The document provides an overview of curve sketching in calculus including objectives, rationale, theorems for determining monotonicity and concavity, and a checklist for graphing functions. It then gives examples of graphing cubic and quartic functions step-by-step, demonstrating how to analyze critical points, inflection points, and asymptotic behavior to create the curve. The examples illustrate applying differentiation rules to determine monotonicity from the derivative sign chart and concavity from the second derivative test.
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.
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.
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.
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.
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.
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.
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...
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.
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
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
- 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.
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
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.
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.
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.
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 (-∞,∞).
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.
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.
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.
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.
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].
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.
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
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.
- 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.
- 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.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
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!"
“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.
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.
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
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.
The document is about quadratic polynomial functions and contains the following information:
1. It discusses investigating relationships between numbers expressed in tables to represent them in the Cartesian plane, identifying patterns and creating conjectures to generalize and algebraically express this generalization, recognizing when this representation is a quadratic polynomial function of the type y = ax^2.
2. It provides examples of converting algebraic representations of quadratic polynomial functions into geometric representations in the Cartesian plane, distinguishing cases in which one variable is directly proportional to the square of the other, using or not using software or dynamic algebra and geometry applications.
3. It discusses characterizing the coefficients of quadratic functions, constructing their graphs in the Cartesian plane, and determining their zeros (
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.
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.
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.
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.
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.
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.
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.
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.
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
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 Cambridgealproelearning
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.
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.
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.
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.
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.
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.
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Mel Anthony Pepito
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.
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.
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.
Lesson 27: Integration by Substitution (Section 041 slides)Mel Anthony Pepito
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.
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.
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.
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.
Lesson 9: The Product and Quotient Rules (Section 21 slides)Mel Anthony Pepito
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.
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.
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.
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.
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.
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
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.
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.
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.
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.
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.
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.
- 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.
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
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Matthew Leingang
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.
Lesson 26: The Fundamental Theorem of Calculus (Section 021 slides)Mel Anthony Pepito
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.
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.
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)Mel Anthony Pepito
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 (-∞,∞).
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.
Lesson 25: Evaluating Definite Integrals (Section 041 slides)Mel Anthony Pepito
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.
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.
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Mel Anthony Pepito
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.
Lesson 26: The Fundamental Theorem of Calculus (Section 041 slides)Matthew Leingang
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].
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)Matthew Leingang
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.
Lesson20 -derivatives_and_the_shape_of_curves_021_slidesMel Anthony Pepito
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
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.
- 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.
- 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.
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.
Lesson 14: Derivatives of Logarithmic and Exponential FunctionsMel Anthony Pepito
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
Quality Patents: Patents That Stand the Test of TimeAurora Consulting
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
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.
How Social Media Hackers Help You to See Your Wife's Message.pdfHackersList
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.
Fluttercon 2024: Showing that you care about security - OpenSSF Scorecards fo...Chris Swan
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.
MYIR Product Brochure - A Global Provider of Embedded SOMs & SolutionsLinda Zhang
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!"
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.
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 ConferenceUiPathCommunity
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
How Netflix Builds High Performance Applications at Global ScaleScyllaDB
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.
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
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