  Raffi Hovasapian

Maximum & Minimum Values of a Function

Slide Duration:

Section 1: Limits and Derivatives
Overview & Slopes of Curves

42m 8s

Intro
0:00
Overview & Slopes of Curves
0:21
Differential and Integral
0:22
Fundamental Theorem of Calculus
6:36
Differentiation or Taking the Derivative
14:24
What Does the Derivative Mean and How do We Find it?
15:18
Example: f'(x)
19:24
Example: f(x) = sin (x)
29:16
General Procedure for Finding the Derivative of f(x)
37:33
More on Slopes of Curves

50m 53s

Intro
0:00
Slope of the Secant Line along a Curve
0:12
Slope of the Tangent Line to f(x) at a Particlar Point
0:13
Slope of the Secant Line along a Curve
2:59
Instantaneous Slope
6:51
Instantaneous Slope
6:52
Example: Distance, Time, Velocity
13:32
Instantaneous Slope and Average Slope
25:42
Slope & Rate of Change
29:55
Slope & Rate of Change
29:56
Example: Slope = 2
33:16
Example: Slope = 4/3
34:32
Example: Slope = 4 (m/s)
39:12
Example: Density = Mass / Volume
40:33
Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change
47:46
Example Problems for Slopes of Curves

59m 12s

Intro
0:00
Example I: Water Tank
0:13
Part A: Which is the Independent Variable and Which is the Dependent?
2:00
Part B: Average Slope
3:18
Part C: Express These Slopes as Rates-of-Change
9:28
Part D: Instantaneous Slope
14:54
Example II: y = √(x-3)
28:26
Part A: Calculate the Slope of the Secant Line
30:39
Part B: Instantaneous Slope
41:26
Part C: Equation for the Tangent Line
43:59
Example III: Object in the Air
49:37
Part A: Average Velocity
50:37
Part B: Instantaneous Velocity
55:30
Desmos Tutorial

18m 43s

Intro
0:00
Desmos Tutorial
1:42
Desmos Tutorial
1:43
Things You Must Learn To Do on Your Particular Calculator
2:39
Things You Must Learn To Do on Your Particular Calculator
2:40
Example I: y=sin x
4:54
Example II: y=x³ and y = d/(dx) (x³)
9:22
Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}
13:15
The Limit of a Function

51m 53s

Intro
0:00
The Limit of a Function
0:14
The Limit of a Function
0:15
Graph: Limit of a Function
12:24
Table of Values
16:02
lim x→a f(x) Does not Say What Happens When x = a
20:05
Example I: f(x) = x²
24:34
Example II: f(x) = 7
27:05
Example III: f(x) = 4.5
30:33
Example IV: f(x) = 1/x
34:03
Example V: f(x) = 1/x²
36:43
The Limit of a Function, Cont.
38:16
Infinity and Negative Infinity
38:17
Does Not Exist
42:45
Summary
46:48
Example Problems for the Limit of a Function

24m 43s

Intro
0:00
Example I: Explain in Words What the Following Symbols Mean
0:10
Example II: Find the Following Limit
5:21
Example III: Use the Graph to Find the Following Limits
7:35
Example IV: Use the Graph to Find the Following Limits
11:48
Example V: Sketch the Graph of a Function that Satisfies the Following Properties
15:25
Example VI: Find the Following Limit
18:44
Example VII: Find the Following Limit
20:06
Calculating Limits Mathematically

53m 48s

Intro
0:00
Plug-in Procedure
0:09
Plug-in Procedure
0:10
Limit Laws
9:14
Limit Law 1
10:05
Limit Law 2
10:54
Limit Law 3
11:28
Limit Law 4
11:54
Limit Law 5
12:24
Limit Law 6
13:14
Limit Law 7
14:38
Plug-in Procedure, Cont.
16:35
Plug-in Procedure, Cont.
16:36
Example I: Calculating Limits Mathematically
20:50
Example II: Calculating Limits Mathematically
27:37
Example III: Calculating Limits Mathematically
31:42
Example IV: Calculating Limits Mathematically
35:36
Example V: Calculating Limits Mathematically
40:58
Limits Theorem
44:45
Limits Theorem 1
44:46
Limits Theorem 2: Squeeze Theorem
46:34
Example VI: Calculating Limits Mathematically
49:26
Example Problems for Calculating Limits Mathematically

