Example
 Here f(x) = x3 + x2 , f′ (x) = 3x2 + 2x, and f′′ (x) = 6x + 2.
                                            .′′ (x)
                                            f                    f
                                                                 .(x)
                                                      .′ (x)
                                                      f




                                                .


                                                                        .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                 June 10, 2010       6 / 45

Recommended for you

Lesson 1: Functions
Lesson 1: FunctionsLesson 1: Functions
Lesson 1: Functions

This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.

v6301212010fv6301210412010f
Lesson 7: The Derivative as a Function
Lesson 7: The Derivative as a FunctionLesson 7: The Derivative as a Function
Lesson 7: The Derivative as a Function

This document contains lecture notes on derivatives from a Calculus I class at New York University. It discusses the derivative as a function, finding the derivative of other functions, and the relationship between a function and its derivative. The notes include examples of finding the derivative of the reciprocal function and state that if a function is decreasing on an interval, its derivative will be nonpositive on that interval, while if it is increasing the derivative will be nonnegative. It also contains proofs and graphs related to derivatives.

functionv6301210022010suderivative
Lesson 23: The Definite Integral (slides)
Lesson 23: The Definite Integral (slides)Lesson 23: The Definite Integral (slides)
Lesson 23: The Definite Integral (slides)

The document is a lecture note from an NYU Calculus I class covering the definite integral. It provides announcements about upcoming quizzes and exams. The content discusses computing definite integrals using Riemann sums and the limit of Riemann sums, as well as properties of the definite integral. Examples are given of calculating Riemann sums using different representative points in each interval.

calculusintegralv6301210162010sp
Graphing Checklist


To graph a function f, follow this plan:
  0. Find when f is positive, negative, zero,
     not defined.
  1. Find f′ and form its sign chart. Conclude
     information about increasing/decreasing
     and local max/min.
  2. Find f′′ and form its sign chart. Conclude
     concave up/concave down and inflection.
  3. Put together a big chart to assemble
     monotonicity and concavity data
  4. Graph!



                                                                      .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010       7 / 45
Outline



 Simple examples
    A cubic function
    A quartic function


 More Examples
   Points of nondifferentiability
   Horizontal asymptotes
   Vertical asymptotes
   Trigonometric and polynomial together
   Logarithmic



                                                                      .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010       8 / 45
Graphing a cubic


 Example
 Graph f(x) = 2x3 − 3x2 − 12x.




                                                                      .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010       9 / 45
Graphing a cubic


 Example
 Graph f(x) = 2x3 − 3x2 − 12x.

 (Step 0) First, let’s find the zeros. We can at least factor out one power
 of x:
                             f(x) = x(2x2 − 3x − 12)
 so f(0) = 0. The other factor is a quadratic, so we the other two roots
 are                        √
                                                     √
                       3 ± 32 − 4(2)(−12)        3 ± 105
                  x=                          =
                                  4                  4
 It’s OK to skip this step for now since the roots are so complicated.


                                                                      .   .   .     .      .      .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010       9 / 45

Recommended for you

Module 11 graph of functions PMR
Module 11 graph of functions PMRModule 11 graph of functions PMR
Module 11 graph of functions PMR

The document is a mathematics exam for Form 3 students on the topic of graphs of functions. It contains 35 multiple choice questions testing students' understanding of key concepts like linear functions, independent and dependent variables, interpreting graphs of functions, and the relationship between algebraic equations and their graphs. It also contains 5 short answer questions asking students to explain terms and describe graphs.

Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...

This document provides an introduction to global sensitivity analysis. It discusses how sensitivity analysis can quantify the sensitivity of a model output to variations in its input parameters. It introduces Sobol' sensitivity indices, which measure the contribution of each input parameter to the variance of the model output. The document outlines how Sobol' indices are defined based on decomposing the model output variance into terms related to individual input parameters and their interactions. It notes that Sobol' indices are generally estimated using Monte Carlo-type sampling approaches due to the high-dimensional integrals involved in their exact calculation.

program on quasi-monte carlo and high-dimensionalqmcglobal
Nonlinear Manifolds in Computer Vision
Nonlinear Manifolds in Computer VisionNonlinear Manifolds in Computer Vision
Nonlinear Manifolds in Computer Vision

This document provides an overview of a 2004 CVPR tutorial on nonlinear manifolds in computer vision. The tutorial is divided into four parts that cover: (1) motivation for studying nonlinear manifolds and how differential geometry can be useful in vision, (2) tools from differential geometry like manifolds, tangent spaces, and geodesics, (3) statistics on manifolds like distributions and estimation, and (4) algorithms and applications in computer vision like pose estimation, tracking, and optimal linear projections. Nonlinear manifolds are important in computer vision as the underlying spaces in problems involving constraints like objects on circles or matrices with orthogonality constraints are nonlinear. Differential geometry provides a framework for generalizing tools from vector spaces to nonlinear

Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                                        .




                                                                          .   .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching               June 10, 2010   10 / 45
Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                                        .                     .               . −2
                                                                              x
                                                            2
                                                            .
                                  .                                           x
                                                                              . +1
                                −
                                . 1
                                                                              .′ (x)
                                                                              f
                                  .                           .
                                −
                                . 1                         2
                                                            .                 f
                                                                              .(x)


                                                                          .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                    June 10, 2010   10 / 45
Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                                  .                                       x
                                                                          . +1
                                −
                                . 1
                                                                          .′ (x)
                                                                          f
                                  .                       .
                                −
                                . 1                     2
                                                        .                 f
                                                                          .(x)


                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45
Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                            −
                            . .          .
                                         +                        .
                                                                  +
                                                                          x
                                                                          . +1
                              −
                              . 1
                                                                          .′ (x)
                                                                          f
                                  .                       .
                                −
                                . 1                     2
                                                        .                 f
                                                                          .(x)


                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45

Recommended for you

Module 10 Graphs Of Functions
Module 10 Graphs Of FunctionsModule 10 Graphs Of Functions
Module 10 Graphs Of Functions

This document contains 6 math problems involving graphing functions. Each problem has parts that involve: 1) Completing a table of values for a function. 2) Graphing the function on graph paper using given scales. 3) Finding specific values from the graph. 4) Drawing and finding values from a linear function related to the original. The problems provide practice graphing and extracting information from graphs of quadratic, cubic, rational, and other polynomial functions. The document demonstrates how to set up and solve multi-step math word problems involving graphing functions.

7.curves Further Mathematics Zimbabwe Zimsec Cambridge
7.curves   Further Mathematics Zimbabwe Zimsec Cambridge7.curves   Further Mathematics Zimbabwe Zimsec Cambridge
7.curves Further Mathematics Zimbabwe Zimsec Cambridge

The document discusses properties of curves defined by functions. It begins by listing objectives for understanding important points on graphs like maxima, minima, and inflection points. It emphasizes using graphing technology to experiment but not substitute for analytical work. Examples are provided to demonstrate finding maximums, minimums, intersections, and asymptotes of various functions. The key points are determining features of a curve from its defining function.

further mathematicsmathematicszimsec
Form 4 formulae and note
Form 4 formulae and noteForm 4 formulae and note
Form 4 formulae and note

This document contains notes and formulae on additional mathematics for Form 4. It covers topics such as functions, quadratic equations, quadratic functions, indices and logarithms, coordinate geometry, statistics, circular measure, differentiation, solutions of triangles, and index numbers. The key points covered include the definition of functions, the formula for the sum and product of roots of a quadratic equation, the axis of symmetry and nature of roots of quadratic functions, and common differentiation rules.

Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                            −
                            . .          .
                                         +                        .
                                                                  +
                                                                          x
                                                                          . +1
                              −
                              . 1
                            . .
                            +                                             .′ (x)
                                                                          f
                                                          .
                                −
                                . 1                     2
                                                        .                 f
                                                                          .(x)


                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45
Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                            −
                            . .          .
                                         +                        .
                                                                  +
                                                                          x
                                                                          . +1
                              −
                              . 1
                            . .
                            +            −
                                         .                                .′ (x)
                                                                          f
                                                          .
                                −
                                . 1                     2
                                                        .                 f
                                                                          .(x)


                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45
Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                            −
                            . .          .
                                         +                        .
                                                                  +
                                                                          x
                                                                          . +1
                              −
                              . 1
                            . .
                            +            −
                                         .                        .
                                                                  +       .′ (x)
                                                                          f
                                                          .
                                −
                                . 1                     2
                                                        .                 f
                                                                          .(x)


                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45
Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                            −
                            . .          .
                                         +                        .
                                                                  +
                                                                          x
                                                                          . +1
                              −
                              . 1
                            . .
                            +            −
                                         .                        .
                                                                  +       .′ (x)
                                                                          f
                                                          .
                            ↗−
                            . . 1                       2
                                                        .                 f
                                                                          .(x)


                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45

Recommended for you

P2 Graphs Function
P2  Graphs FunctionP2  Graphs Function
P2 Graphs Function

The document contains 10 math problems involving graphing functions and inequalities on Cartesian planes. The problems involve sketching graphs of functions, finding coordinates that satisfy equations, drawing lines to solve equations, and shading regions defined by inequalities. Tables are used to list x and y values satisfying equations.

Image Processing 3
Image Processing 3Image Processing 3
Image Processing 3

The document discusses spatial transformations and intensity transformations for image enhancement. Spatial transformations include scaling, rotation, translation, and shear, which can be represented using a matrix. Intensity transformations modify pixel intensities based on a transfer function and are used for contrast enhancement. Common transformations include image negative, powers, and logarithms. The appropriate transformation depends on the image content and which intensity values need enhancement.

Lesson18 -maximum_and_minimum_values_slides
Lesson18 -maximum_and_minimum_values_slidesLesson18 -maximum_and_minimum_values_slides
Lesson18 -maximum_and_minimum_values_slides

This document is a section from a Calculus I course at New York University covering maximum and minimum values. It begins with announcements about exams and assignments. The objectives are to understand the Extreme Value Theorem and Fermat's Theorem, and use the Closed Interval Method to find extreme values. The document then covers the definitions of extreme points/values and the statements of the Extreme Value Theorem and Fermat's Theorem. It provides examples to illustrate the necessity of the hypotheses in the theorems. The focus is on using calculus concepts like continuity and differentiability to determine maximum and minimum values of functions on closed intervals.

Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                            −
                            . .          .
                                         +                        .
                                                                  +
                                                                          x
                                                                          . +1
                              −
                              . 1
                            . .
                            +            −
                                         .                        .
                                                                  +       .′ (x)
                                                                          f
                                                          .
                            ↗−
                            . . 1        ↘
                                         .              2
                                                        .                 f
                                                                          .(x)


                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45
Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                            −
                            . .          .
                                         +                        .
                                                                  +
                                                                          x
                                                                          . +1
                              −
                              . 1
                            . .
                            +            −
                                         .                       .
                                                                 +        .′ (x)
                                                                          f
                                                          .
                            ↗−
                            . . 1        ↘
                                         .              2
                                                        .        ↗
                                                                 .        f
                                                                          .(x)


                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45
Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                            −
                            . .          .
                                         +                        .
                                                                  +
                                                                          x
                                                                          . +1
                               −
                               . 1
                            . .
                            +            −
                                         .                       .
                                                                 +        .′ (x)
                                                                          f
                                                          .
                            ↗−
                            . . 1        ↘
                                         .              2
                                                        .        ↗
                                                                 .        f
                                                                          .(x)
                              m
                              . ax

                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45
Step 1: Monotonicity



                                 f(x) = 2x3 − 3x2 − 12x
                        =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2)

 We can form a sign chart from this:

                            −
                            .             −
                                        . .               .       .
                                                                  +
                                                                          . −2
                                                                          x
                                                        2
                                                        .
                            −
                            . .          .
                                         +                        .
                                                                  +
                                                                          x
                                                                          . +1
                               −
                               . 1
                            . .
                            +            −
                                         .                       .
                                                                 +        .′ (x)
                                                                          f
                                                         .
                            ↗−
                            . . 1        ↘
                                         .             2
                                                       .         ↗
                                                                 .        f
                                                                          .(x)
                              m
                              . ax                    m
                                                      . in

                                                                      .   .        .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                    June 10, 2010   10 / 45

Recommended for you

Lesson 19: The Mean Value Theorem (slides)
Lesson 19: The Mean Value Theorem (slides)Lesson 19: The Mean Value Theorem (slides)
Lesson 19: The Mean Value Theorem (slides)

The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.

increasingconcave upv6301212011sp
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)

This document contains lecture notes from a Calculus I class at New York University on October 21, 2010. It discusses the objectives and outline for sections 3.1-3.2 on exponential and logarithmic functions. Key points include definitions of exponential functions, properties like ax+y = axay, and conventions for extending exponents to non-positive integers and irrational numbers.

v6301210212010fv6301212010ffunction
Lesson 7: The Derivative (Section 21 slide)
Lesson 7: The Derivative (Section 21 slide)Lesson 7: The Derivative (Section 21 slide)
Lesson 7: The Derivative (Section 21 slide)

This document is from a Calculus I course at New York University. It outlines the objectives and content for Sections 2.1-2.2 on the derivative and rates of change. The sections will define the derivative, discuss how it models rates of change, and how to find the derivative of various functions graphically and numerically. It will also cover how to find higher-order derivatives and sketch the derivative graph from a function graph.

v6301210212010fv6301212010ffunction
Step 2: Concavity




                                        f′ (x) = 6x2 − 6x − 12
                                 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1)

 Another sign chart:                     .




                                                                           .   .   .      .      .     .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching               June 10, 2010   11 / 45
Step 2: Concavity




                                        f′ (x) = 6x2 − 6x − 12
                                 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1)

 Another sign chart:                     .
                                                                               .′′ (x)
                                                                               f
                                                .
                                              ./2
                                              1                                f
                                                                               .(x)




                                                                           .   .      .      .      .     .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   11 / 45
Step 2: Concavity




                                        f′ (x) = 6x2 − 6x − 12
                                 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1)

 Another sign chart:                     .

                                −
                                . −                                            .′′ (x)
                                                                               f
                                                .
                                              ./2
                                              1                                f
                                                                               .(x)




                                                                           .   .      .      .      .     .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   11 / 45
Step 2: Concavity




                                        f′ (x) = 6x2 − 6x − 12
                                 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1)

 Another sign chart:                     .

