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.
Materi kuliah tentang Aplikasi Integral. Cari lebih banyak mata kuliah Semester 1 di: http://muhammadhabibielecture.blogspot.com/2014/12/kuliah-semester-1-thp-ftp-ub.html
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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 more good examples.
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.
Materi kuliah tentang Aplikasi Integral. Cari lebih banyak mata kuliah Semester 1 di: http://muhammadhabibielecture.blogspot.com/2014/12/kuliah-semester-1-thp-ftp-ub.html
An antiderivative of a function is a function whose derivative is the given function. The problem of antidifferentiation is interesting, complicated, and useful, especially when discussing motion.
This is the slideshow version from class.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
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 more good examples.
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 19: The Mean Value Theorem (Section 021 slides)Matthew Leingang
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 19: The Mean Value Theorem (Section 041 handout)Matthew Leingang
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 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
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.
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 18: Maximum and Minimum Values (Section 021 handout)Matthew Leingang
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.
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.
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 more good examples.
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.
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 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.
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.
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.
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 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 20: Derivatives and the Shape of Curves (Section 041 handout)
1. Section 4.2
Derivatives and the Shapes of Curves
V63.0121.041, Calculus I
New York University
November 15, 2010
Announcements
Quiz this week in recitation on 3.3, 3.4, 3.5, 3.7
There is class on November 24
Announcements
Quiz this week in recitation
on 3.3, 3.4, 3.5, 3.7
There is class on November
24
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 2 / 31
Objectives
Use the derivative of a
function to determine the
intervals along which the
function is increasing or
decreasing (The
Increasing/Decreasing Test)
Use the First Derivative Test
to classify critical points of a
function as local maxima,
local minima, or neither.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 3 / 31
Notes
Notes
Notes
1
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010
2. Objectives
Use the second derivative of
a function to determine the
intervals along which the
graph of the function is
concave up or concave down
(The Concavity Test)
Use the first and second
derivative of a function to
classify critical points as
local maxima or local
minima, when applicable
(The Second Derivative
Test)
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 4 / 31
Outline
Recall: The Mean Value Theorem
Monotonicity
The Increasing/Decreasing Test
Finding intervals of monotonicity
The First Derivative Test
Concavity
Definitions
Testing for Concavity
The Second Derivative Test
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 5 / 31
Recall: The Mean Value Theorem
Theorem (The Mean Value Theorem)
Let f be continuous on [a, b] and
differentiable on (a, b). Then
there exists a point c in (a, b)
such that
f (b) − f (a)
b − a
= f (c).
a
b
c
Another way to put this is that there exists a point c such that
f (b) = f (a) + f (c)(b − a)
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 6 / 31
Notes
Notes
Notes
2
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010
3. Why the MVT is the MITC
Most Important Theorem In Calculus!
Theorem
Let f = 0 on an interval (a, b). Then f is constant on (a, b).
Proof.
Pick any points x and y in (a, b) with x < y. Then f is continuous on
[x, y] and differentiable on (x, y). By MVT there exists a point z in (x, y)
such that
f (y) = f (x) + f (z)(y − x)
So f (y) = f (x). Since this is true for all x and y in (a, b), then f is
constant.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 7 / 31
Outline
Recall: The Mean Value Theorem
Monotonicity
The Increasing/Decreasing Test
Finding intervals of monotonicity
The First Derivative Test
Concavity
Definitions
Testing for Concavity
The Second Derivative Test
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 8 / 31
What does it mean for a function to be increasing?
Definition
A function f is increasing on (a, b) if
f (x) < f (y)
whenever x and y are two points in (a, b) with x < y.
An increasing function “preserves order.”
Write your own definition (mutatis mutandis) of decreasing,
nonincreasing, nondecreasing
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 9 / 31
Notes
Notes
Notes
3
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010
4. 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).
Proof.
It works the same as the last theorem. Pick two points x and y in (a, b)
with x < y. We must show f (x) < f (y). By MVT there exists a point c
in (x, y) such that
f (y) − f (x) = f (c)(y − x) > 0.
So f (y) > f (x).
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 10 / 31
Finding intervals of monotonicity I
Example
Find the intervals of monotonicity of f (x) = 2x − 5.
Solution
f (x) = 2 is always positive, so f is increasing on (−∞, ∞).
Example
Describe the monotonicity of f (x) = arctan(x).
Solution
Since f (x) =
1
1 + x2
is always positive, f (x) is always increasing.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 11 / 31
Finding intervals of monotonicity II
Example
Find the intervals of monotonicity of f (x) = x2
− 1.
Solution
f (x) = 2x, which is positive when x > 0 and negative when x is.
We can draw a number line:
f
f
−
0
0 +
So f is decreasing on (−∞, 0) and increasing on (0, ∞).
In fact we can say f is decreasing on (−∞, 0] and increasing on [0, ∞)
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 12 / 31
Notes
Notes
Notes
4
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010
5. Finding intervals of monotonicity III
Example
Find the intervals of monotonicity of f (x) = x2/3
(x + 2).
Solution
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 13 / 31
The First Derivative Test
Theorem (The First Derivative Test)
Let f be continuous on [a, b] and c a critical point of f in (a, b).
If f (x) > 0 on (a, c) and f (x) < 0 on (c, b), then c is a local
maximum.
If f (x) < 0 on (a, c) and f (x) > 0 on (c, b), then c is a local
minimum.
