The document contains lecture notes on curve sketching from a Calculus I class. It discusses using the first and second derivative tests to determine properties of a function like monotonicity, concavity, maxima, minima, and points of inflection in order to sketch the graph of the function. It then provides an example of using these tests to sketch the graph of the cubic function f(x) = 2x^3 - 3x^2 - 12x.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Lesson 20: Derivatives and the Shapes of Curves (handout)Matthew Leingang
This document contains lecture notes on calculus from a Calculus I course. It covers determining the monotonicity of functions using the first derivative test. Key points include using the sign of the derivative to determine if a function is increasing or decreasing over an interval, and using the first derivative test to classify critical points as local maxima, minima, or neither. Examples are provided to demonstrate finding intervals of monotonicity for various functions and applying the first derivative test.
The document provides information about an upcoming calculus exam, quiz, and lecture on curve sketching. It outlines the procedure for sketching curves, including using the increasing/decreasing test and concavity test. It then provides an example of sketching a cubic function, showing the steps of finding critical points, inflection points, and putting together a sign chart to determine the function's monotonicity and concavity over intervals.
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 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.
The document provides an overview of curve sketching in calculus. It discusses objectives like sketching a function graph completely by identifying zeros, asymptotes, critical points, maxima/minima and inflection points. Examples are provided to demonstrate the increasing/decreasing test using derivatives and the concavity test to determine concave up/down regions. A step-by-step process is outlined to graph functions which involves analyzing monotonicity using the sign chart of the derivative and concavity using the second derivative sign chart. This is demonstrated on sketching the graph of a cubic function f(x)=2x^3-3x^2-12x.
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
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.
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 (-∞,∞).
Lesson 14: Derivatives of Logarithmic and Exponential Functions (handout)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
This document contains notes from a Calculus I class lecture on the derivative. The lecture covered the definition of the derivative and examples of how it can be used to model rates of change in various contexts like velocity, population growth, and marginal costs. It also discussed properties of the derivative like how the derivative of a function relates to whether the function is increasing or decreasing over an interval.
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
Lesson 13: Exponential and Logarithmic Functions (handout)Matthew Leingang
This document contains lecture notes from a Calculus I class covering exponential and logarithmic functions. The notes discuss definitions and properties of exponential functions, including the number e and the natural exponential function. Conventions for rational, irrational and negative exponents are also defined. Examples are provided to illustrate approximating exponential expressions with irrational exponents using rational exponents. The objectives are to understand exponential functions, their properties, and apply laws of logarithms including the change of base formula.
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 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.
Similar to Lesson 21: Curve Sketching (handout) (20)
This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document contains lecture notes from a Calculus I class discussing optimization problems. It begins with announcements about upcoming exams and courses the professor is teaching. It then presents an example problem about finding the rectangle of a fixed perimeter with the maximum area. The solution uses calculus techniques like taking the derivative to find the critical points and determine that the optimal rectangle is a square. The notes discuss strategies for solving optimization problems and summarize the key steps to take.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
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.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
1. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Sec on 4.4
Curve Sketching
V63.0121.001: Calculus I
Professor Ma hew Leingang
New York University
April 13, 2011
.
.
Notes
Announcements
Quiz 4 on Sec ons 3.3,
3.4, 3.5, and 3.7 this
week (April 14/15)
Quiz 5 on Sec ons
4.1–4.4 April 28/29
Final Exam Thursday May
12, 2:00–3:50pm
I am teaching Calc II MW
2:00pm and Calc III TR
2:00pm both Fall ’11 and
Spring ’12
.
.
Notes
Objectives
given a func on, graph it
completely, indica ng
zeroes (if easy)
asymptotes if applicable
cri cal points
local/global max/min
inflec on points
.
.
. 1
.
2. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Why?
Graphing func ons is like
dissec on … or diagramming
sentences
You can really know a lot
about a func on when you
know all of its anatomy.
.
.
Notes
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
f(x)
f′ (x)
f(x) = x3 + x2
f′ (x) = 3x2 + 2x .
.
.
Notes
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
f′′ (x) f′ (x) f(x)
f(x) = x3 + x2
f′ (x) = 3x2 + 2x .
f′′ (x) = 6x + 2
.
.
. 2
.
3. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Graphing Checklist
To graph a func on f, follow this plan:
0. Find when f is posi ve, nega ve, zero,
not defined.
1. Find f′ and form its sign chart.
Conclude informa on about
increasing/decreasing and local
max/min.
2. Find f′′ and form its sign chart.
Conclude concave up/concave down
and inflec on.
.
.
Notes
Graphing Checklist
To graph a func on f, follow this plan:
3. Put together a big chart to assemble
monotonicity and concavity data
4. Graph!
.