21m 22s

Intro
0:00
Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law
0:16
Example II: Evaluate the Following Limit
1:51
Example III: Evaluate the Following Limit
3:36
Example IV: Evaluate the Following Limit
8:56
Example V: Evaluate the Following Limit
11:19
Example VI: Calculating Limits Mathematically
13:19
Example VII: Calculating Limits Mathematically
14:59
Calculating Limits as x Goes to Infinity

50m 1s

Intro
0:00
Limit as x Goes to Infinity
0:14
Limit as x Goes to Infinity
0:15
Let's Look at f(x) = 1 / (x-3)
1:04
Summary
9:34
Example I: Calculating Limits as x Goes to Infinity
12:16
Example II: Calculating Limits as x Goes to Infinity
21:22
Example III: Calculating Limits as x Goes to Infinity
24:10
Example IV: Calculating Limits as x Goes to Infinity
36:00
Example Problems for Limits at Infinity

36m 31s

Intro
0:00
Example I: Calculating Limits as x Goes to Infinity
0:14
Example II: Calculating Limits as x Goes to Infinity
3:27
Example III: Calculating Limits as x Goes to Infinity
8:11
Example IV: Calculating Limits as x Goes to Infinity
14:20
Example V: Calculating Limits as x Goes to Infinity
20:07
Example VI: Calculating Limits as x Goes to Infinity
23:36
Continuity

53m

Intro
0:00
Definition of Continuity
0:08
Definition of Continuity
0:09
Example: Not Continuous
3:52
Example: Continuous
4:58
Example: Not Continuous
5:52
Procedure for Finding Continuity
9:45
Law of Continuity
13:44
Law of Continuity
13:45
Example I: Determining Continuity on a Graph
15:55
Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous
17:57
Example III: Is the Following Function Continuous at the Given Point?
22:42
Theorem for Composite Functions
25:28
Theorem for Composite Functions
25:29
Example IV: Is cos(x³ + ln x) Continuous at x=π/2?
27:00
Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?
34:04
Types of Discontinuity
39:18
Removable Discontinuity
39:33
Jump Discontinuity
40:06
Infinite Discontinuity
40:32
Intermediate Value Theorem
40:58
Intermediate Value Theorem: Hypothesis & Conclusion
40:59
Intermediate Value Theorem: Graphically
43:40
Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]
47:46
Derivative I

40m 2s

Intro
0:00
Derivative
0:09
Derivative
0:10
Example I: Find the Derivative of f(x)=x³
2:20
Notations for the Derivative
7:32
Notations for the Derivative
7:33
Derivative & Rate of Change
11:14
Recall the Rate of Change
11:15
Instantaneous Rate of Change
17:04
Graphing f(x) and f'(x)
19:10
Example II: Find the Derivative of x⁴ - x²
24:00
Example III: Find the Derivative of f(x)=√x
30:51
Derivatives II

53m 45s

Intro
0:00
Example I: Find the Derivative of (2+x)/(3-x)
0:18
Derivatives II
9:02
f(x) is Differentiable if f'(x) Exists
9:03
Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
17:19
Geometrically: Differentiability Means the Graph is Smooth
18:44
Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0
20:53
Example II: For x > 0
23:53
Example II: For x < 0
25:36
Example II: What is f(0) and What is the lim |x| as x→0?
30:46
Differentiability & Continuity
34:22
Differentiability & Continuity
34:23
How Can a Function Not be Differentiable at a Point?
39:38
How Can a Function Not be Differentiable at a Point?
39:39
Higher Derivatives
41:58
Higher Derivatives
41:59
Derivative Operator
45:12
Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³
49:29
More Example Problems for The Derivative