                                −
                                . −                    . +
                                                       +                       .′′ (x)
                                                                               f
                                                .
                                              ./2
                                              1                                f
                                                                               .(x)




                                                                           .   .      .      .      .     .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   11 / 45

Recommended for you

Lesson 13: Related Rates Problems
Lesson 13: Related Rates ProblemsLesson 13: Related Rates Problems
Lesson 13: Related Rates Problems

This document contains lecture notes on related rates from a Calculus I class at New York University. It begins with announcements about assignments and no class on a holiday. It then outlines the objectives of learning to use derivatives to understand rates of change and model word problems. Examples are provided, including an oil slick problem worked out in detail. Strategies for solving related rates problems are discussed. Further examples on people walking and electrical resistors are presented.

rate of changefunctionv6301210022010su
Lesson 27: Integration by Substitution (Section 041 slides)
Lesson 27: Integration by Substitution (Section 041 slides)Lesson 27: Integration by Substitution (Section 041 slides)
Lesson 27: Integration by Substitution (Section 041 slides)

The document provides notes from a Calculus I class at New York University on December 13, 2010. It includes announcements about upcoming review sessions and the final exam. It then outlines the objectives and topics to be covered, which is integration by substitution. Examples are provided to demonstrate how to use u-substitution to evaluate indefinite integrals. The key steps are to let u equal an expression involving x, find du in terms of dx, and then substitute into the integral.

integrationantiderivativeintegral
Lesson 17: Indeterminate Forms and L'Hôpital's Rule
Lesson 17: Indeterminate Forms and L'Hôpital's RuleLesson 17: Indeterminate Forms and L'Hôpital's Rule
Lesson 17: Indeterminate Forms and L'Hôpital's Rule

This document is from a Calculus I class lecture on L'Hopital's rule given at New York University. It begins with announcements and objectives for the lecture. It then provides examples of limits that are indeterminate forms to motivate L'Hopital's rule. The document explains the rule and when it can be used to evaluate limits. It also introduces the mathematician L'Hopital and applies the rule to examples introduced earlier.

calculusfunctionindeterminate
Step 2: Concavity




                                        f′ (x) = 6x2 − 6x − 12
                                 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1)

 Another sign chart:                     .

                                −
                                . −                    . +
                                                       +                       .′′ (x)
                                                                               f
                                                .
                                 .
                                 ⌢            ./2
                                              1                                f
                                                                               .(x)




                                                                           .   .      .      .      .     .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   11 / 45
Step 2: Concavity




                                        f′ (x) = 6x2 − 6x − 12
                                 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1)

 Another sign chart:                     .

                                −
                                . −                    . +
                                                       +                       .′′ (x)
                                                                               f
                                                .
                                 .
                                 ⌢            ./2
                                              1         .
                                                        ⌣                      f
                                                                               .(x)




                                                                           .   .      .      .      .     .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   11 / 45
Step 2: Concavity




                                        f′ (x) = 6x2 − 6x − 12
                                 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1)

 Another sign chart:                     .

                                −
                                . −                    . +
                                                       +                       .′′ (x)
                                                                               f
                                                .
                                 .
                                 ⌢            ./2
                                              1         .
                                                        ⌣                      f
                                                                               .(x)
                                              I
                                              .P




                                                                           .   .      .      .      .     .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   11 / 45
Step 3: One sign chart to rule them all



 Remember, f(x) = 2x3 − 3x2 − 12x.

                                        .




                                                                          .   .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching               June 10, 2010   12 / 45

Recommended for you

Lesson 11: Implicit Differentiation
Lesson 11: Implicit DifferentiationLesson 11: Implicit Differentiation
Lesson 11: Implicit Differentiation

Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.

v6301210342009fcalculusv6301212009f
Lesson 10: The Chain Rule (Section 21 slides)
Lesson 10: The Chain Rule (Section 21 slides)Lesson 10: The Chain Rule (Section 21 slides)
Lesson 10: The Chain Rule (Section 21 slides)

The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.

v6301210212010fcompositionv6301212010f
Lesson 9: The Product and Quotient Rules (Section 21 slides)
Lesson 9: The Product and Quotient Rules (Section 21 slides)Lesson 9: The Product and Quotient Rules (Section 21 slides)
Lesson 9: The Product and Quotient Rules (Section 21 slides)

The derivative of a sum of functions is the sum of the derivatives of those functions, but the derivative of a product or a quotient of functions is not so simple. We'll derive and use the product and quotient rule for these purposes. It will allow us to find the derivatives of other trigonometric functions, and derivatives of power functions with negative whole number exponents.

v6301210212010fsecantv6301212010f
Step 3: One sign chart to rule them all



 Remember, f(x) = 2x3 − 3x2 − 12x.

                            −
                      . . . .
                      +                   −
                                          .                .
                                                           +              .′ (x)
                                                                          f
                                                    .
                      ↗− ↘
                      . . 1 .             ↘
                                          .       2
                                                  .        ↗
                                                           .              m
                                                                          . onotonicity




                                                                      .            .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                        June 10, 2010   12 / 45
Step 3: One sign chart to rule them all



 Remember, f(x) = 2x3 − 3x2 − 12x.

                     . .
                     +            −
                                  . .       −
                                            .                .
                                                             +              .′ (x)
                                                                            f
                                                .
                     ↗−
                     . . 1        ↘
                                  .         ↘ .
                                            .  2             ↗
                                                             .              m
                                                                            .′′ onotonicity
                    −
                    . −          −
                                 . − .     . +
                                           +                . +
                                                            +               f
                                                                            . (x)
                     .
                     ⌢            .
                                  ⌢ 1/2
                                      .     .
                                            ⌣                .
                                                             ⌣              c
                                                                            . oncavity




                                                                        .            .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)     Section 4.4 Curve Sketching                        June 10, 2010   12 / 45
Step 3: One sign chart to rule them all



 Remember, f(x) = 2x3 − 3x2 − 12x.

                     . .
                     +         −
                               . .       −
                                         .        .
                                                  +                       .′ (x)
                                                                          f
                                               .
                     ↗−
                     . . 1     ↘
                               .         ↘ .
                                         .   2    ↗
                                                  .                       m
                                                                          .′′ onotonicity
                    −
                    . −       −
                              . − . . + +        . +
                                                 +                        f
                                                                          . (x)
                     .
                     ⌢         ⌢ ./2 .
                               .   1     ⌣        .
                                                  ⌣                       c
                                                                          . oncavity
                           7
                           ..    −
                                 . 6 1/2   −.
                                           . 20                           f
                                                                          .(x)
                                     .
                          −
                          . 1      .
                                   1/2       2
                                             .                            s
                                                                          . hape of f
                         m
                         . ax      I
                                   .P       m
                                            . in




                                                                      .            .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                        June 10, 2010   12 / 45
Combinations of monotonicity and concavity




                                        I
                                        .I                              I
                                                                        .


                                                       .



                                        I
                                        .II                        I
                                                                   .V




                                                                            .   .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)     Section 4.4 Curve Sketching                   June 10, 2010   13 / 45

Recommended for you

Lesson 4: Calculating Limits (Section 41 slides)
Lesson 4: Calculating Limits (Section 41 slides)Lesson 4: Calculating Limits (Section 41 slides)
Lesson 4: Calculating Limits (Section 41 slides)

This document provides an overview of calculating limits from a Calculus I course at New York University. It begins with announcements about homework being due and includes sections on recalling the concept of limit, basic limits, limit laws, limits with algebra, and important trigonometric limits. The document uses examples, definitions, proofs, and the error-tolerance game to explain how to calculate limits, limit laws, and applying algebra to limits. It provides the essential information and objectives for understanding how to compute elementary limits.

limit lawsv6301212010fv6301210412010f
Lesson 7: The Derivative (Section 41 slides)
Lesson 7: The Derivative (Section 41 slides)Lesson 7: The Derivative (Section 41 slides)
Lesson 7: The Derivative (Section 41 slides)

The document outlines topics to be covered in Calculus I class sessions on the derivative and rates of change, including: defining the derivative at a point and using it to find the slope of the tangent line to a curve at that point; examples of derivatives modeling rates of change; and how to find the derivative function and second derivative of a given function. It provides learning objectives, an outline of topics, and an example problem worked out graphically and numerically to illustrate finding the slope of the tangent line.

v6301212010fv6301210412010ffunction
Lecture7
Lecture7Lecture7
Lecture7

This lecture covers various solution methods for unconstrained optimization problems, including: 1) Line search methods like dichotomous search and the Fibonacci and golden section methods for one-dimensional problems. 2) Newton's method and the false position method for curve fitting to minimize functions in one dimension. 3) Descent methods for multidimensional problems like the gradient method, which follows the negative gradient direction at each step, and how scaling can improve convergence rates.

Combinations of monotonicity and concavity
                                                                                .
                                                                                decreasing,
                                                                                concave
                                                                                down

                                        I
                                        .I                              I
                                                                        .


                                                       .



                                        I
                                        .II                        I
                                                                   .V




                                                                            .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)     Section 4.4 Curve Sketching                      June 10, 2010   13 / 45
Combinations of monotonicity and concavity
                     .                                                          .
                     increasing,                                                decreasing,
                     concave                                                    concave
                     down                                                       down

                                        I
                                        .I                              I
                                                                        .


                                                       .



                                        I
                                        .II                        I
                                                                   .V




                                                                            .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)     Section 4.4 Curve Sketching                      June 10, 2010   13 / 45
Combinations of monotonicity and concavity
                     .                                                          .
                     increasing,                                                decreasing,
                     concave                                                    concave
                     down                                                       down

                                        I
                                        .I                              I
                                                                        .


                                                       .



                                        I
                                        .II                        I
                                                                   .V

                     .
                     decreasing,
                     concave up
                                                                            .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)     Section 4.4 Curve Sketching                      June 10, 2010   13 / 45
Combinations of monotonicity and concavity
                     .                                                          .
                     increasing,                                                decreasing,
                     concave                                                    concave
                     down                                                       down

                                        I
                                        .I                              I
                                                                        .


                                                       .



                                        I
                                        .II                        I
                                                                   .V

                     .                                                          .
                     decreasing,                                                increasing,
                     concave up                                                 concave up
                                                                            .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)     Section 4.4 Curve Sketching                      June 10, 2010   13 / 45

Recommended for you

Lesson 3: Limits (Section 21 slides)
Lesson 3: Limits (Section 21 slides)Lesson 3: Limits (Section 21 slides)
Lesson 3: Limits (Section 21 slides)

This document contains lecture notes from a Calculus I class at New York University on September 14, 2010. The notes cover announcements, guidelines for written homework, a rubric for grading homework, examples of good and bad homework, and objectives for the concept of limits. The bulk of the document discusses the heuristic definition of a limit using an error-tolerance game approach, provides examples to illustrate the game, and outlines the path to a precise definition of a limit.

v6301210212010fv6301212010ffunction
Lesson 1: Functions
Lesson 1: FunctionsLesson 1: Functions
Lesson 1: Functions

No, this is not a function because the same input of 2 is mapped to two different outputs of 4 and 5. For a relation to be a function, each input must map to a unique output. V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33

v6301212010fv6301210412010f
Lesson 28: Integration by Subsitution
Lesson 28: Integration by SubsitutionLesson 28: Integration by Subsitution
Lesson 28: Integration by Subsitution

The document is a section from a Calculus I course at NYU from June 22, 2010. It discusses using the substitution method to evaluate indefinite integrals. Specifically, it provides an example of using the substitution u=x^2+3 to evaluate the integral of (x^2+3)^3 4x dx. The solution transforms the integral into an integral of u^3 du and evaluates it to be 1/4 u^4 + C.

integrationcalculusfunction
Step 3: One sign chart to rule them all



 Remember, f(x) = 2x3 − 3x2 − 12x.

                     . .
                     +        −
                              . .       −
                                        .        .
                                                 +                        .′ (x)
                                                                          f
                                              .
                     ↗−
                     . . 1    ↘
                              .         ↘ .
                                        .   2    ↗
                                                 .                        m
                                                                          .′′ onotonicity
                    −
                    . −      −
                             . − . . + +        . +
                                                +                         f
                                                                          . (x)
                     .
                     ⌢        ⌢ ./2 .
                              .   1     ⌣        .
                                                 ⌣                        c
                                                                          . oncavity
                          7
                          ..    −
                                . 6 1/2   −.
                                          . 20                            f
                                                                          .(x)
                                    .
                      . . 1
                         −        .
                                  1/2       2
                                            .                             s
                                                                          . hape of f
                        m
                        . ax      I
                                  .P       m
                                           . in




                                                                      .            .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                        June 10, 2010   14 / 45
Step 3: One sign chart to rule them all



 Remember, f(x) = 2x3 − 3x2 − 12x.

                     . .
                     +        −
                              . .       −
                                        .        .
                                                 +                        .′ (x)
                                                                          f
                                              .
                     ↗−
                     . . 1    ↘
                              .         ↘ .
                                        .   2    ↗
                                                 .                        m
                                                                          .′′ onotonicity
                    −
                    . −      −
                             . − . . + +        . +
                                                +                         f
                                                                          . (x)
                     .
                     ⌢        ⌢ ./2 .
                              .   1     ⌣        .
                                                 ⌣                        c
                                                                          . oncavity
                          7
                          ..    −
                                . 6 1/2   −.
                                          . 20                            f
                                                                          .(x)
                                    .
                      . . 1 . ./2
                         −        1         2
                                            .                             s
                                                                          . hape of f
                        m
                        . ax      I
                                  .P       m
                                           . in




                                                                      .            .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                        June 10, 2010   14 / 45
Step 3: One sign chart to rule them all



 Remember, f(x) = 2x3 − 3x2 − 12x.

                     . .
                     +        −
                              . .       −
                                        .        .
                                                 +                        .′ (x)
                                                                          f
                                              .
                     ↗−
                     . . 1    ↘
                              .         ↘ .
                                        .   2    ↗
                                                 .                        m
                                                                          .′′ onotonicity
                    −
                    . −      −
                             . − . . + +        . +
                                                +                         f
                                                                          . (x)
                     .
                     ⌢        ⌢ ./2 .
                              .   1     ⌣        .
                                                 ⌣                        c
                                                                          . oncavity
                          7
                          ..    −
                                . 6 1/2   −.
                                          . 20                            f
                                                                          .(x)
                                    .
                      . . 1 . ./2 .
                         −        1         2
                                            .                             s
                                                                          . hape of f
                        m
                        . ax      I
                                  .P       m
                                           . in




                                                                      .            .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                        June 10, 2010   14 / 45
Step 3: One sign chart to rule them all



 Remember, f(x) = 2x3 − 3x2 − 12x.