.
Notes
Outline
Simple examples
A cubic func on
A quar c func on
More Examples
Points of nondifferen ability
Horizontal asymptotes
Ver cal asymptotes
Trigonometric and polynomial together
Logarithmic
.
.
. 3
.
4. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
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 quadra c, 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.
.
Notes
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:
− .− +
x−2
2
− + +
x+1
−1 f′ (x)
+ − +
↗−1 ↘ 2 ↗ f(x)
max min
.
.
Notes
Step 2: Concavity
f′ (x) = 6x2 − 6x − 12
=⇒ f′′ (x) = 12x − 6 = 6(2x − 1)
Another sign chart:
.
−− ++ f′′ (x)
⌢ 1/2 ⌣ f(x)
IP
.
.
. 4
.
5. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
+ −. − + f′ (x)
↗−1 ↘ ↘ 2 ↗ monotonicity
−− −− ++ ++ f′′ (x)
⌢ ⌢ 1/2 ⌣ ⌣ concavity
7 −61/2 −20 f(x)
−1 1/2 2 shape of f
max IP min
.
.
Notes
monotonicity and concavity
increasing, decreasing,
concave concave
down down
II I
.
III IV
decreasing, increasing,
concave concave
up up
.
.
Notes
Step 3: One sign chart to rule them all
Remember, f(x) = 2x3 − 3x2 − 12x.
+ −. − + f′ (x)
↗−1 ↘ ↘ 2 ↗ monotonicity
−− −− ++ ++ f′′ (x)
⌢ ⌢ 1/2 ⌣ ⌣ concavity
7 −61/2 −20 f(x)
−1 1/2 2 shape of f
max IP min
.
.
. 5
.
6. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
f(x) Notes
Step 4: Graph
f(x) = 2x3 − 3x2 − 12x
( √ ) (−1, 7)
3− 105
4 ,0 (0, 0)
. ( x√ )
(1/2, −61/2)
3+ 105
4 ,0
(2, −20)
7 −61/2 −20 f(x)
−1 1/2 2 shape of f
max IP min
.
.
Notes
Graphing a quartic
Example
Graph f(x) = x4 − 4x3 + 10
(Step 0) We know f(0) = 10 and lim f(x) = +∞. Not too many
x→±∞
other points on the graph are evident.
.
.
Notes
Step 1: Monotonicity
f(x) = x4 − 4x3 + 10
=⇒ f′ (x) = 4x3 − 12x2 = 4x2 (x − 3)
We make its sign chart.
+0 . + +
4x2
0
− − 0+
(x − 3)
3 ′
−0 − 0 + f (x)
↘0 ↘ 3 ↗ f(x)
min
.
.
. 6
.
7. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Step 2: Concavity
f′ (x) = 4x3 − 12x2
=⇒ f′′ (x) = 12x2 − 24x = 12x(x − 2)
Here is its sign chart:
−0. + +
12x
0
− 0 +−
x−2
2 f′′ (x)
++0 −− 0 ++
⌣0 ⌢ 2 ⌣ f(x)
IP IP
.
.
Notes
Step 3: Grand Unified Sign Chart
.
Remember, f(x) = x4 − 4x3 + 10.
−0 − −0+ f′ (x)
↘0 ↘ ↘3↗ monotonicity
f′′ (x)
++0 −− 0++ ++
⌣0 ⌢ 2⌣ ⌣ concavity
10 −6 −17 f(x)
0 2 3 shape
IP IP min
.
.
y
Notes
Step 4: Graph
f(x) = x4 − 4x3 + 10
(0, 10)
. x
(2, −6)
(3, −17)
10 −6 −17 f(x)
0 2 3 shape
IP IP min
.
.
. 7
.
8. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Outline
Simple examples
A cubic func on
A quar c func on
More Examples
Points of nondifferen ability
Horizontal asymptotes
Ver cal asymptotes
Trigonometric and polynomial together
Logarithmic
.
.
Notes
Graphing a function with a cusp
Example
√
Graph f(x) = x + |x|
This func on looks strange because of the absolute value. But
whenever we become nervous, we can just take cases.
.
.
Notes
Step 0: Finding Zeroes
√
f(x) = x + |x|
First, look at f by itself. We can tell that f(0) = 0 and that
f(x) > 0 if x is posi ve.
Are there nega ve numbers which are zeroes for f?
√ √
x + −x = 0 =⇒ −x = −x
−x = x2 =⇒ x2 + x = 0
The only solu ons are x = 0 and x = −1.
.
.
. 8
.
9. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Step 0: Asymptotic behavior
√
f(x) = x + |x|
lim f(x) = ∞, because both terms tend to ∞.
x→∞
lim f(x) is indeterminate of the form −∞ + ∞. It’s the same
x→−∞ √
as lim (−y + y)
y→+∞
√
√ √ y+y
lim (−y + y) = lim ( y − y) · √
y→+∞ y→∞ y+y
y − y2
= lim √ = −∞
y→∞ y+y
.
.
Notes
Step 1: The derivative
√
Remember, f(x) = x + |x|.
To find f′ , first assume x > 0. Then
d ( √ ) 1
f′ (x) = x+ x =1+ √
dx 2 x
No ce
f′ (x) > 0 when x > 0 (so no cri cal points here)
lim+ f′ (x) = ∞ (so 0 is a cri cal point)
x→0
lim f′ (x) = 1 (so the graph is asympto c to a line of slope 1)
x→∞
.
.
Notes
Step 1: The derivative
√
Remember, f(x) = x + |x|.
If x is nega ve, we have
d ( √ ) 1
f′ (x) = x + −x = 1 − √
dx 2 −x
No ce
lim− f′ (x) = −∞ (other side of the cri cal point)
x→0
lim f′ (x) = 1 (asympto c to a line of slope 1)
x→−∞
′
f (x) = 0 when
1 √ 1 1 1
1− √ = 0 =⇒ −x = =⇒ −x = =⇒ x = −
2 −x 2 4 4
.
.
. 9
.
10. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Step 1: Monotonicity
1
1 + √
if x > 0
′
f (x) = 2 x
1 − √ 1
if x < 0
2 −x
We can’t make a mul -factor sign chart because of the absolute
value, but we can test points in between cri cal points.
+ 0− ∞ + f′ (x)
.
↗ − 1↘ 0
4
↗ f(x)
max min
.
.
Notes
Step 2: Concavity
( )
d 1 1
If x > 0, then f′′ (x) = 1 + x−1/2 = − x−3/2 This is
dx 2 4
nega ve whenever x > 0. ( )
d 1 1
If x < 0, then f′′ (x) = 1 − (−x)−1/2 = − (−x)−3/2
dx 2 4
which is also always nega ve for nega ve x.
1
In other words, f′′ (x) = − |x|−3/2 .
4
Here is the sign chart:
−− −∞ −− f′′ (x)
.
⌢ 0 ⌢ f(x)
.
.
Notes
Step 3: Synthesis
Now we can put these things together.
√
f(x) = x + |x|
+1 + 0− ∞ + f′
+1 (x)
.
↗ ↗ − 1↘ 0 ↗ ↗monotonicity
f′′
−∞ −− −− 4 −∞ −− −∞ (x)
⌢ ⌢ 1 ⌢0 ⌢ ⌢concavity
−∞ 0 4 0 +∞f(x)
−1 −4 0
1 shape
zero max min
.
.
. 10
.
11. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Graph
√
f(x) = x + |x|
(− 1 , 1 )
4 4
(−1, 0)
. x
(0, 0)
1
−∞ 0 4 0 +∞ x
−1 −1 0 shape
4
zero max min
.
.
Example with Horizontal Notes
Asymptotes
Example
Graph f(x) = xe−x
2
.
.
Notes
Step 1: Monotonicity
.
.
. 11
.
12. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Step 2: Concavity
.
.
Notes
Step 3: Synthesis
.
.
Notes
Step 4: Graph
f(x)
x
.
.
.
. 12
.
13. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Example with Vertical Asymptotes
Example
1 1
Graph f(x) = +
x x2
.
.
Notes
Step 0
.
.
Notes
Step 1: Monotonicity
.
.
. 13
.
14. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Step 2: Concavity
.
.
Notes
Step 3: Synthesis
.
.
Notes
Step 4: Graph
y
. x
.
.
. 14
.
15. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Trigonometric and polynomial Notes
together
Problem
Graph f(x) = cos x − x
.
.
Notes
Step 0: intercepts and asymptotes
.
.
Notes
Step 1: Monotonicity
.
.
. 15
.
16. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Step 2: Concavity
.
.
Notes
Step 3: Synthesis
.
.
Notes
Step 4: Graph
f(x) = cos x − x
y
.
x
.
.
. 16
.
17. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Logarithmic
Problem
Graph f(x) = x ln x2
.
.
Step 0: Intercepts and Notes
Asymptotes
.
.
Notes
Step 1: Monotonicity
.
.
. 17
.
18. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Step 2: Concavity
.
.
Notes
Step 3: Synthesis
.
.
Notes
Step 4: Graph
y
. x
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19. . V63.0121.001: Calculus I
. Sec on 4.4: Curve Sketching
. April 13, 2011
Notes
Summary
Graphing is a procedure that gets easier with prac ce.
Remember to follow the checklist.
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Notes
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Notes
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. 19
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