31m 38s

Intro
0:00
Example I: Sketch f'(x)
0:10
Example II: Sketch f'(x)
2:14
Example III: Find the Derivative of the Following Function sing the Definition
3:49
Example IV: Determine f, f', and f'' on a Graph
12:43
Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value
13:40
Example VI: Distance vs. Time
20:15
Example VII: Displacement, Velocity, and Acceleration
23:56
Example VIII: Graph the Displacement Function
28:20
Section 2: Differentiation
Differentiation of Polynomials & Exponential Functions

47m 35s

Intro
0:00
Differentiation of Polynomials & Exponential Functions
0:15
Derivative of a Function
0:16
Derivative of a Constant
2:35
Power Rule
3:08
If C is a Constant
4:19
Sum Rule
5:22
Exponential Functions
6:26
Example I: Differentiate
7:45
Example II: Differentiate
12:38
Example III: Differentiate
15:13
Example IV: Differentiate
16:20
Example V: Differentiate
19:19
Example VI: Find the Equation of the Tangent Line to a Function at a Given Point
12:18
Example VII: Find the First & Second Derivatives
25:59
Example VIII
27:47
Part A: Find the Velocity & Acceleration Functions as Functions of t
27:48
Part B: Find the Acceleration after 3 Seconds
30:12
Part C: Find the Acceleration when the Velocity is 0
30:53
Part D: Graph the Position, Velocity, & Acceleration Graphs
32:50
Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents
34:53
Example X: Find a Point on a Graph
42:31
The Product, Power & Quotient Rules

47m 25s

Intro
0:00
The Product, Power and Quotient Rules
0:19
Differentiate Functions
0:20
Product Rule
5:30
Quotient Rule
9:15
Power Rule
10:00
Example I: Product Rule
13:48
Example II: Quotient Rule
16:13
Example III: Power Rule
18:28
Example IV: Find dy/dx
19:57
Example V: Find dy/dx
24:53
Example VI: Find dy/dx
28:38
Example VII: Find an Equation for the Tangent to the Curve
34:54
Example VIII: Find d²y/dx²
38:08
Derivatives of the Trigonometric Functions

41m 8s

Intro
0:00
Derivatives of the Trigonometric Functions
0:09
Let's Find the Derivative of f(x) = sin x
0:10
Important Limits to Know
4:59
d/dx (sin x)
6:06
d/dx (cos x)
6:38
d/dx (tan x)
6:50
d/dx (csc x)
7:02
d/dx (sec x)
7:15
d/dx (cot x)
7:27
Example I: Differentiate f(x) = x² - 4 cos x
7:56
Example II: Differentiate f(x) = x⁵ tan x
9:04
Example III: Differentiate f(x) = (cos x) / (3 + sin x)
10:56
Example IV: Differentiate f(x) = e^x / (tan x - sec x)
14:06
Example V: Differentiate f(x) = (csc x - 4) / (cot x)
15:37
Example VI: Find an Equation of the Tangent Line
21:48
Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?
25:17
28:23
Example IX: Evaluate
33:22
Example X: Evaluate
36:38
The Chain Rule

24m 56s

Intro
0:00
The Chain Rule
0:13
Recall the Composite Functions
0:14
Derivatives of Composite Functions
1:34
Example I: Identify f(x) and g(x) and Differentiate
6:41
Example II: Identify f(x) and g(x) and Differentiate
9:47
Example III: Differentiate
11:03
Example IV: Differentiate f(x) = -5 / (x² + 3)³
12:15
Example V: Differentiate f(x) = cos(x² + c²)
14:35
Example VI: Differentiate f(x) = cos⁴x +c²
15:41
Example VII: Differentiate
17:03
Example VIII: Differentiate f(x) = sin(tan x²)
19:01
Example IX: Differentiate f(x) = sin(tan² x)
21:02
More Chain Rule Example Problems