                     . .
                     +        −
                              . .       −
                                        .        .
                                                 +                        .′ (x)
                                                                          f
                                              .
                     ↗−
                     . . 1    ↘
                              .         ↘ .
                                        .   2    ↗
                                                 .                        m
                                                                          .′′ onotonicity
                    −
                    . −      −
                             . − . . + +        . +
                                                +                         f
                                                                          . (x)
                     .
                     ⌢        ⌢ ./2 .
                              .   1     ⌣        .
                                                 ⌣                        c
                                                                          . oncavity
                          7
                          ..    −
                                . 6 1/2   −.
                                          . 20                            f
                                                                          .(x)
                                    .
                      . . 1 . ./2 .
                         −        1         2
                                            .     .                       s
                                                                          . hape of f
                        m
                        . ax      I
                                  .P       m
                                           . in




                                                                      .            .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching                        June 10, 2010   14 / 45

Recommended for you

Lesson 15: Exponential Growth and Decay
Lesson 15: Exponential Growth and DecayLesson 15: Exponential Growth and Decay
Lesson 15: Exponential Growth and Decay

This document contains lecture notes on exponential growth and decay from a Calculus I class at New York University. It begins with announcements about an upcoming review session, office hours, and midterm exam. It then outlines the topics to be covered, including the differential equation y=ky, modeling population growth, radioactive decay including carbon-14 dating, Newton's law of cooling, and continuously compounded interest. Examples are provided of solving various differential equations representing exponential growth or decay. The document explains that many real-world situations exhibit exponential behavior due to proportional growth rates.

lesson 15: exponential growth and decay
Lesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential FunctionsLesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential Functions

linear function, quadratic functions, cubic functions, power functions, transformations, and compositions.

v6301210212010fcompositiongraph
Lesson 12: Linear Approximation
Lesson 12: Linear ApproximationLesson 12: Linear Approximation
Lesson 12: Linear Approximation

This document summarizes a calculus lecture on linear approximations. It provides examples of using the tangent line to approximate the sine function at different points. Specifically, it estimates sin(61°) by taking linear approximations about 0 and about 60°. The linear approximation about 0 is x, giving a value of 1.06465. The linear approximation about 60° uses the fact that the sine is √3/2 and the derivative is √3/2 at π/3, giving a better approximation than using 0.

calculusfunctionv6301210162010sp
Step 4: Graph
                                                   f
                                                   .(x)




                      .(x) = 2x3 − 3x2 − 12x
                      f

                     ( √          )         . −1, 7)
                                            (
                                                .
                     . 3− 4105 , 0                            . 0, 0)
                                                              (
                                        .                 .                                 .
                                                                    . 1/2, −61/2)
                                                                    (                           ( . x
                                                                                                    √        )
                                                                .                               . 3+ 4105 , 0

                                                                        . 2, −20)
                                                                        (
                                                                              .

                                                7
                                                ..  −
                                                    . 61/2 −.
                                                           . 20                                         f
                                                                                                        .(x)
                                                        .
                                            . . 1 . ./2 .
                                               −      1      2
                                                             .                          .               s
                                                                                                        . hape of f
                                              m
                                              . ax    I
                                                      .P    m
                                                            . in            .       .       .       .          .   .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                         June 10, 2010     15 / 45
Step 4: Graph
                                                   f
                                                   .(x)




                      .(x) = 2x3 − 3x2 − 12x
                      f

                     ( √          )         . −1, 7)
                                            (
                                                .
                     . 3− 4105 , 0                            . 0, 0)
                                                              (
                                        .                 .                                 .
                                                                    . 1/2, −61/2)
                                                                    (                           ( . x
                                                                                                    √        )
                                                                .                               . 3+ 4105 , 0

                                                                        . 2, −20)
                                                                        (
                                                                              .

                                                7
                                                ..  −
                                                    . 61/2 −.
                                                           . 20                                         f
                                                                                                        .(x)
                                                        .
                                            . . 1 . ./2 .
                                               −      1      2
                                                             .                          .               s
                                                                                                        . hape of f
                                              m
                                              . ax    I
                                                      .P    m
                                                            . in            .       .       .       .          .   .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                         June 10, 2010     15 / 45
Step 4: Graph
                                                   f
                                                   .(x)




                      .(x) = 2x3 − 3x2 − 12x
                      f

                     ( √          )         . −1, 7)
                                            (
                                                .
                     . 3− 4105 , 0                            . 0, 0)
                                                              (
                                        .                 .                                 .
                                                                    . 1/2, −61/2)
                                                                    (                           ( . x
                                                                                                    √        )
                                                                .                               . 3+ 4105 , 0

                                                                        . 2, −20)
                                                                        (
                                                                              .

                                                7
                                                ..  −
                                                    . 61/2 −.
                                                           . 20                                         f
                                                                                                        .(x)
                                                        .
                                            . . 1 . ./2 .
                                               −      1      2
                                                             .                          .               s
                                                                                                        . hape of f
                                              m
                                              . ax    I
                                                      .P    m
                                                            . in            .       .       .       .          .   .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                         June 10, 2010     15 / 45
Step 4: Graph
                                                   f
                                                   .(x)




                      .(x) = 2x3 − 3x2 − 12x
                      f

                     ( √          )         . −1, 7)
                                            (
                                                .
                     . 3− 4105 , 0                            . 0, 0)
                                                              (
                                        .                 .                                 .
                                                                    . 1/2, −61/2)
                                                                    (                           ( . x
                                                                                                    √        )
                                                                .                               . 3+ 4105 , 0

                                                                        . 2, −20)
                                                                        (
                                                                              .

                                                7
                                                ..  −
                                                    . 61/2 −.
                                                           . 20                                         f
                                                                                                        .(x)
                                                        .
                                            . . 1 . ./2 .
                                               −      1      2
                                                             .                          .               s
                                                                                                        . hape of f
                                              m
                                              . ax    I
                                                      .P    m
                                                            . in            .       .       .       .          .   .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                         June 10, 2010     15 / 45

Recommended for you

Lesson 6: Limits Involving ∞ (Section 21 slides)
Lesson 6: Limits Involving ∞ (Section 21 slides)Lesson 6: Limits Involving ∞ (Section 21 slides)
Lesson 6: Limits Involving ∞ (Section 21 slides)

This document is from a Calculus I course at New York University and covers limits involving infinity. It defines infinite limits, both limits approaching positive and negative infinity, and limits at vertical asymptotes. Examples are provided of known infinite limits, like limits of 1/x as x approaches 0. The document demonstrates finding one-sided limits at points where a function is not continuous using a number line to determine the sign of factors in the denominator.

infinityv6301210212010fv6301212010f
Lesson 21: Curve Sketching (Section 041 slides)
Lesson 21: Curve Sketching (Section 041 slides)Lesson 21: Curve Sketching (Section 041 slides)
Lesson 21: Curve Sketching (Section 041 slides)

This document provides an example of graphing a cubic function step-by-step. It begins by finding the derivative to determine where the function is increasing and decreasing. The derivative is factored to yield a sign chart showing the function is decreasing on (-∞,-1) and (2,∞) and increasing on (-1,2). This information will be used to sketch the graph of the cubic function.

v6301210212010fv6301212010ffunction
Lesson 21: Curve Sketching (Section 041 handout)
Lesson 21: Curve Sketching (Section 041 handout)Lesson 21: Curve Sketching (Section 041 handout)
Lesson 21: Curve Sketching (Section 041 handout)

The document provides an overview of curve sketching techniques for functions. It discusses objectives like graphing functions completely and indicating key features. It then covers topics like the increasing/decreasing test, concavity test, a checklist for graphing functions, and examples of graphing cubic, quartic, and other types of functions. Examples are worked through step-by-step to demonstrate the process of analyzing monotonicity, concavity, and asymptotes to fully sketch the graph of various functions.

v6301212010fv6301210412010ffunction
Step 4: Graph
                                                   f
                                                   .(x)




                      .(x) = 2x3 − 3x2 − 12x
                      f

                     ( √          )         . −1, 7)
                                            (
                                                .
                     . 3− 4105 , 0                            . 0, 0)
                                                              (
                                        .                 .                                 .
                                                                    . 1/2, −61/2)
                                                                    (                           ( . x
                                                                                                    √        )
                                                                .                               . 3+ 4105 , 0

                                                                        . 2, −20)
                                                                        (
                                                                              .

                                                7
                                                ..  −
                                                    . 61/2 −.
                                                           . 20                                         f
                                                                                                        .(x)
                                                        .
                                            . . 1 . ./2 .
                                               −      1      2
                                                             .                          .               s
                                                                                                        . hape of f
                                              m
                                              . ax    I
                                                      .P    m
                                                            . in            .       .       .       .          .   .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                         June 10, 2010     15 / 45
Graphing a quartic




 Example
 Graph f(x) = x4 − 4x3 + 10




                                                                      .   .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010   16 / 45
Graphing a quartic




 Example
 Graph f(x) = x4 − 4x3 + 10

 (Step 0) We know f(0) = 10 and lim f(x) = +∞. Not too many other
                                                 x→±∞
 points on the graph are evident.




                                                                      .   .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)   Section 4.4 Curve Sketching               June 10, 2010   16 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)




                                                                          .   .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching               June 10, 2010   17 / 45

Recommended for you

Lesson20 -derivatives_and_the_shape_of_curves_021_slides
Lesson20  -derivatives_and_the_shape_of_curves_021_slidesLesson20  -derivatives_and_the_shape_of_curves_021_slides
Lesson20 -derivatives_and_the_shape_of_curves_021_slides

- The document is a lecture on calculus from an NYU course. It discusses using derivatives to determine the monotonicity and concavity of functions. - There will be a quiz this week covering sections 3.3, 3.4, 3.5, and 3.7. Homework is due November 24. - The lecture covers using the first derivative to determine if a function is increasing or decreasing over an interval, and using the second derivative to determine if a graph is concave up or down.

Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)

We can put all of our graph-description techniques into a single picture. (The problem I did on the sketchpad is now prettified.)

v6301212010ffunctionv6301210412010f
Lesson 18: Derivatives and the Shapes of Curves
Lesson 18: Derivatives and the Shapes of CurvesLesson 18: Derivatives and the Shapes of Curves
Lesson 18: Derivatives and the Shapes of Curves

The document is a lecture note on derivatives and the shapes of curves. It discusses the mean value theorem and its applications to determining monotonicity and concavity of functions. Specifically, it covers: - Using the first derivative test to find intervals where a function is increasing or decreasing by determining where the derivative is positive or negative - Examples of applying this process to functions like x2 - 1 and x2/3(x + 2) - Definitions of increasing, decreasing, and concavity - How the second derivative test can determine concavity by examining the sign of the second derivative

increasingcalculusmonotonicity
Step 1: Monotonicity



                                            f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                                        .




                                                                              .   .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)           Section 4.4 Curve Sketching               June 10, 2010   17 / 45
Step 1: Monotonicity



                                             f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                                        0
                                        ..
                                                                               . x2
                                                                               4
                                        0
                                        .




                                                                               .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)            Section 4.4 Curve Sketching                  June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0
                                                                          . x2
                                                                          4
                                 0
                                 .




                                                                          .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                  June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +
                                                                          . x2
                                                                          4
                                 0
                                 .




                                                                          .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                  June 10, 2010   17 / 45

Recommended for you

Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)

The document outlines a calculus class lecture on the fundamental theorem of calculus, including recalling the second fundamental theorem, stating the first fundamental theorem, and providing examples of differentiating functions defined by integrals. It gives announcements for upcoming class sections and exam dates, lists the objectives of the current section, and provides an outline of topics to be covered including area as a function, statements and proofs of the theorems, and applications to differentiation.

antiderivativev6301210212010fintegral
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)

The document provides an overview of Section 5.4 on the Fundamental Theorem of Calculus from a Calculus I course at New York University. It outlines topics to be covered, including recalling the Second Fundamental Theorem, stating the First Fundamental Theorem, and differentiating functions defined by integrals. Examples are provided to illustrate using the theorems to find derivatives and integrals.

antiderivativev6301210212010fintegral
Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)

The document provides notes on curve sketching from a Calculus I class at New York University. It includes announcements about an upcoming homework assignment and class, objectives of learning how to graph functions, and notes on techniques for curve sketching such as using the increasing/decreasing test and concavity test to determine properties of a function's graph. Examples are provided of graphing cubic and quartic functions as well as functions with absolute values.

v6301210212010fv6301212010ffunction
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                   .
                                                                    +
                                                                          . x2
                                                                          4
                                 0
                                 .




                                                                          .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                  June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                    .
                                                                     +
                                                                          . x2
                                                                          4
                                 0
                                 .
                                                                0
                                                                ..
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .




                                                                          .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                  June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                    .
                                                                     +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                                0
                                                                ..
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .




                                                                          .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                  June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                    .
                                                                     +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                −
                                                .               0
                                                                ..
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .




                                                                          .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                  June 10, 2010   17 / 45

Recommended for you

Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)

Describe the monotonicity of f(x) = arctan(x). The derivative of arctan(x) is f'(x) = 1/(1+x^2), which is always positive. Therefore, f(x) = arctan(x) is an increasing function on (-∞,∞).

increasingconcave upv6301212010f
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: Antiderivatives

This document is from a Calculus I class at New York University and covers antiderivatives. It begins with announcements about an upcoming quiz. The objectives are to find antiderivatives of simple functions, remember that a function whose derivative is zero must be constant, and solve rectilinear motion problems. It then outlines finding antiderivatives through tabulation, graphically, and with rectilinear motion examples. The document provides examples of finding antiderivatives of power functions by using the power rule in reverse.

antiderivativecalculusfunction
Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)

The document summarizes key points from Section 5.3 of a calculus class at New York University on evaluating definite integrals. It outlines the objectives of the section, provides an example of using the midpoint rule to estimate a definite integral, and reviews properties and comparison properties of definite integrals, including how to interpret definite integrals as the net change of a function over an interval.

antiderivativeintegralv6301212010f
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                   .
                                                                    +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                −
                                                .               .. .
                                                                0 +
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .




                                                                          .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                  June 10, 2010   17 / 45
Step 1: Monotonicity



                                             f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0                   .
                                                     +                   .
                                                                         +
                                                                               . x2
                                                                               4
                                 0
                                 .
                               −
                               .                     −
                                                     .               .. .
                                                                     0 +
                                                                               . x − 3)
                                                                               (
                                                                     3
                                                                     .
                                        0
                                        ..                           0
                                                                     ..        .′ (x)
                                                                               f
                                        0
                                        .                            3
                                                                     .         f
                                                                               .(x)


                                                                               .        .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)            Section 4.4 Curve Sketching                    June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                   .
                                                                    +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                −
                                                .               .. .
                                                                0 +
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .
                               − 0
                               . ..                             0
                                                                ..        .′ (x)
                                                                          f
                                 0
                                 .                              3
                                                                .         f
                                                                          .(x)


                                                                          .        .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                    June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                   .
                                                                    +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                −
                                                .               .. .
                                                                0 +
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .
                               − 0
                               . ..             −
                                                .               0
                                                                ..        .′ (x)
                                                                          f
                                 0
                                 .                              3
                                                                .         f
                                                                          .(x)


                                                                          .        .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                    June 10, 2010   17 / 45

Recommended for you

Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)

This document outlines a calculus lecture on evaluating definite integrals. The lecture will: 1) Use the Evaluation Theorem to evaluate definite integrals and write antiderivatives as indefinite integrals. 2) Interpret definite integrals as the "net change" of a function over an interval. 3) Provide examples of evaluating definite integrals and estimating integrals using the midpoint rule. 4) Discuss properties of integrals such as additivity and illustrate how a definite integral from a to c can be broken into integrals from a to b and b to c.

antiderivativeintegralv6301212010f
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)

The document discusses the fundamental theorem of calculus. It begins by outlining the topics to be covered, including a review of the second fundamental theorem of calculus, an explanation of the first fundamental theorem of calculus, and examples of differentiating functions defined by integrals. It then provides more details on these topics, such as defining the integral as a limit, stating the second fundamental theorem, using integrals to represent concepts like distance traveled and mass, and working through an example of finding the derivative of a function defined by an integral.

antiderivativeintegralv6301212010f
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)

g(x) represents the area under the curve of f(t) from 0 to x. As x increases from 0 to 10, g(x) will increase, representing the accumulating area under f(t) over the interval [0,x].

antiderivativeintegralv6301212010f
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                   .
                                                                    +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                −
                                                .               .. .
                                                                0 +
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .
                               − 0
                               . ..             −
                                                .               .. .
                                                                0 +       .′ (x)
                                                                          f
                                 0
                                 .                              3
                                                                .         f
                                                                          .(x)


                                                                          .        .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                    June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                   .
                                                                    +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                −
                                                .               .. .
                                                                0 +
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .
                               − 0
                               . ..             −
                                                .               .. .
                                                                0 +       .′ (x)
                                                                          f
                               ↘ 0
                               . .                              3
                                                                .         f
                                                                          .(x)


                                                                          .        .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                    June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                   .
                                                                    +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                −
                                                .               .. .
                                                                0 +
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .
                               − 0
                               . ..             −
                                                .               .. .
                                                                0 +       .′ (x)
                                                                          f
                               ↘ 0
                               . .              ↘
                                                .               3
                                                                .         f
                                                                          .(x)


                                                                          .        .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                    June 10, 2010   17 / 45
Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                   .
                                                                    +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                −
                                                .               .. .
                                                                0 +
                                                                          . x − 3)
                                                                          (
                                                                3
                                                                .
                               − 0
                               . ..             −
                                                .               .. .
                                                                0 +       .′ (x)
                                                                          f
                               ↘ 0
                               . .              ↘
                                                .               3 ↗
                                                                . .       f
                                                                          .(x)


                                                                          .        .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                    June 10, 2010   17 / 45

Recommended for you

Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)

The Mean Value Theorem gives us tests for determining the shape of curves between critical points.

increasingconcave upv6301210212010f
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)

1) The document discusses using derivatives to determine the shapes of curves, including intervals of increasing/decreasing behavior, local extrema, and concavity. 2) Key tests covered are the increasing/decreasing test, first derivative test, concavity test, and second derivative test. These allow determining monotonicity, local extrema, and concavity from the signs of the first and second derivatives. 3) Examples demonstrate applying these tests to find the intervals of monotonicity, points of local extrema, and intervals of concavity for various functions.

v6301212010fv6301210412010ffunction
Lesson20 -derivatives_and_the_shape_of_curves_021_slides
Lesson20 -derivatives_and_the_shape_of_curves_021_slidesLesson20 -derivatives_and_the_shape_of_curves_021_slides
Lesson20 -derivatives_and_the_shape_of_curves_021_slides

f is decreasing on (−∞, −4/5] and increasing on [−4/5, ∞). . . . . . . V63.0121.021, Calculus I (NYU) Section 4.2 The Shapes of Curves November 16, 2010 13 / 32

Step 1: Monotonicity



                                        f(x) = x4 − 4x3 + 10
                              =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)

 We make its sign chart.

                               . ..
                               + 0              .
                                                +                   .
                                                                    +
                                                                          . x2
                                                                          4
                                 0
                                 .
                               −
                               .                −
                                                .              .. .
                                                               0 +
                                                                          . x − 3)
                                                                          (
                                                               3
                                                               .
                               − 0
                               . ..             −
                                                .              .. .
                                                               0 +        .′ (x)
                                                                          f
                               ↘ 0
                               . .              ↘
                                                .              3 ↗
                                                               . .        f
                                                                          .(x)
                                                              m
                                                              . in

                                                                          .        .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)       Section 4.4 Curve Sketching                    June 10, 2010   17 / 45
Step 2: Concavity



                                        f′ (x) = 4x3 − 12x2
                             =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)



                                        .




                                                                           .   .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching               June 10, 2010   18 / 45
Step 2: Concavity



                                        f′ (x) = 4x3 − 12x2
                             =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)

 Here is its sign chart:

                                        .




                                                                           .   .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching               June 10, 2010   18 / 45
Step 2: Concavity



                                        f′ (x) = 4x3 − 12x2
                             =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)

 Here is its sign chart:

                                        0
                                        ..
                                                                               1
                                                                               . 2x
                                        0
                                        .




                                                                           .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   18 / 45

Recommended for you

Lesson 27: The Fundamental Theorem of Calculus
Lesson 27: The Fundamental Theorem of Calculus Lesson 27: The Fundamental Theorem of Calculus
Lesson 27: The Fundamental Theorem of Calculus

Here are the key points about g given f: - g represents the area under the curve of f over successive intervals of the x-axis - As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative - The concavity (convexity or concavity) of g will match the concavity of f over each interval In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.

calculusfunctionarea
Lesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite IntegralsLesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite Integrals

- The document is from a Calculus I class at New York University and covers evaluating definite integrals. - It discusses using the Evaluation Theorem to evaluate definite integrals, writing antiderivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. - Examples are provided of using the midpoint rule to estimate a definite integral, and properties of definite integrals like additivity and comparison properties are reviewed.

calculusfunctionarea
Lesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite IntegralsLesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite Integrals

- The document is about evaluating definite integrals and contains sections on: evaluating definite integrals using the evaluation theorem, writing antiderivatives as indefinite integrals, interpreting definite integrals as net change over an interval, and examples of computing definite integrals. - It discusses properties of definite integrals such as additivity and comparison properties, and provides examples of definite integrals that can be evaluated using known area formulas or by direct computation of antiderivatives.

calculusareafunction
Step 2: Concavity



                                        f′ (x) = 4x3 − 12x2
                             =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)

 Here is its sign chart:

                                 − 0
                                 . ..
                                                                               1
                                                                               . 2x
                                   0
                                   .




                                                                           .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   18 / 45
Step 2: Concavity



                                        f′ (x) = 4x3 − 12x2
                             =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)

 Here is its sign chart:

                                 − 0
                                 . ..         .
                                              +
                                                                               1
                                                                               . 2x
                                   0
                                   .




                                                                           .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   18 / 45
Step 2: Concavity



                                        f′ (x) = 4x3 − 12x2
                             =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)

 Here is its sign chart:

                                 − 0
                                 . ..         .
                                              +                   .
                                                                  +
                                                                               1
                                                                               . 2x
                                   0
                                   .




                                                                           .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   18 / 45
Step 2: Concavity



                                        f′ (x) = 4x3 − 12x2
                             =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)

 Here is its sign chart:

                                 − 0
                                 . ..         .
                                              +                   .
                                                                  +
                                                                               1
                                                                               . 2x
                                   0
                                   .
                                                        0
                                                        ..
                                                                               . −2
                                                                               x
                                                        2
                                                        .




                                                                           .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   18 / 45

Recommended for you

Lesson 16: Inverse Trigonometric Functions
Lesson 16: Inverse Trigonometric FunctionsLesson 16: Inverse Trigonometric Functions
Lesson 16: Inverse Trigonometric Functions

The document is a lecture on inverse trigonometric functions from a Calculus I class at New York University. It defines inverse trigonometric functions as the inverses of restricted trigonometric functions, gives their domains and ranges, and discusses their derivatives. The document also provides examples of evaluating inverse trigonometric functions.

calculusinverse trigonometricfunction
Lesson 14: Derivatives of Logarithmic and Exponential Functions
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsLesson 14: Derivatives of Logarithmic and Exponential Functions
Lesson 14: Derivatives of Logarithmic and Exponential Functions

The document discusses conventions for defining exponential functions with exponents other than positive integers, such as negative exponents, fractional exponents, and exponents of zero. It defines exponential functions with these exponent types in a way that maintains important properties like ax+y = ax * ay. The goal is to extend the definition of exponential functions beyond positive integer exponents in a principled way.

calculusfunctionv6301210022010su
Lesson 19: The Mean Value Theorem
Lesson 19: The Mean Value TheoremLesson 19: The Mean Value Theorem
Lesson 19: The Mean Value Theorem

This document is the notes from a Calculus I class at New York University covering Section 4.2 on the Mean Value Theorem. The notes include objectives, an outline, explanations of Rolle's Theorem and the Mean Value Theorem, examples of using the theorems, and a food for thought question. The key points are that Rolle's Theorem states that if a function is continuous on an interval and differentiable inside the interval, and the function values at the endpoints are equal, then there exists a point in the interior where the derivative is 0. The Mean Value Theorem similarly states that if a function is continuous on an interval and differentiable inside, there exists a point where the average rate of change equals the instantaneous rate of

functionmean value theoremv6301210022010su
Step 2: Concavity



                                        f′ (x) = 4x3 − 12x2
                             =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)

 Here is its sign chart:

                                 − 0
                                 . ..         .
                                              +                   .
                                                                  +
                                                                               1
                                                                               . 2x
                                   0
                                   .
                                 −
                                 .                      0
                                                        ..
                                                                               . −2
                                                                               x
                                                        2
                                                        .




                                                                           .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   18 / 45
Step 2: Concavity



                                        f′ (x) = 4x3 − 12x2
                             =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)

 Here is its sign chart:

                                 − 0
                                 . ..         .
                                              +                   .
                                                                  +
                                                                               1
                                                                               . 2x
                                   0
                                   .
                                 −
                                 .            −
                                              .         0
                                                        ..
                                                                               . −2
                                                                               x
                                                        2
                                                        .




                                                                           .      .   .      .       .    .

V63.0121.002.2010Su, Calculus I (NYU)        Section 4.4 Curve Sketching                  June 10, 2010   18 / 45
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 25: The Definite Integral
Lesson 25: The Definite IntegralLesson 25: The Definite Integral
Lesson 25: The Definite Integral

This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.

calculusintegralfunction
Lesson22 -optimization_problems_slides
Lesson22 -optimization_problems_slidesLesson22 -optimization_problems_slides
Lesson22 -optimization_problems_slides

The document summarizes the steps to solve optimization problems using calculus. It begins with an example of finding the rectangle with maximum area given a fixed perimeter. It works through the solution, identifying the objective function, variables, constraints, and using calculus techniques like taking the derivative to find critical points. The document then outlines Polya's 4-step method for problem solving and provides guidance on setting up optimization problems by understanding the problem, introducing notation, drawing diagrams, and eliminating variables using given constraints. It emphasizes using the Closed Interval Method, evaluating the function at endpoints and critical points to determine maximums and minimums.

