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Calculus AB Minimum & Maximum

Section 4: Applications of Derivatives: Lecture 4 | 12:22 min

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Lecture Comments (9)

	0 answers Post by Dev Ghan on November 21, 2015 The video keeps cutting off at 11:30. Can anyone fix this? Thanks
	0 answers Post by hamed alsubhi on March 6, 2015 I got the answer for both values positive in example three. how come you got -23.14?
	1 answer Last reply by: Jimmy Wu Sat Oct 19, 2013 10:56 PM Post by Si Jia Wen on August 27, 2013 I can't view the end of the video. It keeps cutting me off. Can someone tell me the answer to example 4? Is it 0 and 3/2?
	1 answer Last reply by: Ben Schalck Thu Dec 20, 2012 5:38 PM Post by abbas esmailzadeh on November 3, 2012 is it e^pi/4 and cos^pi/4 or cos of pi/4
	0 answers Post by help me on October 25, 2012 I love your videos :) So easy to understand, unlike Jishi.
	1 answer Last reply by: Arshin Jain Sat Jan 4, 2014 6:23 PM Post by Steve Denton on October 12, 2012 on example 3 at 10:28, my graph does not show a minimum at pi, rather at 3.9 or so. The slope is not 0 at pi. Why were not the tests done with the derivative function?

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Minimum & Maximum

Critical value (maximum or minimum) exists when derivative function = 0
Finding critical value

Take derivative of appropriate function
Set derivative function = 0
Solve for x
Examine derivative behavior around critical value to determine maximum or minimum

Make sure to distinguish between relative and absolute max and min

Minimum & Maximum

Find the minimum or maximum of the following function f(x) = 2x² − 8x + 8

f′(x) = 4x − 8
Find where the derivative is zero
4x − 8 = 0
4x = 8
x = 2
Is this a minimum or a maximum? Test nearby points
f′(0) = 4(1) − 8 = −4
f′(1) = 4(3) − 8 = 4
The derivative goes from negative to positive, so it's a minimum

Minimum at x = 2

Find the relative minimum or maximum of the following function f(x) = −x³ + x² + x − 1

f′(x) = −3x² + 2x + 1
Use quadratic equation to find zeros or roots
x = [(−2 ±√{2² − 4(−3)(1)})/(2(−3))]
x = [(−2 ±√{16})/(−6)]
x = [(−2 ±4)/(−6)]
x = −[1/3], 1
f′(−1) = −3(−1)² + 2(−1) + 1 = −4
f′(0) = −3(0)² + 2(0) + 1 = 1
f′(2) = −3(2)² + 2(2) + 1 = −7

Relative minimum at x = −[1/3] and relative maximum at x = 1

Find the minimums and maximums of f(x) = sinx on the interval [0, 2 π]

f′(x) = cosx
When does cosx equal zero in that interval?
f′([(π)/2]) = cos[(π)/2] = 0
f′([(3 π)/2]) = cos[(3 π)/2] = 0
Test points
f′([(π)/4]) = [(√2)/2]
f′(π) = −1
f′(2 π) = 1

Maximum at x = [(π)/2], minimum at x = [(3 π)/2]

Find the critical points of f(x) = [4/5]x⁵ − [4/3]x³ + x

f′(x) = 4x⁴ − 4x² + 1
f′(x) = (2x² − 1)²
(2x² − 1)² = 0
2x² − 1 = 0
2x² = 1
x = ±[1/(√2)]

Critical points are x = −[1/(√2)] and x = [1/(√2)]

Find the critical points of f(x) = lnx − x

f′(x) = [1/x] − 1
f′(1) = [1/1] − 1 = 0
We know there's a relative minimum or maximum at x = 1, but there's a variable in the denominator, so it's possible there's an undefined value.
f′(0) = [1/0] − 1 is undefined.

Critical points are x = 1 and x = 0

A ball is thrown from the ground into the air. Its position is given by x(t) = −9.8t² + 19.6t. Find the maximum height the ball reaches.

x′(t) = −19.6t + 19.6
−19.6t + 19.6 = 0
19.6t = 19.6
t = 1
It reaches a maximum at t = 1. What position is the ball at at t = 1?
x(1) = −9.8(1)² + 19.6(1) = 9.8

The ball reaches a height of 9.8

Find the maximum and minimum of the following function f(x) = √{1 − x²} in the interval (−1,1)

f′(x) = [1/2] [1/(√{1 − x²})] [d/dx] (1 − x²)
f′(x) = [(−x)/(√{1 − x²})]
f′(0) = [(−0)/(√{1 − 0})] = 0
f′(−.5) = [.5/(√{1 − (−.5)²})] > 0
f′(.5) = [(−.5)/(√{1 − (.5)²})] < 0

Maximum at x = 0

Find the critical points of the following function f(x) = 2x³ − 6x² + 6x + 9

f′(x) = 6x² − 12x + 6
f′(x) = 6(x² − 2x + 1)
f′(x) = 6(x − 1)²
6(x − 1)² = 0
x − 1 = 0
x = 1

Critical point at x = 1

Find the critical points of the following function f(x) = 2x³ − 6x² + 3x + 99

f′(x) = 6x² − 12x + 3
f′(x) = 3(2x² − 4x + 1)
3(2x² − 4x + 1) = 0
(2x² − 4x + 1) = 0
Quadratic equation
x = [(−(−4) ±√{(−4)² − 4(2)(1)})/2(2)]
x = [(4 ±√{16 − 8})/4]
x = [(4 ±2√2)/4]
x = 1 ±[(√2)/2]