25m 32s

Intro
0:00
Example I: Differentiate f(x) = sin(cos(tanx))
0:38
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
2:25
Example III: F(x) = f(g(x)), Find F' (6)
4:22
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
5:35
Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
10:18
Example VI: Differentiate f(x) = sec²(12x)
12:28
Example VII: Differentiate
14:41
Example VIII: Differentiate
19:25
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time
21:13
Implicit Differentiation

52m 31s

Intro
0:00
Implicit Differentiation
0:09
Implicit Differentiation
0:10
Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y
12:15
Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14
19:18
Example III: Find (dy)/(dx) of x³y² + y³x² = 4x
21:43
Example IV: Find (dy)/(dx) of the Following Equation
24:13
Example V: Find (dy)/(dx) of 6sin x cos y = 1
29:00
Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y
31:02
Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x
37:36
Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)
41:03
Example IX: Find (d²y)/(dx²) of x² + y² = 25
44:05
Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)
47:48
Section 3: Applications of the Derivative
Linear Approximations & Differentials

47m 34s

Intro
0:00
Linear Approximations & Differentials
0:09
Linear Approximations & Differentials
0:10
Example I: Linear Approximations & Differentials
11:27
Example II: Linear Approximations & Differentials
20:19
Differentials
30:32
Differentials
30:33
Example III: Linear Approximations & Differentials
34:09
Example IV: Linear Approximations & Differentials
35:57
Example V: Relative Error
38:46
Related Rates

45m 33s

Intro
0:00
Related Rates
0:08
Strategy for Solving Related Rates Problems #1
0:09
Strategy for Solving Related Rates Problems #2
1:46
Strategy for Solving Related Rates Problems #3
2:06
Strategy for Solving Related Rates Problems #4
2:50
Strategy for Solving Related Rates Problems #5
3:38
Example I: Radius of a Balloon
5:15
12:52
Example III: Water Tank
19:08
Example IV: Distance between Two Cars
29:27
Example V: Line-of-Sight
36:20
More Related Rates Examples

37m 17s

Intro
0:00
0:14
Example II: Particle
4:45
Example III: Water Level
10:28
Example IV: Clock
20:47
Example V: Distance between a House and a Plane
29:11
Maximum & Minimum Values of a Function

40m 44s

Intro
0:00
Maximum & Minimum Values of a Function, Part 1
0:23
Absolute Maximum
2:20
Absolute Minimum
2:52
Local Maximum
3:38
Local Minimum
4:26
Maximum & Minimum Values of a Function, Part 2
6:11
Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
7:18
Function with Local Max & Min but No Absolute Max & Min
8:48
Formal Definitions
10:43
Absolute Maximum
11:18
Absolute Minimum
12:57
Local Maximum
14:37
Local Minimum
16:25
Extreme Value Theorem
18:08
Theorem: f'(c) = 0
24:40
Critical Number (Critical Value)
26:14
Procedure for Finding the Critical Values of f(x)
28:32
Example I: Find the Critical Values of f(x) x + sinx
29:51
Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]
35:31
Example Problems for Max & Min

40m 44s

Intro
0:00
Example I: Identify Absolute and Local Max & Min on the Following Graph
0:11
Example II: Sketch the Graph of a Continuous Function
3:11
Example III: Sketch the Following Graphs
4:40
Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²
6:13
Example V: Find the Critical Values of f(x) = |2x - 5|
8:42
Example VI: Find the Critical Values
11:42
Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]
16:57
Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]
20:08
Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]
24:39
The Mean Value Theorem

25m 54s

Intro
0:00
Rolle's Theorem
0:08
Rolle's Theorem: If & Then
0:09
Rolle's Theorem: Geometrically
2:06
There May Be More than 1 c Such That f'( c ) = 0
3:30
Example I: Rolle's Theorem
4:58
The Mean Value Theorem
9:12
The Mean Value Theorem: If & Then
9:13
The Mean Value Theorem: Geometrically
11:07
Example II: Mean Value Theorem
13:43
Example III: Mean Value Theorem
21:19
Using Derivatives to Graph Functions, Part I