Lesson 24: Area and Distances
Lesson 24: Area and DistancesLesson 24: Area and Distances
Lesson 24: Area and Distances

The document is about calculating areas and distances using calculus. It discusses approximating areas of curved regions by dividing them into rectangles and letting the number of rectangles approach infinity. It provides examples of calculating areas of basic shapes like rectangles, parallelograms, and triangles. It then discusses Archimedes' work approximating the area under a parabola by inscribing sequences of triangles. The objectives are to compute areas using limits of approximating rectangles and to compute distances as limits of approximating time intervals.

calculusareafunction
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Introduction
IntroductionIntroduction
Introduction

This document provides information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences at NYU. The course will include weekly lectures, recitations, homework assignments, quizzes, a midterm exam, and a final exam. Grades will be determined based on exam, homework, and quiz scores. The required textbook is available in hardcover, looseleaf, or online formats through the campus bookstore or WebAssign. Students are encouraged to contact the professor or TAs with any questions.

v6301210212010fv6301212010f
Introduction
IntroductionIntroduction
Introduction

This document outlines information about a Calculus I course taught by Professor Matthew Leingang at the Courant Institute of Mathematical Sciences. It provides details on the course staff, contact information for the professor, an overview of assessments including homework, quizzes, a midterm and final exam. Grading breakdown is also included, as well as information on purchasing the required textbook and accessing the course on Blackboard. The document aims to provide students with essential logistical information to succeed in the Calculus I course.

v6301212010fv6301210412010f
Lesson 3: Limits
Lesson 3: LimitsLesson 3: Limits
Lesson 3: Limits

This document contains lecture notes on the concept of limit from a Calculus I course at New York University. It includes announcements about homework and deadlines. It then discusses guidelines for written homework assignments and a rubric for grading. The bulk of the document explains the concept of limit intuitively through an "error-tolerance game" and provides examples to illustrate the idea. It aims to build an informal understanding of limits before providing a precise definition.

v6301212010ffunctionv6301210412010f
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 5: Continuity (Section 41 slides)
Lesson 5: Continuity (Section 41 slides)Lesson 5: Continuity (Section 41 slides)
Lesson 5: Continuity (Section 41 slides)

This document is a section from a Calculus I course at New York University dated September 20, 2010. It discusses continuity of functions, beginning with definitions and examples of determining continuity. Key points covered include the definition of continuity as a function having a limit equal to its value, and theorems stating polynomials, rational functions, and combinations of continuous functions are also continuous. Trigonometric functions like sin, cos, tan, and cot are shown to be continuous on their domains. The document provides examples, explanations, and questions to illustrate the concept of continuity.

v6301212010ffunctionv6301210412010f
Lesson 5: Continuity (Section 21 slides)
Lesson 5: Continuity (Section 21 slides)Lesson 5: Continuity (Section 21 slides)
Lesson 5: Continuity (Section 21 slides)

This document is from a Calculus I class at New York University and covers continuity. It provides announcements about office hours and homework deadlines. It then discusses the objectives of understanding the definition of continuity and applying it to piecewise functions. Examples are provided to demonstrate how to show a function is continuous at a point by evaluating the limit as x approaches the point and showing it equals the function value. The students are asked to determine at which other points the example function is continuous.

v6301210212010fv6301212010ffunction
Lesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential FunctionsLesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential Functions

linear function, quadratic functions, cubic functions, power functions, transformations, and compositions.

v6301212010fv6301210412010f
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 4: Calculating Limits (Section 21 slides)
Lesson 4: Calculating Limits (Section 21 slides)Lesson 4: Calculating Limits (Section 21 slides)
Lesson 4: Calculating Limits (Section 21 slides)

basic limits, limit laws, limits of algebraic and rational functions, limits of piecewise functions, limits of trigonometric quotients.

limit lawsv6301210212010fv6301212010f
Lesson 6: Limits Involving ∞ (Section 41 slides)
Lesson 6: Limits Involving ∞ (Section 41 slides)Lesson 6: Limits Involving ∞ (Section 41 slides)
Lesson 6: Limits Involving ∞ (Section 41 slides)

This document is from a Calculus I course at New York University and covers limits involving infinity. It discusses definitions of limits approaching positive or negative infinity. Examples are provided of functions with infinite limits, such as 1/x as x approaches 0. The document outlines techniques for finding limits at points where a function is not continuous, such as using a number line to determine the signs of factors in a rational function's denominator.

infinityv6301212010ffunction
20240705 QFM024 Irresponsible AI Reading List June 2024
20240705 QFM024 Irresponsible AI Reading List June 202420240705 QFM024 Irresponsible AI Reading List June 2024
20240705 QFM024 Irresponsible AI Reading List June 2024

Everything that I found interesting last month about the irresponsible use of machine intelligence

quantumfaxmachine
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Quality Patents: Patents That Stand the Test of Time
Quality Patents: Patents That Stand the Test of TimeQuality Patents: Patents That Stand the Test of Time
Quality Patents: Patents That Stand the Test of Time

Is your patent a vanity piece of paper for your office wall? Or is it a reliable, defendable, assertable, property right? The difference is often quality. Is your patent simply a transactional cost and a large pile of legal bills for your startup? Or is it a leverageable asset worthy of attracting precious investment dollars, worth its cost in multiples of valuation? The difference is often quality. Is your patent application only good enough to get through the examination process? Or has it been crafted to stand the tests of time and varied audiences if you later need to assert that document against an infringer, find yourself litigating with it in an Article 3 Court at the hands of a judge and jury, God forbid, end up having to defend its validity at the PTAB, or even needing to use it to block pirated imports at the International Trade Commission? The difference is often quality. Quality will be our focus for a good chunk of the remainder of this season. What goes into a quality patent, and where possible, how do you get it without breaking the bank? ** Episode Overview ** In this first episode of our quality series, Kristen Hansen and the panel discuss: ⦿ What do we mean when we say patent quality? ⦿ Why is patent quality important? ⦿ How to balance quality and budget ⦿ The importance of searching, continuations, and draftsperson domain expertise ⦿ Very practical tips, tricks, examples, and Kristen’s Musts for drafting quality applications https://www.aurorapatents.com/patently-strategic-podcast.html

patentspatent applicationpatent prosecution
find out more about the role of autonomous vehicles in facing global challenges
find out more about the role of autonomous vehicles in facing global challengesfind out more about the role of autonomous vehicles in facing global challenges
find out more about the role of autonomous vehicles in facing global challenges

accommodate the strengths, weaknesses, threats and opportunities of autonomous vehicles

automotive self-driving car technology
[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf
[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf
[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf

Kief Morris rethinks the infrastructure code delivery lifecycle, advocating for a shift towards composable infrastructure systems. We should shift to designing around deployable components rather than code modules, use more useful levels of abstraction, and drive design and deployment from applications rather than bottom-up, monolithic architecture and delivery.

infrastructure as codeclouddevops
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Recent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS InfrastructureRecent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS Infrastructure

Recent advancements in the NIST-JARVIS infrastructure: JARVIS-Overview, JARVIS-DFT, AtomGPT, ALIGNN, JARVIS-Leaderboard

jarvisjarvis-dftalignn
How Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdfHow Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdf

In the modern digital era, social media platforms have become integral to our daily lives. These platforms, including Facebook, Instagram, WhatsApp, and Snapchat, offer countless ways to connect, share, and communicate.

social media hackerfacebook hackerhire a instagram hacker
K2G - Insurtech Innovation EMEA Award 2024
K2G - Insurtech Innovation EMEA Award 2024K2G - Insurtech Innovation EMEA Award 2024
K2G - Insurtech Innovation EMEA Award 2024

K2G - Insurtech Innovation Award 2024

k2gawards
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

What's New in Copilot for Microsoft365 May 2024.pptx
What's New in Copilot for Microsoft365 May 2024.pptxWhat's New in Copilot for Microsoft365 May 2024.pptx
What's New in Copilot for Microsoft365 May 2024.pptx

This is a slide deck that showcases the updates in Microsoft Copilot for May 2024

microsoftmicrosoft copilot
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...

Have you noticed the OpenSSF Scorecard badges on the official Dart and Flutter repos? It's Google's way of showing that they care about security. Practices such as pinning dependencies, branch protection, required reviews, continuous integration tests etc. are measured to provide a score and accompanying badge. You can do the same for your projects, and this presentation will show you how, with an emphasis on the unique challenges that come up when working with Dart and Flutter. The session will provide a walkthrough of the steps involved in securing a first repository, and then what it takes to repeat that process across an organization with multiple repos. It will also look at the ongoing maintenance involved once scorecards have been implemented, and how aspects of that maintenance can be better automated to minimize toil.

dartflutteropenssf
MYIR Product Brochure - A Global Provider of Embedded SOMs & Solutions
MYIR Product Brochure - A Global Provider of Embedded SOMs & SolutionsMYIR Product Brochure - A Global Provider of Embedded SOMs & Solutions
MYIR Product Brochure - A Global Provider of Embedded SOMs & Solutions

This brochure gives introduction of MYIR Electronics company and MYIR's products and services. MYIR Electronics Limited (MYIR for short), established in 2011, is a global provider of embedded System-On-Modules (SOMs) and comprehensive solutions based on various architectures such as ARM, FPGA, RISC-V, and AI. We cater to customers' needs for large-scale production, offering customized design, industry-specific application solutions, and one-stop OEM services. MYIR, recognized as a national high-tech enterprise, is also listed among the "Specialized and Special new" Enterprises in Shenzhen, China. Our core belief is that "Our success stems from our customers' success" and embraces the philosophy of "Make Your Idea Real, then My Idea Realizing!"

fpgaembedded systemembedded som
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理
一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理
一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理

原版一模一样【微信:741003700 】【(msvu毕业证书)圣文森山大学毕业证成绩单】【微信:741003700 】学位证,留信学历认证(真实可查,永久存档)原件一模一样纸张工艺/offer、在读证明、外壳等材料/诚信可靠,可直接看成品样本,帮您解决无法毕业带来的各种难题!外壳,原版制作,诚信可靠,可直接看成品样本。行业标杆!精益求精,诚心合作,真诚制作!多年品质 ,按需精细制作,24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题,包您满意。 本公司拥有海外各大学样板无数,能完美还原。 1:1完美还原海外各大学毕业材料上的工艺:水印,阴影底纹,钢印LOGO烫金烫银,LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700 【主营项目】 一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等! 二.真实使馆公证(即留学回国人员证明,不成功不收费) 三.真实教育部学历学位认证(教育部存档!教育部留服网站永久可查) 四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度) 如果您处于以下几种情况: ◇在校期间,因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】 ◇面对父母的压力,希望尽快拿到; ◇不清楚认证流程以及材料该如何准备; ◇回国时间很长,忘记办理; ◇回国马上就要找工作,办给用人单位看; ◇企事业单位必须要求办理的 ◇需要报考公务员、购买免税车、落转户口 ◇申请留学生创业基金 留信网认证的作用: 1:该专业认证可证明留学生真实身份 2:同时对留学生所学专业登记给予评定 3:国家专业人才认证中心颁发入库证书 4:这个认证书并且可以归档倒地方 5:凡事获得留信网入网的信息将会逐步更新到个人身份内,将在公安局网内查询个人身份证信息后,同步读取人才网入库信息 6:个人职称评审加20分 7:个人信誉贷款加10分 8:在国家人才网主办的国家网络招聘大会中纳入资料,供国家高端企业选择人才 办理(msvu毕业证书)圣文森山大学毕业证【微信:741003700 】外观非常简单,由纸质材料制成,上面印有校徽、校名、毕业生姓名、专业等信息。 办理(msvu毕业证书)圣文森山大学毕业证【微信:741003700 】格式相对统一,各专业都有相应的模板。通常包括以下部分: 校徽:象征着学校的荣誉和传承。 校名:学校英文全称 授予学位:本部分将注明获得的具体学位名称。 毕业生姓名:这是最重要的信息之一,标志着该证书是由特定人员获得的。 颁发日期:这是毕业正式生效的时间,也代表着毕业生学业的结束。 其他信息:根据不同的专业和学位,可能会有一些特定的信息或章节。 办理(msvu毕业证书)圣文森山大学毕业证【微信:741003700 】价值很高,需要妥善保管。一般来说,应放置在安全、干燥、防潮的地方,避免长时间暴露在阳光下。如需使用,最好使用复印件而不是原件,以免丢失。 综上所述,办理(msvu毕业证书)圣文森山大学毕业证【微信:741003700 】是证明身份和学历的高价值文件。外观简单庄重,格式统一,包括重要的个人信息和发布日期。对持有人来说,妥善保管是非常重要的。

(msvu毕业证书)圣文森山大学毕业证
20240702 QFM021 Machine Intelligence Reading List June 2024
20240702 QFM021 Machine Intelligence Reading List June 202420240702 QFM021 Machine Intelligence Reading List June 2024
20240702 QFM021 Machine Intelligence Reading List June 2024

Everything that I found interesting about machines behaving intelligently during June 2024

quantumfaxmachine
Verti - EMEA Insurer Innovation Award 2024
Verti - EMEA Insurer Innovation Award 2024Verti - EMEA Insurer Innovation Award 2024
Verti - EMEA Insurer Innovation Award 2024

Verti - Insurer Innovation Award 2024

vertiawards
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Running a Go App in Kubernetes: CPU Impacts
Running a Go App in Kubernetes: CPU ImpactsRunning a Go App in Kubernetes: CPU Impacts
Running a Go App in Kubernetes: CPU Impacts

Understanding the impacts of running a containerized Go application inside Kubernetes with a focus on the CPU.

WhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdf
WhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdfWhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdf
WhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdf

Profile portofolio

“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...
“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...
“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...

For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/07/intels-approach-to-operationalizing-ai-in-the-manufacturing-sector-a-presentation-from-intel/ Tara Thimmanaik, AI Systems and Solutions Architect at Intel, presents the “Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” tutorial at the May 2024 Embedded Vision Summit. AI at the edge is powering a revolution in industrial IoT, from real-time processing and analytics that drive greater efficiency and learning to predictive maintenance. Intel is focused on developing tools and assets to help domain experts operationalize AI-based solutions in their fields of expertise. In this talk, Thimmanaik explains how Intel’s software platforms simplify labor-intensive data upload, labeling, training, model optimization and retraining tasks. She shows how domain experts can quickly build vision models for a wide range of processes—detecting defective parts on a production line, reducing downtime on the factory floor, automating inventory management and other digitization and automation projects. And she introduces Intel-provided edge computing assets that empower faster localized insights and decisions, improving labor productivity through easy-to-use AI tools that democratize AI.

tara thimmanaikintelcomputer vision
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

HTTP Adaptive Streaming – Quo Vadis (2024)
HTTP Adaptive Streaming – Quo Vadis (2024)HTTP Adaptive Streaming – Quo Vadis (2024)
HTTP Adaptive Streaming – Quo Vadis (2024)

Video traffic on the Internet is constantly growing; networked multimedia applications consume a predominant share of the available Internet bandwidth. A major technical breakthrough and enabler in multimedia systems research and of industrial networked multimedia services certainly was the HTTP Adaptive Streaming (HAS) technique. This resulted in the standardization of MPEG Dynamic Adaptive Streaming over HTTP (MPEG-DASH) which, together with HTTP Live Streaming (HLS), is widely used for multimedia delivery in today’s networks. Existing challenges in multimedia systems research deal with the trade-off between (i) the ever-increasing content complexity, (ii) various requirements with respect to time (most importantly, latency), and (iii) quality of experience (QoE). Optimizing towards one aspect usually negatively impacts at least one of the other two aspects if not both. This situation sets the stage for our research work in the ATHENA Christian Doppler (CD) Laboratory (Adaptive Streaming over HTTP and Emerging Networked Multimedia Services; https://athena.itec.aau.at/), jointly funded by public sources and industry. In this talk, we will present selected novel approaches and research results of the first year of the ATHENA CD Lab’s operation. We will highlight HAS-related research on (i) multimedia content provisioning (machine learning for video encoding); (ii) multimedia content delivery (support of edge processing and virtualized network functions for video networking); (iii) multimedia content consumption and end-to-end aspects (player-triggered segment retransmissions to improve video playout quality); and (iv) novel QoE investigations (adaptive point cloud streaming). We will also put the work into the context of international multimedia systems research.

