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College Calculus: Level I First Derivative Test, Second Derivative Test

Section 5: Application of Derivatives: Lecture 3 | 27:11 min

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Lecture Comments (6)

	1 answer Last reply by: Jamison Czech Fri Jul 11, 2014 1:51 AM Post by Jorge Sardinas on March 16, 2014 For the second derivative test wouldn't f''(c) >0 be concave down and f''(c)<0 be concave up, please correct me if I am wrong.
	0 answers Post by Shahaz Shajahan on August 22, 2012 how would you work out the stationary points of sinh(x)?
	0 answers Post by abrar degnh on May 30, 2012 in example two, sorry .
	1 answer Last reply by: alister guerrero Thu Nov 15, 2012 5:32 PM Post by abrar degnh on May 30, 2012 i don't think 0 is a critical point because it will be undefined in this example, please correct me if i'm wrong.

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First Derivative Test, Second Derivative Test

The tests are two options for determining whether a critical point is a local minimum, a local maximum, or neither.
In general the Second Derivative test is easier to use.
Note that the Second Derivative test can be inconclusive; in this case, switch to the First Derivative Test.
Understanding the geometry behind these tests makes them easy to remember and use!

First Derivative Test, Second Derivative Test

Find the points of inflection of f(x) = [1/2] x³ − 3x² + 7x + 11

f′(x) = [3/2] x² − 6x + 7
f"(x) = 3x − 6
3x − 6 = 0
3x = 6
x = 2

There is a point of inflection at x = 2

Find the points of inflection of f(x) = [1/12] x⁴ − [1/2] x² + 12

f′(x) = [1/3] x³ − x
f"(x) = x² − 1
x² − 1 = 0
(x + 1)(x − 1) = 0
x = −1, 1

The points of inflection are x = −1 and x = 1

Determine the concavity of f(x) = sinx on the interval (0, π)

f′(x) = cosx
f"(x) = −sinx
−sinx = 0
sinx = 0
x = 0, π
These two points of inflection are at the end of the interval meaning everything in between must have the same concavity.
f"([(π)/2]) = −sin[(π)/2] = −1
f"(x) < 0 on the interval

f(x) = sinx is concave down on the interval (0, π)

Determine the concavity of f(x) = e^x

f′(x) = e^x
f"(x) = e^x
There are no points of inflection, e^x> 0

f(x) = e^x is concave up.

Determine the concavity of f(x) = 3x² −7x + 21

f′(x) = 6x − 7
f"(x) = 6
No points of inflection and f"(x) > 0

f(x) = 3x² − 7x + 21 is concave up.

Determine the concavity of f(x) = x³ − 12x

f′(x) = 3x² − 12
f"(x) = 6x
6x = 0
x = 0
Inflection point at x = 0
f"(x) < 0 for x < 0
f"(x) > 0 for x > 0

f(x) = x³ − 12x is concave down on x < 0 and concave up on x > 0

Find any critical points of f(x) = x³ − 12x and determine whether they represent local minimums or maximums.

f′(x) = 3x² − 12
3x² − 12 = 0
x² − 4 = 0
x² = 4
x = ±2
The critical points are x = −2 and x = 2
From the previous problem we know that the function is concave down at x = −2 and concave up at x = 2. This tells us that x = −2 is a local maximum and x = 2 is a local minimum.

Local maximum at x = −2 and a local minimum at x = 2

Find the critical points and the concavity of f(x) = lnx

f′(x) = [1/x]
f′(x) is undefined at x = 0. Critical point at x = 0
f"(x) = −[1/(x²)]
No change in concavity, f"(x) never reaches zero.
f"(x) < 0

f(x) = lnx is concave down and has a critical point at x = 0

Determine the concavity of f(x) = x^[1/3]

f′(x) = [1/3] x^−[2/3]
f"(x) = −[2/9] x^−[5/3]
f"(0) and f′(0) are undefined and f"(x) ≠ 0.
For x < 0, f"(x) > 0
For x > 0, f"(x) < 0

f(x) = x^[1/3] is concave up when x < 0 and concave down when x > 0

Determine the concavity of f(x) = [1/12] x⁴ − [1/6] x³ − x² + 17x + 23

f′(x) = [1/3] x³ − [1/2] x² − 2x + 17
f"(x) = x² − x − 2
x² − x − 2 = 0
(x − 2)(x + 1) = 0
x = −1 and x = 2 are the points of inflection
Test points
f"(−2) = (−2)² − (−2) − 2 = 4
f"(0) = (0)² − 0 − 2 = −2
f"(3) = 3² − 3 − 2 = 4

f(x) = [1/12] x⁴ − [1/6] x³ − x² + 17x + 23 is concave up when x <−1, concave down when −1 < x < 2, and concave up when x > 2