25m 54s

Intro
0:00
Using Derivatives to Graph Functions, Part I
0:12
Increasing/ Decreasing Test
0:13
Example I: Find the Intervals Over Which the Function is Increasing & Decreasing
3:26
Example II: Find the Local Maxima & Minima of the Function
19:18
Example III: Find the Local Maxima & Minima of the Function
31:39
Using Derivatives to Graph Functions, Part II

44m 58s

Intro
0:00
Using Derivatives to Graph Functions, Part II
0:13
Concave Up & Concave Down
0:14
What Does This Mean in Terms of the Derivative?
6:14
Point of Inflection
8:52
Example I: Graph the Function
13:18
Example II: Function x⁴ - 5x²
19:03
Intervals of Increase & Decrease
19:04
Local Maxes and Mins
25:01
Intervals of Concavity & X-Values for the Points of Inflection
29:18
Intervals of Concavity & Y-Values for the Points of Inflection
34:18
Graphing the Function
40:52
Example Problems I

49m 19s

Intro
0:00
Example I: Intervals, Local Maxes & Mins
0:26
Example II: Intervals, Local Maxes & Mins
5:05
Example III: Intervals, Local Maxes & Mins, and Inflection Points
13:40
Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
23:02
Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
34:36
Example Problems III

59m 1s

Intro
0:00
Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
0:11
Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
21:24
Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D
37:56
Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
46:19
L'Hospital's Rule

30m 9s

Intro
0:00
L'Hospital's Rule
0:19
Indeterminate Forms
0:20
L'Hospital's Rule
3:38
Example I: Evaluate the Following Limit Using L'Hospital's Rule
8:50
Example II: Evaluate the Following Limit Using L'Hospital's Rule
10:30
Indeterminate Products
11:54
Indeterminate Products
11:55
Example III: L'Hospital's Rule & Indeterminate Products
13:57
Indeterminate Differences
17:00
Indeterminate Differences
17:01
Example IV: L'Hospital's Rule & Indeterminate Differences
18:57
Indeterminate Powers
22:20
Indeterminate Powers
22:21
Example V: L'Hospital's Rule & Indeterminate Powers
25:13
Example Problems for L'Hospital's Rule

38m 14s

Intro
0:00
Example I: Evaluate the Following Limit
0:17
Example II: Evaluate the Following Limit
2:45
Example III: Evaluate the Following Limit
6:54
Example IV: Evaluate the Following Limit
8:43
Example V: Evaluate the Following Limit
11:01
Example VI: Evaluate the Following Limit
14:48
Example VII: Evaluate the Following Limit
17:49
Example VIII: Evaluate the Following Limit
20:37
Example IX: Evaluate the Following Limit
25:16
Example X: Evaluate the Following Limit
32:44
Optimization Problems I

49m 59s

Intro
0:00
Example I: Find the Dimensions of the Box that Gives the Greatest Volume
1:23
Fundamentals of Optimization Problems
18:08
Fundamental #1
18:33
Fundamental #2
19:09
Fundamental #3
19:19
Fundamental #4
20:59
Fundamental #5
21:55
Fundamental #6
23:44
Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square
24:36
Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)
35:13
Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R
43:10
Optimization Problems II

55m 10s

Intro
0:00
Example I: Optimization Problem
0:13
Example II: Optimization Problem
17:34
Example III: Optimization Problem
35:06
Example IV: Revenue, Cost, and Profit
43:22
Newton's Method

30m 22s

Intro
0:00
Newton's Method
0:45
Newton's Method
0:46
Example I: Find x2 and x3
13:18
Example II: Use Newton's Method to Approximate
15:48
Example III: Find the Root of the Following Equation to 6 Decimal Places
19:57
Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point
23:11
Section 4: Integrals
Antiderivatives