UiPath Community Day Kraków: Devs4Devs Conference
UiPath Community Day Kraków: Devs4Devs ConferenceUiPath Community Day Kraków: Devs4Devs Conference
UiPath Community Day Kraków: Devs4Devs Conference

We are honored to launch and host this event for our UiPath Polish Community, with the help of our partners - Proservartner! We certainly hope we have managed to spike your interest in the subjects to be presented and the incredible networking opportunities at hand, too! Check out our proposed agenda below 👇👇 08:30 ☕ Welcome coffee (30') 09:00 Opening note/ Intro to UiPath Community (10') Cristina Vidu, Global Manager, Marketing Community @UiPath Dawid Kot, Digital Transformation Lead @Proservartner 09:10 Cloud migration - Proservartner & DOVISTA case study (30') Marcin Drozdowski, Automation CoE Manager @DOVISTA Pawel Kamiński, RPA developer @DOVISTA Mikolaj Zielinski, UiPath MVP, Senior Solutions Engineer @Proservartner 09:40 From bottlenecks to breakthroughs: Citizen Development in action (25') Pawel Poplawski, Director, Improvement and Automation @McCormick & Company Michał Cieślak, Senior Manager, Automation Programs @McCormick & Company 10:05 Next-level bots: API integration in UiPath Studio (30') Mikolaj Zielinski, UiPath MVP, Senior Solutions Engineer @Proservartner 10:35 ☕ Coffee Break (15') 10:50 Document Understanding with my RPA Companion (45') Ewa Gruszka, Enterprise Sales Specialist, AI & ML @UiPath 11:35 Power up your Robots: GenAI and GPT in REFramework (45') Krzysztof Karaszewski, Global RPA Product Manager 12:20 🍕 Lunch Break (1hr) 13:20 From Concept to Quality: UiPath Test Suite for AI-powered Knowledge Bots (30') Kamil Miśko, UiPath MVP, Senior RPA Developer @Zurich Insurance 13:50 Communications Mining - focus on AI capabilities (30') Thomasz Wierzbicki, Business Analyst @Office Samurai 14:20 Polish MVP panel: Insights on MVP award achievements and career profiling

#uipathcommunity#automation#automationdeveloper
What’s New in Teams Calling, Meetings and Devices May 2024
What’s New in Teams Calling, Meetings and Devices May 2024What’s New in Teams Calling, Meetings and Devices May 2024
What’s New in Teams Calling, Meetings and Devices May 2024

This is a powerpoint that features Microsoft Teams Devices and everything that is new including updates to its software and devices for May 2024

microsoft teamsmicrosoft
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

How Netflix Builds High Performance Applications at Global Scale
How Netflix Builds High Performance Applications at Global ScaleHow Netflix Builds High Performance Applications at Global Scale
How Netflix Builds High Performance Applications at Global Scale

We all want to build applications that are blazingly fast. We also want to scale them to users all over the world. Can the two happen together? Can users in the slowest of environments also get a fast experience? Learn how we do this at Netflix: how we understand every user's needs and preferences and build high performance applications that work for every user, every time.

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

Recommended for you

Lesson 21: Curve Sketching
Lesson 21: Curve Sketching

More Related Content

What's hot

Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Matthew Leingang
 
Função quadrática
Função quadráticaFunção quadrática
Função quadrática
CARLOSROBERTORODRIGU30
 
Lesson 21: Curve Sketching (Section 4 version)
Lesson 21: Curve Sketching (Section 4 version)Lesson 21: Curve Sketching (Section 4 version)
Lesson 21: Curve Sketching (Section 4 version)
Matthew Leingang
 
A basic introduction to learning
A basic introduction to learningA basic introduction to learning
A basic introduction to learning
Andres Mendez-Vazquez
 
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: Antiderivatives
Matthew Leingang
 
April 3, 2014
April 3, 2014April 3, 2014
April 3, 2014
khyps13
 
Lesson 1: Functions
Lesson 1: FunctionsLesson 1: Functions
Lesson 1: Functions
Matthew Leingang
 
Lesson 7: The Derivative as a Function
Lesson 7: The Derivative as a FunctionLesson 7: The Derivative as a Function
Lesson 7: The Derivative as a Function
Matthew Leingang
 
Lesson 23: The Definite Integral (slides)
Lesson 23: The Definite Integral (slides)Lesson 23: The Definite Integral (slides)
Lesson 23: The Definite Integral (slides)
Matthew Leingang
 
Module 11 graph of functions PMR
Module 11 graph of functions PMRModule 11 graph of functions PMR
Module 11 graph of functions PMR
roszelan
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
The Statistical and Applied Mathematical Sciences Institute
 
Nonlinear Manifolds in Computer Vision
Nonlinear Manifolds in Computer VisionNonlinear Manifolds in Computer Vision
Nonlinear Manifolds in Computer Vision
zukun
 
Module 10 Graphs Of Functions
Module 10 Graphs Of FunctionsModule 10 Graphs Of Functions
Module 10 Graphs Of Functions
guestcc333c
 
7.curves Further Mathematics Zimbabwe Zimsec Cambridge
7.curves   Further Mathematics Zimbabwe Zimsec Cambridge7.curves   Further Mathematics Zimbabwe Zimsec Cambridge
7.curves Further Mathematics Zimbabwe Zimsec Cambridge
alproelearning
 
Form 4 formulae and note
Form 4 formulae and noteForm 4 formulae and note
Form 4 formulae and note
smktsj2
 
P2 Graphs Function
P2  Graphs FunctionP2  Graphs Function
P2 Graphs Function
guestcc333c
 
Image Processing 3
Image Processing 3Image Processing 3
Image Processing 3
jainatin
 

What's hot (17)

Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
 
Função quadrática
Função quadráticaFunção quadrática
Função quadrática
 
Lesson 21: Curve Sketching (Section 4 version)
Lesson 21: Curve Sketching (Section 4 version)Lesson 21: Curve Sketching (Section 4 version)
Lesson 21: Curve Sketching (Section 4 version)
 
A basic introduction to learning
A basic introduction to learningA basic introduction to learning
A basic introduction to learning
 
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: Antiderivatives
 
April 3, 2014
April 3, 2014April 3, 2014
April 3, 2014
 
Lesson 1: Functions
Lesson 1: FunctionsLesson 1: Functions
Lesson 1: Functions
 
Lesson 7: The Derivative as a Function
Lesson 7: The Derivative as a FunctionLesson 7: The Derivative as a Function
Lesson 7: The Derivative as a Function
 
Lesson 23: The Definite Integral (slides)
Lesson 23: The Definite Integral (slides)Lesson 23: The Definite Integral (slides)
Lesson 23: The Definite Integral (slides)
 
Module 11 graph of functions PMR
Module 11 graph of functions PMRModule 11 graph of functions PMR
Module 11 graph of functions PMR
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
 
Nonlinear Manifolds in Computer Vision
Nonlinear Manifolds in Computer VisionNonlinear Manifolds in Computer Vision
Nonlinear Manifolds in Computer Vision
 
Module 10 Graphs Of Functions
Module 10 Graphs Of FunctionsModule 10 Graphs Of Functions
Module 10 Graphs Of Functions
 
7.curves Further Mathematics Zimbabwe Zimsec Cambridge
7.curves   Further Mathematics Zimbabwe Zimsec Cambridge7.curves   Further Mathematics Zimbabwe Zimsec Cambridge
7.curves Further Mathematics Zimbabwe Zimsec Cambridge
 
Form 4 formulae and note
Form 4 formulae and noteForm 4 formulae and note
Form 4 formulae and note
 
P2 Graphs Function
P2  Graphs FunctionP2  Graphs Function
P2 Graphs Function
 
Image Processing 3
Image Processing 3Image Processing 3
Image Processing 3
 

Viewers also liked

Lesson18 -maximum_and_minimum_values_slides
Lesson18 -maximum_and_minimum_values_slidesLesson18 -maximum_and_minimum_values_slides
Lesson18 -maximum_and_minimum_values_slides
Mel Anthony Pepito
 
Lesson 19: The Mean Value Theorem (slides)
Lesson 19: The Mean Value Theorem (slides)Lesson 19: The Mean Value Theorem (slides)
Lesson 19: The Mean Value Theorem (slides)
Mel Anthony Pepito
 
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
Mel Anthony Pepito
 
Lesson 7: The Derivative (Section 21 slide)
Lesson 7: The Derivative (Section 21 slide)Lesson 7: The Derivative (Section 21 slide)
Lesson 7: The Derivative (Section 21 slide)
Mel Anthony Pepito
 
Lesson 13: Related Rates Problems
Lesson 13: Related Rates ProblemsLesson 13: Related Rates Problems
Lesson 13: Related Rates Problems
Mel Anthony Pepito
 
Lesson 27: Integration by Substitution (Section 041 slides)
Lesson 27: Integration by Substitution (Section 041 slides)Lesson 27: Integration by Substitution (Section 041 slides)
Lesson 27: Integration by Substitution (Section 041 slides)
Mel Anthony Pepito
 
Lesson 17: Indeterminate Forms and L'Hôpital's Rule
Lesson 17: Indeterminate Forms and L'Hôpital's RuleLesson 17: Indeterminate Forms and L'Hôpital's Rule
Lesson 17: Indeterminate Forms and L'Hôpital's Rule
Mel Anthony Pepito
 
Lesson 11: Implicit Differentiation
Lesson 11: Implicit DifferentiationLesson 11: Implicit Differentiation
Lesson 11: Implicit Differentiation
Mel Anthony Pepito
 
Lesson 10: The Chain Rule (Section 21 slides)
Lesson 10: The Chain Rule (Section 21 slides)Lesson 10: The Chain Rule (Section 21 slides)
Lesson 10: The Chain Rule (Section 21 slides)
Mel Anthony Pepito
 
Lesson 9: The Product and Quotient Rules (Section 21 slides)
Lesson 9: The Product and Quotient Rules (Section 21 slides)Lesson 9: The Product and Quotient Rules (Section 21 slides)
Lesson 9: The Product and Quotient Rules (Section 21 slides)
Mel Anthony Pepito
 
Lesson 4: Calculating Limits (Section 41 slides)
Lesson 4: Calculating Limits (Section 41 slides)Lesson 4: Calculating Limits (Section 41 slides)
Lesson 4: Calculating Limits (Section 41 slides)
Mel Anthony Pepito
 
Lesson 7: The Derivative (Section 41 slides)
Lesson 7: The Derivative (Section 41 slides)Lesson 7: The Derivative (Section 41 slides)
Lesson 7: The Derivative (Section 41 slides)
Mel Anthony Pepito
 
Lecture7
Lecture7Lecture7
Lecture7
Bharathvajan .k
 
Lesson 3: Limits (Section 21 slides)
Lesson 3: Limits (Section 21 slides)Lesson 3: Limits (Section 21 slides)
Lesson 3: Limits (Section 21 slides)
Mel Anthony Pepito
 
Lesson 1: Functions
Lesson 1: FunctionsLesson 1: Functions
Lesson 1: Functions
Mel Anthony Pepito
 
Lesson 28: Integration by Subsitution
Lesson 28: Integration by SubsitutionLesson 28: Integration by Subsitution
Lesson 28: Integration by Subsitution
Mel Anthony Pepito
 
Lesson 15: Exponential Growth and Decay
Lesson 15: Exponential Growth and DecayLesson 15: Exponential Growth and Decay
Lesson 15: Exponential Growth and Decay
Mel Anthony Pepito
 
Lesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential FunctionsLesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential Functions
Mel Anthony Pepito
 
Lesson 12: Linear Approximation
Lesson 12: Linear ApproximationLesson 12: Linear Approximation
Lesson 12: Linear Approximation
Mel Anthony Pepito
 
Lesson 6: Limits Involving ∞ (Section 21 slides)
Lesson 6: Limits Involving ∞ (Section 21 slides)Lesson 6: Limits Involving ∞ (Section 21 slides)
Lesson 6: Limits Involving ∞ (Section 21 slides)
Mel Anthony Pepito
 

Viewers also liked (20)

Lesson18 -maximum_and_minimum_values_slides
Lesson18 -maximum_and_minimum_values_slidesLesson18 -maximum_and_minimum_values_slides
Lesson18 -maximum_and_minimum_values_slides
 
Lesson 19: The Mean Value Theorem (slides)
Lesson 19: The Mean Value Theorem (slides)Lesson 19: The Mean Value Theorem (slides)
Lesson 19: The Mean Value Theorem (slides)
 
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)
 
Lesson 7: The Derivative (Section 21 slide)
Lesson 7: The Derivative (Section 21 slide)Lesson 7: The Derivative (Section 21 slide)
Lesson 7: The Derivative (Section 21 slide)
 
Lesson 13: Related Rates Problems
Lesson 13: Related Rates ProblemsLesson 13: Related Rates Problems
Lesson 13: Related Rates Problems
 
Lesson 27: Integration by Substitution (Section 041 slides)
Lesson 27: Integration by Substitution (Section 041 slides)Lesson 27: Integration by Substitution (Section 041 slides)
Lesson 27: Integration by Substitution (Section 041 slides)
 
Lesson 17: Indeterminate Forms and L'Hôpital's Rule
Lesson 17: Indeterminate Forms and L'Hôpital's RuleLesson 17: Indeterminate Forms and L'Hôpital's Rule
Lesson 17: Indeterminate Forms and L'Hôpital's Rule
 
Lesson 11: Implicit Differentiation
Lesson 11: Implicit DifferentiationLesson 11: Implicit Differentiation
Lesson 11: Implicit Differentiation
 
Lesson 10: The Chain Rule (Section 21 slides)
Lesson 10: The Chain Rule (Section 21 slides)Lesson 10: The Chain Rule (Section 21 slides)
Lesson 10: The Chain Rule (Section 21 slides)
 
Lesson 9: The Product and Quotient Rules (Section 21 slides)
Lesson 9: The Product and Quotient Rules (Section 21 slides)Lesson 9: The Product and Quotient Rules (Section 21 slides)
Lesson 9: The Product and Quotient Rules (Section 21 slides)
 