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Answer

First Derivative Test, Second Derivative Test

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

Intro

Local Minimum and Local Maximum

Example

First and Second Derivative Test

Lecture Example 1

Lecture Example 2

Lecture Example 3

Additional Example 4

Additional Example 5

Intro 0:00
Local Minimum and Local Maximum 0:14

Example

First and Second Derivative Test 1:26

First Derivative Test
Example
Second Derivative Test (Concavity)
Example: Concave Down
Example: Concave Up
Inconclusive

Lecture Example 1 5:23
Lecture Example 2 12:03
Lecture Example 3 15:54
Additional Example 4
Additional Example 5

College Calculus 1 Online Course

Section 1: Overview of Functions
	Review of Functions	26:29
	Compositions of Functions	12:29
Section 2: Limits
	Average and Instantaneous Rates of Change	20:59
	Limit Investigations	22:37
	Algebraic Evaluation of Limits	28:19
	Formal Definition of a Limit	23:39
	Continuity and the Intermediate Value Theorem	19:09
Section 3: Derivatives, part 1
	Limit Definition of the Derivative	22:52
	The Power Rule	26:01
	The Product Rule	14:54
	The Quotient Rule	19:17
	Applications of Rates of Change	17:43
	Trigonometric Derivatives	26:58
	The Chain Rule	23:47
	Inverse Trigonometric Functions	27:05
	Equation of a Tangent Line	15:52
Section 4: Derivatives, part 2
	Implicit Differentiation	30:05
	Higher Derivatives	13:16
	Logarithmic and Exponential Function Derivatives	17:42
	Hyperbolic Trigonometric Function Derivatives	14:30
	Related Rates	29:05
	Linear Approximation	23:52
Section 5: Application of Derivatives
	Absolute Minima and Maxima	18:57
	Mean Value Theorem and Rolle's Theorem	20:00
	First Derivative Test, Second Derivative Test	27:11
	L'Hopital's Rule	23:09
	Curve Sketching	40:16
	Applied Optimization	25:37
	Newton's Method	25:13
Section 6: Integrals
	Approximating Areas and Distances	36:50
	Riemann Sums, Definite Integrals, Fundamental Theorem of Calculus	22:02
	Substitution Method for Integration	23:19
Section 7: Application of Integrals, part 1
	Area Between Curves	19:59
	Volume by Method of Disks and Washers	24:22
	Volume by Method of Cylindrical Shells	30:29
	Average Value of a Function	16:31
Section 8: Extra
	Graphs of f, f', f''	23:58
	Slope Fields for Differential Equations	18:32
	Separable Differential Equations	17:04

Related Books

	Calculus by Larson, 9th Edition Authors: Ron Larson, Bruce H. Edwards ISBN: 0547167024 Publisher: Brooks Cole Year: 2009 The book is filled with examples and detailed explanations. Capstone exercises are also included for each section and synthesize the main concepts into a single example.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 1 Author: Ron Larson ISBN: 547213093 Publisher: Brooks Cole Year: 2008 This solutions manual includes worked out solutions to every odd-numbered exercise in Calculus of a Single Variable, 9th edition.
	Calculus by Larson, 9th Edition, Solutions Manual, Volume 2 Authors: Ron Larson, Bruce H. Edwards ISBN: 547213107 Publisher: Brooks Cole Year: 2009 This solutions manual includes worked out solutions to every odd-numbered exercise in Multivariable Calculus, 9th edition.

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Professor Switkes

First Derivative Test, Second Derivative Test

Slide Duration:

Table of Contents

Section 1: Overview of Functions

Review of Functions

26m 29s

Intro