55m 26s

Intro
0:00
Antiderivatives
0:23
Definition of an Antiderivative
0:24
Antiderivative Theorem
7:58
Function & Antiderivative
12:10
x^n
12:30
1/x
13:00
e^x
13:08
cos x
13:18
sin x
14:01
sec² x
14:11
secxtanx
14:18
1/√(1-x²)
14:26
1/(1+x²)
14:36
-1/√(1-x²)
14:45
Example I: Find the Most General Antiderivative for the Following Functions
15:07
Function 1: f(x) = x³ -6x² + 11x - 9
15:42
Function 2: f(x) = 14√(x) - 27 4√x
19:12
Function 3: (fx) = cos x - 14 sinx
20:53
Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )
22:10
Function 5: f(x) = (3e^x) - 2/(1+x²)
25:42
Example II: Given the Following, Find the Original Function f(x)
26:37
Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 40
27:55
Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 5
30:34
Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.2
32:54
Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 7
37:54
Example III: Falling Object
41:58
Problem 1: Find an Equation for the Height of the Ball after t Seconds
42:48
Problem 2: How Long Will It Take for the Ball to Strike the Ground?
48:30
Problem 3: What is the Velocity of the Ball as it Hits the Ground?
49:52
Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?
50:46
The Area Under a Curve

51m 3s

Intro
0:00
The Area Under a Curve
0:13
Approximate Using Rectangles
0:14
Let's Do This Again, Using 4 Different Rectangles
9:40
Approximate with Rectangles
16:10
Left Endpoint
18:08
Right Endpoint
25:34
Left Endpoint vs. Right Endpoint
30:58
Number of Rectangles
34:08
True Area
37:36
True Area
37:37
Sigma Notation & Limits
43:32
When You Have to Explicitly Solve Something
47:56
Example Problems for Area Under a Curve

33m 7s

Intro
0:00
Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve
0:10
Example II: Using 5 Rectangles, Approximate the Area Under the Curve
11:32
Example III: Find the True Area by Evaluating the Limit Expression
16:07
Example IV: Find the True Area by Evaluating the Limit Expression
24:52
The Definite Integral

43m 19s

Intro
0:00
The Definite Integral
0:08
Definition to Find the Area of a Curve
0:09
Definition of the Definite Integral
4:08
Symbol for Definite Integral
8:45
Regions Below the x-axis
15:18
Associating Definite Integral to a Function
19:38
Integrable Function
27:20
Evaluating the Definite Integral
29:26
Evaluating the Definite Integral
29:27
Properties of the Definite Integral
35:24
Properties of the Definite Integral
35:25
Example Problems for The Definite Integral

32m 14s

Intro
0:00
Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
0:11
Example II: Express the Following Limit as a Definite Integral
5:28
Example III: Evaluate the Following Definite Integral Using the Definition
6:28
Example IV: Evaluate the Following Integral Using the Definition
17:06
Example V: Evaluate the Following Definite Integral by Using Areas
25:41
Example VI: Definite Integral
30:36
The Fundamental Theorem of Calculus

24m 17s

Intro
0:00
The Fundamental Theorem of Calculus
0:17
Evaluating an Integral
0:18
Lim as x → ∞
12:19
Taking the Derivative
14:06
Differentiation & Integration are Inverse Processes
15:04
1st Fundamental Theorem of Calculus
20:08
1st Fundamental Theorem of Calculus
20:09
2nd Fundamental Theorem of Calculus
22:30
2nd Fundamental Theorem of Calculus
22:31
Example Problems for the Fundamental Theorem

25m 21s

Intro
0:00
Example I: Find the Derivative of the Following Function
0:17
Example II: Find the Derivative of the Following Function
1:40
Example III: Find the Derivative of the Following Function
2:32
Example IV: Find the Derivative of the Following Function
5:55
Example V: Evaluate the Following Integral
7:13
Example VI: Evaluate the Following Integral
9:46
Example VII: Evaluate the Following Integral
12:49
Example VIII: Evaluate the Following Integral
13:53
Example IX: Evaluate the Following Graph
15:24
Local Maxs and Mins for g(x)
15:25
Where Does g(x) Achieve Its Absolute Max on [0,8]
20:54
On What Intervals is g(x) Concave Up/Down?
22:20
Sketch a Graph of g(x)
24:34
More Example Problems, Including Net Change Applications