Lesson 4: Calculating Limits (Section 41 slides)
Lesson 4: Calculating Limits (Section 41 slides)Lesson 4: Calculating Limits (Section 41 slides)
Lesson 4: Calculating Limits (Section 41 slides)
 
Lesson 7: The Derivative (Section 41 slides)
Lesson 7: The Derivative (Section 41 slides)Lesson 7: The Derivative (Section 41 slides)
Lesson 7: The Derivative (Section 41 slides)
 
Lecture7
Lecture7Lecture7
Lecture7
 
Lesson 3: Limits (Section 21 slides)
Lesson 3: Limits (Section 21 slides)Lesson 3: Limits (Section 21 slides)
Lesson 3: Limits (Section 21 slides)
 
Lesson 1: Functions
Lesson 1: FunctionsLesson 1: Functions
Lesson 1: Functions
 
Lesson 28: Integration by Subsitution
Lesson 28: Integration by SubsitutionLesson 28: Integration by Subsitution
Lesson 28: Integration by Subsitution
 
Lesson 15: Exponential Growth and Decay
Lesson 15: Exponential Growth and DecayLesson 15: Exponential Growth and Decay
Lesson 15: Exponential Growth and Decay
 
Lesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential FunctionsLesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential Functions
 
Lesson 12: Linear Approximation
Lesson 12: Linear ApproximationLesson 12: Linear Approximation
Lesson 12: Linear Approximation
 
Lesson 6: Limits Involving ∞ (Section 21 slides)
Lesson 6: Limits Involving ∞ (Section 21 slides)Lesson 6: Limits Involving ∞ (Section 21 slides)
Lesson 6: Limits Involving ∞ (Section 21 slides)
 

Similar to Lesson 21: Curve Sketching

Lesson 21: Curve Sketching (Section 041 slides)
Lesson 21: Curve Sketching (Section 041 slides)Lesson 21: Curve Sketching (Section 041 slides)
Lesson 21: Curve Sketching (Section 041 slides)
Mel Anthony Pepito
 
Lesson 21: Curve Sketching (Section 041 handout)
Lesson 21: Curve Sketching (Section 041 handout)Lesson 21: Curve Sketching (Section 041 handout)
Lesson 21: Curve Sketching (Section 041 handout)
Matthew Leingang
 
Lesson20 -derivatives_and_the_shape_of_curves_021_slides
Lesson20  -derivatives_and_the_shape_of_curves_021_slidesLesson20  -derivatives_and_the_shape_of_curves_021_slides
Lesson20 -derivatives_and_the_shape_of_curves_021_slides
Matthew Leingang
 
Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)
Mel Anthony Pepito
 
Lesson 18: Derivatives and the Shapes of Curves
Lesson 18: Derivatives and the Shapes of CurvesLesson 18: Derivatives and the Shapes of Curves
Lesson 18: Derivatives and the Shapes of Curves
Matthew Leingang
 
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Matthew Leingang
 
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Mel Anthony Pepito
 
Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)
Matthew Leingang
 
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Mel Anthony Pepito
 
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: Antiderivatives
Mel Anthony Pepito
 
Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Mel Anthony Pepito
 
Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Matthew Leingang
 
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Mel Anthony Pepito
 
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Matthew Leingang
 
Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)
Matthew Leingang
 
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)
Matthew Leingang
 
Lesson20 -derivatives_and_the_shape_of_curves_021_slides
Lesson20 -derivatives_and_the_shape_of_curves_021_slidesLesson20 -derivatives_and_the_shape_of_curves_021_slides
Lesson20 -derivatives_and_the_shape_of_curves_021_slides
Mel Anthony Pepito
 
Lesson 27: The Fundamental Theorem of Calculus
Lesson 27: The Fundamental Theorem of Calculus Lesson 27: The Fundamental Theorem of Calculus
Lesson 27: The Fundamental Theorem of Calculus
Mel Anthony Pepito
 
Lesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite IntegralsLesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite Integrals
Matthew Leingang
 
Lesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite IntegralsLesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite Integrals
Mel Anthony Pepito
 

Similar to Lesson 21: Curve Sketching (20)

Lesson 21: Curve Sketching (Section 041 slides)
Lesson 21: Curve Sketching (Section 041 slides)Lesson 21: Curve Sketching (Section 041 slides)
Lesson 21: Curve Sketching (Section 041 slides)
 
Lesson 21: Curve Sketching (Section 041 handout)
Lesson 21: Curve Sketching (Section 041 handout)Lesson 21: Curve Sketching (Section 041 handout)
Lesson 21: Curve Sketching (Section 041 handout)
 
Lesson20 -derivatives_and_the_shape_of_curves_021_slides
Lesson20  -derivatives_and_the_shape_of_curves_021_slidesLesson20  -derivatives_and_the_shape_of_curves_021_slides
Lesson20 -derivatives_and_the_shape_of_curves_021_slides
 
Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)
 
Lesson 18: Derivatives and the Shapes of Curves
Lesson 18: Derivatives and the Shapes of CurvesLesson 18: Derivatives and the Shapes of Curves
Lesson 18: Derivatives and the Shapes of Curves
 
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
 
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)
 
Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)Lesson 21: Curve Sketching (Section 021 handout)
Lesson 21: Curve Sketching (Section 021 handout)
 
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)
 
Lesson 23: Antiderivatives
Lesson 23: AntiderivativesLesson 23: Antiderivatives
Lesson 23: Antiderivatives
 
Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)
 
Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)Lesson 25: Evaluating Definite Integrals (Section 041 slides)
Lesson 25: Evaluating Definite Integrals (Section 041 slides)
 
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
 
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)
 
Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 021 handout)
 
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)
 
Lesson20 -derivatives_and_the_shape_of_curves_021_slides
Lesson20 -derivatives_and_the_shape_of_curves_021_slidesLesson20 -derivatives_and_the_shape_of_curves_021_slides
Lesson20 -derivatives_and_the_shape_of_curves_021_slides
 
Lesson 27: The Fundamental Theorem of Calculus
Lesson 27: The Fundamental Theorem of Calculus Lesson 27: The Fundamental Theorem of Calculus
Lesson 27: The Fundamental Theorem of Calculus
 
Lesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite IntegralsLesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite Integrals
 
Lesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite IntegralsLesson 26: Evaluating Definite Integrals
Lesson 26: Evaluating Definite Integrals
 

More from Mel Anthony Pepito

Lesson 16: Inverse Trigonometric Functions
Lesson 16: Inverse Trigonometric FunctionsLesson 16: Inverse Trigonometric Functions
Lesson 16: Inverse Trigonometric Functions
Mel Anthony Pepito
 
Lesson 14: Derivatives of Logarithmic and Exponential Functions
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsLesson 14: Derivatives of Logarithmic and Exponential Functions
Lesson 14: Derivatives of Logarithmic and Exponential Functions
Mel Anthony Pepito
 
Lesson 19: The Mean Value Theorem
Lesson 19: The Mean Value TheoremLesson 19: The Mean Value Theorem
Lesson 19: The Mean Value Theorem
Mel Anthony Pepito
 
Lesson 25: The Definite Integral
Lesson 25: The Definite IntegralLesson 25: The Definite Integral
Lesson 25: The Definite Integral
Mel Anthony Pepito
 
Lesson22 -optimization_problems_slides
Lesson22 -optimization_problems_slidesLesson22 -optimization_problems_slides
Lesson22 -optimization_problems_slides
Mel Anthony Pepito
 
Lesson 24: Area and Distances
Lesson 24: Area and DistancesLesson 24: Area and Distances
Lesson 24: Area and Distances
Mel Anthony Pepito
 
Introduction
IntroductionIntroduction
Introduction
Mel Anthony Pepito
 
Introduction
IntroductionIntroduction
Introduction
Mel Anthony Pepito
 
Lesson 3: Limits
Lesson 3: LimitsLesson 3: Limits
Lesson 3: Limits
Mel Anthony Pepito
 
Lesson 5: Continuity (Section 41 slides)
Lesson 5: Continuity (Section 41 slides)Lesson 5: Continuity (Section 41 slides)
Lesson 5: Continuity (Section 41 slides)
Mel Anthony Pepito
 
Lesson 5: Continuity (Section 21 slides)
Lesson 5: Continuity (Section 21 slides)Lesson 5: Continuity (Section 21 slides)
Lesson 5: Continuity (Section 21 slides)
Mel Anthony Pepito
 
Lesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential FunctionsLesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential Functions
Mel Anthony Pepito
 
Lesson 4: Calculating Limits (Section 21 slides)
Lesson 4: Calculating Limits (Section 21 slides)Lesson 4: Calculating Limits (Section 21 slides)
Lesson 4: Calculating Limits (Section 21 slides)
Mel Anthony Pepito
 
Lesson 6: Limits Involving ∞ (Section 41 slides)
Lesson 6: Limits Involving ∞ (Section 41 slides)Lesson 6: Limits Involving ∞ (Section 41 slides)
Lesson 6: Limits Involving ∞ (Section 41 slides)
Mel Anthony Pepito
 

More from Mel Anthony Pepito (14)

Lesson 16: Inverse Trigonometric Functions
Lesson 16: Inverse Trigonometric FunctionsLesson 16: Inverse Trigonometric Functions
Lesson 16: Inverse Trigonometric Functions
 
Lesson 14: Derivatives of Logarithmic and Exponential Functions
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsLesson 14: Derivatives of Logarithmic and Exponential Functions
Lesson 14: Derivatives of Logarithmic and Exponential Functions
 
Lesson 19: The Mean Value Theorem
Lesson 19: The Mean Value TheoremLesson 19: The Mean Value Theorem
Lesson 19: The Mean Value Theorem
 
Lesson 25: The Definite Integral
Lesson 25: The Definite IntegralLesson 25: The Definite Integral
Lesson 25: The Definite Integral
 
Lesson22 -optimization_problems_slides
Lesson22 -optimization_problems_slidesLesson22 -optimization_problems_slides
Lesson22 -optimization_problems_slides
 
Lesson 24: Area and Distances
Lesson 24: Area and DistancesLesson 24: Area and Distances
Lesson 24: Area and Distances
 
Introduction
IntroductionIntroduction
Introduction
 
Introduction
IntroductionIntroduction
Introduction
 
Lesson 3: Limits
Lesson 3: LimitsLesson 3: Limits
Lesson 3: Limits
 
Lesson 5: Continuity (Section 41 slides)
Lesson 5: Continuity (Section 41 slides)Lesson 5: Continuity (Section 41 slides)
Lesson 5: Continuity (Section 41 slides)
 
Lesson 5: Continuity (Section 21 slides)
Lesson 5: Continuity (Section 21 slides)Lesson 5: Continuity (Section 21 slides)
Lesson 5: Continuity (Section 21 slides)
 
Lesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential FunctionsLesson 2: A Catalog of Essential Functions
Lesson 2: A Catalog of Essential Functions
 
Lesson 4: Calculating Limits (Section 21 slides)
Lesson 4: Calculating Limits (Section 21 slides)Lesson 4: Calculating Limits (Section 21 slides)
Lesson 4: Calculating Limits (Section 21 slides)
 
Lesson 6: Limits Involving ∞ (Section 41 slides)
Lesson 6: Limits Involving ∞ (Section 41 slides)Lesson 6: Limits Involving ∞ (Section 41 slides)
Lesson 6: Limits Involving ∞ (Section 41 slides)
 

Recently uploaded

20240705 QFM024 Irresponsible AI Reading List June 2024
20240705 QFM024 Irresponsible AI Reading List June 202420240705 QFM024 Irresponsible AI Reading List June 2024
20240705 QFM024 Irresponsible AI Reading List June 2024
Matthew Sinclair
 
Quality Patents: Patents That Stand the Test of Time
Quality Patents: Patents That Stand the Test of TimeQuality Patents: Patents That Stand the Test of Time
Quality Patents: Patents That Stand the Test of Time
Aurora Consulting
 
find out more about the role of autonomous vehicles in facing global challenges
find out more about the role of autonomous vehicles in facing global challengesfind out more about the role of autonomous vehicles in facing global challenges
find out more about the role of autonomous vehicles in facing global challenges
huseindihon
 
[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf
[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf
[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf
Kief Morris
 
Recent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS InfrastructureRecent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS Infrastructure
KAMAL CHOUDHARY
 
How Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdfHow Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdf
HackersList
 
K2G - Insurtech Innovation EMEA Award 2024
K2G - Insurtech Innovation EMEA Award 2024K2G - Insurtech Innovation EMEA Award 2024
K2G - Insurtech Innovation EMEA Award 2024
The Digital Insurer
 
What's New in Copilot for Microsoft365 May 2024.pptx
What's New in Copilot for Microsoft365 May 2024.pptxWhat's New in Copilot for Microsoft365 May 2024.pptx
What's New in Copilot for Microsoft365 May 2024.pptx
Stephanie Beckett
 
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...
Chris Swan
 
MYIR Product Brochure - A Global Provider of Embedded SOMs & Solutions
MYIR Product Brochure - A Global Provider of Embedded SOMs & SolutionsMYIR Product Brochure - A Global Provider of Embedded SOMs & Solutions
MYIR Product Brochure - A Global Provider of Embedded SOMs & Solutions
Linda Zhang
 
一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理
一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理
一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理
uuuot
 
20240702 QFM021 Machine Intelligence Reading List June 2024
20240702 QFM021 Machine Intelligence Reading List June 202420240702 QFM021 Machine Intelligence Reading List June 2024
20240702 QFM021 Machine Intelligence Reading List June 2024
Matthew Sinclair
 
Verti - EMEA Insurer Innovation Award 2024
Verti - EMEA Insurer Innovation Award 2024Verti - EMEA Insurer Innovation Award 2024
Verti - EMEA Insurer Innovation Award 2024
The Digital Insurer
 
Running a Go App in Kubernetes: CPU Impacts
Running a Go App in Kubernetes: CPU ImpactsRunning a Go App in Kubernetes: CPU Impacts
Running a Go App in Kubernetes: CPU Impacts
ScyllaDB
 
WhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdf
WhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdfWhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdf
WhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdf
ArgaBisma
 
“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...
“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...
“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...
Edge AI and Vision Alliance
 