The critical points are x = 1 + [(√2)/2] and x = 1 − [(√2)/2]

Find the relative minimum and maximum of f(x) = x³ − 12x

f′(x) = 3x² − 12
3x² − 12x = 0
x² − 4x = 0
(x + 2)(x − 2) = 0
x = −2, 2
Test points
f′(−3) = 3(−3)² − 12 = 15
f′(0) = 3(0)² − 12 = −12
f′(3) = 3(3)² − 12 = 15

Relative maximum at x = −2 relative minimum at x = 2

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Answer

Minimum & Maximum

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Intro

Extrema: First Derivative Test

Example 1

Example 2: Real World Application/ Cost Function

Example 3: Minimums & Maximums

Example 4: Find Critical Points

Intro 0:00
Extrema: First Derivative Test 0:09

Example 1

Example 2: Real World Application/ Cost Function 4:05
Example 3: Minimums & Maximums 7:10
Example 4: Find Critical Points 10:52

Mathematics: AP Calculus AB

Section 1: Functions
	Definitions & Properties of Functions	11:26
	Graphing	13:58
	Inverse Functions	6:47
	Polynomial Functions	5:04
	Trigonometric Functions	6:45
	Inverse Trigonometric Functions	5:58
	Trigonometric Identities	17:42
	Exponential Functions	5:53
	Logarithmic Functions	7:08
	Rational Functions	15:36
	Conic Sections	14:58
Section 2: Limits and Continuity
	Limit Definition & Properties	7:15
	Solving Limits with Algebra	8:01
	Rational Limit Rules	3:16
	One Sided Limits	9:57
	Special Trigonometric Limits	8:28
	Limits & Continuity	10:14
	Limits: Multiple Choice Practice	6:16
Section 3: Derivatives
	Derivative Definition & Properties	4:11
	Basic Rules of Differentiation	7:07
	Power Rule	7:14
	Trigonometric Rules	7:53
	Product Rule	11:11
	Quotient Rule	16:50
	Chain Rule	19:48
	Higher Order Derivatives	15:00
	Derivatives of Exponential Functions	13:14
	Derivatives of Logarithmic Functions	11:30
	Derivatives of Inverse Trigonometric Functions	16:54
	Implicit Differentiation	16:53
	Multiple Choice Practice: Derivatives	11:07
Section 4: Applications of Derivatives
	Tangent & Normal Lines	22:36
	Position Velocity & Acceleration	18:42
	Related Rates	26:22
	Minimum & Maximum	12:22
	Concavity	11:43
	Rolles Theorem	8:28
	Mean Value Theorem	9:39
	Differentials	12:25
	Applications of Derivatives: Multiple Choice Practice	13:21
	Applications of Derivatives: Free Response Practice	10:22
Section 5: Integrals
	Definition of Integrals	1:08
	Integrals of Power Rule	8:50
	Integrals Basic Rules of Integration	9:43
	Trigonometric Rules of Integrals	8:58
	Chain Rule	13:59
	Integrals of Exponential Functions	12:52
	Integrals of Natural Logarithmic Functions	13:00
	Integrals of Inverse Trigonometric Functions	8:29
	Integrals: Multiple Choice Practice	15:37
Section 6: Applications of Integrals
	Fundamental Theorem of Calculus	15:55
	Area Under A Curve	18:34
	Reimann Sums	10:35
	Trapezoid Rule	12:46
	Mean Value Theorem	11:22
	Second Fundamental Theorem of Calculus	4:44
	Area Between Curves	16:39
	Revolving Solids Washer Disk Methods	21:09
	Revolving Solids Cylindrical Shells Method	26:46
	Revolving Solids Known Cross Sections	27:41
	Differential Equations Eulers Method	17:54
	Differential Equations Slope Fields	16:30
	Application of Integrals: Multiple Choice Practice	14:19
	Application of Integrals: Free Response Practice	9:14
Section 7: Sample AP Test
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 1	17:50
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 2	17:32
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 3	22:14
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 4	19:35
	AP Calculus AB Practice test: Section 1: Multiple Choice Part 5	25:43
	AP Calculus AB Practice Test: Section 2: Free Response Part 1	16:50
	AP Calculus AB Practice Test: Section 2: Free Response Part 2	21:36
	AP Calculus AB Practice Test: Section 2: Free Response Part 3	13:39

Related Books

	Cracking the AP Calculus AB Exam, 2015 Edition Author: Princeton Review ISBN: 804124809 Publisher: Princeton Review Year: 2014 The book includes 3 full length practice tests with detailed explanations, a review of all the key concepts, and targeted strateeies to ace th exam for your highest score.
	Cliffs AP: Calculus AB & BC, 3rd Edition Authors: Dale W Johnson, Kerry J King ISBN: 764586831 Publisher: Cliffs Notes Year: 2001 The book has a topic-by topic breakdown and lots of problem approach suggestions for both free response and multiple choice calculus questions. The book also includes two full length tests.

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Minimum & Maximum

Slide Duration:

Table of Contents

Section 1: Functions

Definitions & Properties of Functions

11m 26s

Intro