If f (x) has the same sign on (a, c) and (c, b), then c is not a local
extremum.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 14 / 31
Outline
Recall: The Mean Value Theorem
Monotonicity
The Increasing/Decreasing Test
Finding intervals of monotonicity
The First Derivative Test
Concavity
Definitions
Testing for Concavity
The Second Derivative Test
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 17 / 31
Notes
Notes
Notes
5
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010
6. Concavity
Definition
The graph of f is called concave up on an interval I if it lies above all its
tangents on I. The graph of f is called concave down on I if it lies below
all its tangents on I.
concave up concave down
We sometimes say a concave up graph “holds water” and a concave down
graph “spills water”.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 18 / 31
Inflection points indicate a change in concavity
Definition
A point P on a curve y = f (x) is called an inflection point if f is
continuous at P and the curve changes from concave upward to concave
downward at P (or vice versa).
concave
down
concave up
inflection point
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 19 / 31
Theorem (Concavity Test)
If f (x) > 0 for all x in an interval I, then the graph of f is concave
upward on I.
If f (x) < 0 for all x in I, then the graph of f is concave downward
on I.
Proof.
Suppose f (x) > 0 on I. This means f is increasing on I. Let a and x be
in I. The tangent line through (a, f (a)) is the graph of
L(x) = f (a) + f (a)(x − a)
By MVT, there exists a c between a and x with
f (x) = f (a) + f (c)(x − a)
Since f is increasing, f (x) > L(x).
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 20 / 31
Notes
Notes
Notes
6
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010
7. Finding Intervals of Concavity I
Example
Find the intervals of concavity for the graph of f (x) = x3
+ x2
.
Solution
We have f (x) = 3x2
+ 2x, so f (x) = 6x + 2.
This is negative when x < −1/3, positive when x > −1/3, and 0 when
x = −1/3
So f is concave down on (−∞, −1/3), concave up on (−1/3, ∞), and
has an inflection point at (−1/3, 2/27)
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 21 / 31
Finding Intervals of Concavity II
Example
Find the intervals of concavity of the graph of f (x) = x2/3
(x + 2).
Solution
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 22 / 31
The Second Derivative Test
Theorem (The Second Derivative Test)
Let f , f , and f be continuous on [a, b]. Let c be be a point in (a, b)
with f (c) = 0.
If f (c) < 0, then c is a local maximum.
If f (c) > 0, then c is a local minimum.
Remarks
If f (c) = 0, the second derivative test is inconclusive (this does not
mean c is neither; we just don’t know yet).
We look for zeroes of f and plug them into f to determine if their f
values are local extreme values.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 23 / 31
Notes
Notes
Notes
7
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010
8. Proof of the Second Derivative Test
Proof.
Suppose f (c) = 0 and f (c) > 0. Since f is continuous, f (x) > 0 for
all x sufficiently close to c. Since f = (f ) , we know f is increasing near
c. Since f (c) = 0 and f is increasing, f (x) < 0 for x close to c and less
than c, and f (x) > 0 for x close to c and more than c. This means f
changes sign from negative to positive at c, which means (by the First
Derivative Test) that f has a local minimum at c.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 24 / 31
Using the Second Derivative Test I
Example
Find the local extrema of f (x) = x3
+ x2
.
Solution
f (x) = 3x2
+ 2x = x(3x + 2) is 0 when x = 0 or x = −2/3.
Remember f (x) = 6x + 2
Since f (−2/3) = −2 < 0, −2/3 is a local maximum.
Since f (0) = 2 > 0, 0 is a local minimum.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 25 / 31
Using the Second Derivative Test II
Example
Find the local extrema of f (x) = x2/3
(x + 2)
Solution
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 26 / 31
Notes
Notes
Notes
8
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010
9. Using the Second Derivative Test II: Graph
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 27 / 31
When the second derivative is zero
At inflection points c, if f is differentiable at c, then f (c) = 0
Is it necessarily true, though?
Consider these examples:
f (x) = x4
g(x) = −x4
h(x) = x3
All of them have critical points at zero with a second derivative of zero.
But the first has a local min at 0, the second has a local max at 0, and the
third has an inflection point at 0. This is why we say 2DT has nothing to
say when f (c) = 0.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 28 / 31
When first and second derivative are zero
function derivatives graph type
f (x) = x4
f (x) = 4x3, f (0) = 0
min
f (x) = 12x2, f (0) = 0
g(x) = −x4
g (x) = −4x3, g (0) = 0
max
g (x) = −12x2, g (0) = 0
h(x) = x3
h (x) = 3x2, h (0) = 0
infl.
h (x) = 6x, h (0) = 0
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 29 / 31
Notes
Notes
Notes
9
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010
10. When the second derivative is zero
At inflection points c, if f is differentiable at c, then f (c) = 0
Is it necessarily true, though?
Consider these examples:
f (x) = x4
g(x) = −x4
h(x) = x3
All of them have critical points at zero with a second derivative of zero.
But the first has a local min at 0, the second has a local max at 0, and the
third has an inflection point at 0. This is why we say 2DT has nothing to
say when f (c) = 0.
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 30 / 31
Summary
Concepts: Mean Value Theorem, monotonicity, concavity
Facts: derivatives can detect monotonicity and concavity
Techniques for drawing curves: the Increasing/Decreasing Test and
the Concavity Test
Techniques for finding extrema: the First Derivative Test and the
Second Derivative Test
V63.0121.041, Calculus I (NYU) Section 4.2 The Shapes of Curves November 15, 2010 31 / 31
Notes
Notes
Notes
10
Section 4.2 : The Shapes of CurvesV63.0121.041, Calculus I November 15, 2010