HTTP Adaptive Streaming – Quo Vadis (2024)
HTTP Adaptive Streaming – Quo Vadis (2024)HTTP Adaptive Streaming – Quo Vadis (2024)
HTTP Adaptive Streaming – Quo Vadis (2024)
Alpen-Adria-Universität
 
UiPath Community Day Kraków: Devs4Devs Conference
UiPath Community Day Kraków: Devs4Devs ConferenceUiPath Community Day Kraków: Devs4Devs Conference
UiPath Community Day Kraków: Devs4Devs Conference
UiPathCommunity
 
What’s New in Teams Calling, Meetings and Devices May 2024
What’s New in Teams Calling, Meetings and Devices May 2024What’s New in Teams Calling, Meetings and Devices May 2024
What’s New in Teams Calling, Meetings and Devices May 2024
Stephanie Beckett
 
How Netflix Builds High Performance Applications at Global Scale
How Netflix Builds High Performance Applications at Global ScaleHow Netflix Builds High Performance Applications at Global Scale
How Netflix Builds High Performance Applications at Global Scale
ScyllaDB
 

Recently uploaded (20)

20240705 QFM024 Irresponsible AI Reading List June 2024
20240705 QFM024 Irresponsible AI Reading List June 202420240705 QFM024 Irresponsible AI Reading List June 2024
20240705 QFM024 Irresponsible AI Reading List June 2024
 
Quality Patents: Patents That Stand the Test of Time
Quality Patents: Patents That Stand the Test of TimeQuality Patents: Patents That Stand the Test of Time
Quality Patents: Patents That Stand the Test of Time
 
find out more about the role of autonomous vehicles in facing global challenges
find out more about the role of autonomous vehicles in facing global challengesfind out more about the role of autonomous vehicles in facing global challenges
find out more about the role of autonomous vehicles in facing global challenges
 
[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf
[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf
[Talk] Moving Beyond Spaghetti Infrastructure [AOTB] 2024-07-04.pdf
 
Recent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS InfrastructureRecent Advancements in the NIST-JARVIS Infrastructure
Recent Advancements in the NIST-JARVIS Infrastructure
 
How Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdfHow Social Media Hackers Help You to See Your Wife's Message.pdf
How Social Media Hackers Help You to See Your Wife's Message.pdf
 
K2G - Insurtech Innovation EMEA Award 2024
K2G - Insurtech Innovation EMEA Award 2024K2G - Insurtech Innovation EMEA Award 2024
K2G - Insurtech Innovation EMEA Award 2024
 
What's New in Copilot for Microsoft365 May 2024.pptx
What's New in Copilot for Microsoft365 May 2024.pptxWhat's New in Copilot for Microsoft365 May 2024.pptx
What's New in Copilot for Microsoft365 May 2024.pptx
 
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...
 
MYIR Product Brochure - A Global Provider of Embedded SOMs & Solutions
MYIR Product Brochure - A Global Provider of Embedded SOMs & SolutionsMYIR Product Brochure - A Global Provider of Embedded SOMs & Solutions
MYIR Product Brochure - A Global Provider of Embedded SOMs & Solutions
 
一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理
一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理
一比一原版(msvu毕业证书)圣文森山大学毕业证如何办理
 
20240702 QFM021 Machine Intelligence Reading List June 2024
20240702 QFM021 Machine Intelligence Reading List June 202420240702 QFM021 Machine Intelligence Reading List June 2024
20240702 QFM021 Machine Intelligence Reading List June 2024
 
Verti - EMEA Insurer Innovation Award 2024
Verti - EMEA Insurer Innovation Award 2024Verti - EMEA Insurer Innovation Award 2024
Verti - EMEA Insurer Innovation Award 2024
 
Running a Go App in Kubernetes: CPU Impacts
Running a Go App in Kubernetes: CPU ImpactsRunning a Go App in Kubernetes: CPU Impacts
Running a Go App in Kubernetes: CPU Impacts
 
WhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdf
WhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdfWhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdf
WhatsApp Image 2024-03-27 at 08.19.52_bfd93109.pdf
 
“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...
“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...
“Intel’s Approach to Operationalizing AI in the Manufacturing Sector,” a Pres...
 
HTTP Adaptive Streaming – Quo Vadis (2024)
HTTP Adaptive Streaming – Quo Vadis (2024)HTTP Adaptive Streaming – Quo Vadis (2024)
HTTP Adaptive Streaming – Quo Vadis (2024)
 
UiPath Community Day Kraków: Devs4Devs Conference
UiPath Community Day Kraków: Devs4Devs ConferenceUiPath Community Day Kraków: Devs4Devs Conference
UiPath Community Day Kraków: Devs4Devs Conference
 
What’s New in Teams Calling, Meetings and Devices May 2024
What’s New in Teams Calling, Meetings and Devices May 2024What’s New in Teams Calling, Meetings and Devices May 2024
What’s New in Teams Calling, Meetings and Devices May 2024
 
How Netflix Builds High Performance Applications at Global Scale
How Netflix Builds High Performance Applications at Global ScaleHow Netflix Builds High Performance Applications at Global Scale
How Netflix Builds High Performance Applications at Global Scale
 

Lesson 21: Curve Sketching

  • 1. Section 4.4 Curve Sketching V63.0121.002.2010Su, Calculus I New York University June 10, 2010 Announcements Homework 4 due Tuesday . . . . . .
  • 2. Announcements Homework 4 due Tuesday . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 2 / 45
  • 3. Objectives given a function, graph it completely, indicating zeroes (if easy) asymptotes if applicable critical points local/global max/min inflection points . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 3 / 45
  • 4. Why? Graphing functions is like dissection . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
  • 5. Why? Graphing functions is like dissection … or diagramming sentences . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
  • 6. Why? Graphing functions is like dissection … or diagramming sentences You can really know a lot about a function when you know all of its anatomy. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 4 / 45
  • 7. The Increasing/Decreasing Test Theorem (The Increasing/Decreasing Test) If f′ > 0 on (a, b), then f is increasing on (a, b). If f′ < 0 on (a, b), then f is decreasing on (a, b). Example Here f(x) = x3 + x2 , and f′ (x) = 3x2 + 2x. f .(x) .′ (x) f . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 5 / 45
  • 8. Testing for Concavity Theorem (Concavity Test) If f′′ (x) > 0 for all x in (a, b), then the graph of f is concave upward on (a, b) If f′′ (x) < 0 for all x in (a, b), then the graph of f is concave downward on (a, b). Example Here f(x) = x3 + x2 , f′ (x) = 3x2 + 2x, and f′′ (x) = 6x + 2. .′′ (x) f f .(x) .′ (x) f . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 6 / 45
  • 9. Graphing Checklist To graph a function f, follow this plan: 0. Find when f is positive, negative, zero, not defined. 1. Find f′ and form its sign chart. Conclude information about increasing/decreasing and local max/min. 2. Find f′′ and form its sign chart. Conclude concave up/concave down and inflection. 3. Put together a big chart to assemble monotonicity and concavity data 4. Graph! . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 7 / 45
  • 10. Outline Simple examples A cubic function A quartic function More Examples Points of nondifferentiability Horizontal asymptotes Vertical asymptotes Trigonometric and polynomial together Logarithmic . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 8 / 45
  • 11. Graphing a cubic Example Graph f(x) = 2x3 − 3x2 − 12x. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 9 / 45
  • 12. Graphing a cubic Example Graph f(x) = 2x3 − 3x2 − 12x. (Step 0) First, let’s find the zeros. We can at least factor out one power of x: f(x) = x(2x2 − 3x − 12) so f(0) = 0. The other factor is a quadratic, so we the other two roots are √ √ 3 ± 32 − 4(2)(−12) 3 ± 105 x= = 4 4 It’s OK to skip this step for now since the roots are so complicated. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 9 / 45
  • 13. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 14. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: . . . −2 x 2 . . x . +1 − . 1 .′ (x) f . . − . 1 2 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 15. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . . x . +1 − . 1 .′ (x) f . . − . 1 2 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 16. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . − . . . + . + x . +1 − . 1 .′ (x) f . . − . 1 2 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 17. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . − . . . + . + x . +1 − . 1 . . + .′ (x) f . − . 1 2 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 18. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . − . . . + . + x . +1 − . 1 . . + − . .′ (x) f . − . 1 2 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 19. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . − . . . + . + x . +1 − . 1 . . + − . . + .′ (x) f . − . 1 2 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 20. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . − . . . + . + x . +1 − . 1 . . + − . . + .′ (x) f . ↗− . . 1 2 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 21. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . − . . . + . + x . +1 − . 1 . . + − . . + .′ (x) f . ↗− . . 1 ↘ . 2 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 22. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . − . . . + . + x . +1 − . 1 . . + − . . + .′ (x) f . ↗− . . 1 ↘ . 2 . ↗ . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 23. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . − . . . + . + x . +1 − . 1 . . + − . . + .′ (x) f . ↗− . . 1 ↘ . 2 . ↗ . f .(x) m . ax . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 24. Step 1: Monotonicity f(x) = 2x3 − 3x2 − 12x =⇒ f′ (x) = 6x2 − 6x − 12 = 6(x + 1)(x − 2) We can form a sign chart from this: − . − . . . . + . −2 x 2 . − . . . + . + x . +1 − . 1 . . + − . . + .′ (x) f . ↗− . . 1 ↘ . 2 . ↗ . f .(x) m . ax m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 10 / 45
  • 25. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
  • 26. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . .′′ (x) f . ./2 1 f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
  • 27. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − .′′ (x) f . ./2 1 f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
  • 28. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − . + + .′′ (x) f . ./2 1 f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
  • 29. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − . + + .′′ (x) f . . ⌢ ./2 1 f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
  • 30. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − . + + .′′ (x) f . . ⌢ ./2 1 . ⌣ f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
  • 31. Step 2: Concavity f′ (x) = 6x2 − 6x − 12 =⇒ f′′ (x) = 12x − 6 = 6(2x − 1) Another sign chart: . − . − . + + .′′ (x) f . . ⌢ ./2 1 . ⌣ f .(x) I .P . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 11 / 45
  • 32. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
  • 33. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. − . . . . + − . . + .′ (x) f . ↗− ↘ . . 1 . ↘ . 2 . ↗ . m . onotonicity . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
  • 34. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ . ⌢ 1/2 . . ⌣ . ⌣ c . oncavity . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
  • 35. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . − . 1 . 1/2 2 . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 12 / 45
  • 36. Combinations of monotonicity and concavity I .I I . . I .II I .V . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
  • 37. Combinations of monotonicity and concavity . decreasing, concave down I .I I . . I .II I .V . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
  • 38. Combinations of monotonicity and concavity . . increasing, decreasing, concave concave down down I .I I . . I .II I .V . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
  • 39. Combinations of monotonicity and concavity . . increasing, decreasing, concave concave down down I .I I . . I .II I .V . decreasing, concave up . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
  • 40. Combinations of monotonicity and concavity . . increasing, decreasing, concave concave down down I .I I . . I .II I .V . . decreasing, increasing, concave up concave up . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 13 / 45
  • 41. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . . . 1 − . 1/2 2 . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
  • 42. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . . . 1 . ./2 − 1 2 . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
  • 43. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
  • 44. Step 3: One sign chart to rule them all Remember, f(x) = 2x3 − 3x2 − 12x. . . + − . . − . . + .′ (x) f . ↗− . . 1 ↘ . ↘ . . 2 ↗ . m .′′ onotonicity − . − − . − . . + + . + + f . (x) . ⌢ ⌢ ./2 . . 1 ⌣ . ⌣ c . oncavity 7 .. − . 6 1/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 14 / 45
  • 45. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
  • 46. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
  • 47. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
  • 48. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
  • 49. Step 4: Graph f .(x) .(x) = 2x3 − 3x2 − 12x f ( √ ) . −1, 7) ( . . 3− 4105 , 0 . 0, 0) ( . . . . 1/2, −61/2) ( ( . x √ ) . . 3+ 4105 , 0 . 2, −20) ( . 7 .. − . 61/2 −. . 20 f .(x) . . . 1 . ./2 . − 1 2 . . s . hape of f m . ax I .P m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 15 / 45
  • 50. Graphing a quartic Example Graph f(x) = x4 − 4x3 + 10 . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 16 / 45
  • 51. Graphing a quartic Example Graph f(x) = x4 − 4x3 + 10 (Step 0) We know f(0) = 10 and lim f(x) = +∞. Not too many other x→±∞ points on the graph are evident. . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 16 / 45
  • 52. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 53. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 54. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. 0 .. . x2 4 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 55. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . x2 4 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 56. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . x2 4 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 57. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 58. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . 0 .. . x − 3) ( 3 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 59. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . 0 .. . x − 3) ( 3 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 60. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . 0 .. . x − 3) ( 3 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 61. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 62. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . 0 .. 0 .. .′ (x) f 0 . 3 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 63. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. 0 .. .′ (x) f 0 . 3 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 64. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . 0 .. .′ (x) f 0 . 3 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 65. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f 0 . 3 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 66. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f ↘ 0 . . 3 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 67. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f ↘ 0 . . ↘ . 3 . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 68. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f ↘ 0 . . ↘ . 3 ↗ . . f .(x) . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 69. Step 1: Monotonicity f(x) = x4 − 4x3 + 10 =⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3) We make its sign chart. . .. + 0 . + . + . x2 4 0 . − . − . .. . 0 + . x − 3) ( 3 . − 0 . .. − . .. . 0 + .′ (x) f ↘ 0 . . ↘ . 3 ↗ . . f .(x) m . in . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 17 / 45
  • 70. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
  • 71. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
  • 72. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: 0 .. 1 . 2x 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
  • 73. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. 1 . 2x 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
  • 74. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. . + 1 . 2x 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
  • 75. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. . + . + 1 . 2x 0 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
  • 76. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. . + . + 1 . 2x 0 . 0 .. . −2 x 2 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
  • 77. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. . + . + 1 . 2x 0 . − . 0 .. . −2 x 2 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45
  • 78. Step 2: Concavity f′ (x) = 4x3 − 12x2 =⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2) Here is its sign chart: − 0 . .. . + . + 1 . 2x 0 . − . − . 0 .. . −2 x 2 . . . . . . . V63.0121.002.2010Su, Calculus I (NYU) Section 4.4 Curve Sketching June 10, 2010 18 / 45