34m 22s

Intro
0:00
Example I: Evaluate the Following Indefinite Integral
0:10
Example II: Evaluate the Following Definite Integral
0:59
Example III: Evaluate the Following Integral
2:59
Example IV: Velocity Function
7:46
Part A: Net Displacement
7:47
Part B: Total Distance Travelled
13:15
Example V: Linear Density Function
20:56
Example VI: Acceleration Function
25:10
Part A: Velocity Function at Time t
25:11
Part B: Total Distance Travelled During the Time Interval
28:38
Solving Integrals by Substitution

27m 20s

Intro
0:00
Table of Integrals
0:35
Example I: Evaluate the Following Indefinite Integral
2:02
Example II: Evaluate the Following Indefinite Integral
7:27
Example IIII: Evaluate the Following Indefinite Integral
10:57
Example IV: Evaluate the Following Indefinite Integral
12:33
Example V: Evaluate the Following
14:28
Example VI: Evaluate the Following
16:00
Example VII: Evaluate the Following
19:01
Example VIII: Evaluate the Following
21:49
Example IX: Evaluate the Following
24:34
Section 5: Applications of Integration
Areas Between Curves

34m 56s

Intro
0:00
Areas Between Two Curves: Function of x
0:08
Graph 1: Area Between f(x) & g(x)
0:09
Graph 2: Area Between f(x) & g(x)
4:07
Is It Possible to Write as a Single Integral?
8:20
Area Between the Curves on [a,b]
9:24
Absolute Value
10:32
Formula for Areas Between Two Curves: Top Function - Bottom Function
17:03
Areas Between Curves: Function of y
17:49
What if We are Given Functions of y?
17:50
Formula for Areas Between Two Curves: Right Function - Left Function
21:48
Finding a & b
22:32
Example Problems for Areas Between Curves

42m 55s

Intro
0:00
Instructions for the Example Problems
0:10
Example I: y = 7x - x² and y=x
0:37
Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=2
6:25
Example III: y=(1/x), y=(1/x³), and x=4
12:25
Example IV: 15-2x² and y=x²-5
15:52
Example V: x=(1/8)y³ and x=6-y²
20:20
Example VI: y=cos x, y=sin(2x), [0,π/2]
24:34
Example VII: y=2x², y=10x², 7x+2y=10
29:51
Example VIII: Velocity vs. Time
33:23
Part A: At 2.187 Minutes, Which care is Further Ahead?
33:24
Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
36:32
Part C: At 4 Minutes Which Car is Ahead?
37:11
Part D: At What Time Will the Cars be Side by Side?
37:50
Volumes I: Slices

34m 15s

Intro
0:00
Volumes I: Slices
0:18
Rotate the Graph of y=√x about the x-axis
0:19
How can I use Integration to Find the Volume?
3:16
Slice the Solid Like a Loaf of Bread
5:06
Volumes Definition
8:56
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
12:18
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
19:05
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
25:28
Volumes II: Volumes by Washers

51m 43s

Intro
0:00
Volumes II: Volumes by Washers
0:11
Rotating Region Bounded by y=x³ & y=x around the x-axis
0:12
Equation for Volumes by Washer
11:14
Process for Solving Volumes by Washer
13:40
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
15:58
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
25:07
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
34:20
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
44:05
Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

Intro
0:00
Solids That Are Not Solids-of-Revolution
0:11
Cross-Section Area Review
0:12
Cross-Sections That Are Not Solids-of-Revolution
7:36
Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H
10:54
Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles
20:39
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H
29:27
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64
36:47
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis
46:13
Volumes IV: Volumes By Cylindrical Shells

50m 2s

Intro
0:00
Volumes by Cylindrical Shells
0:11
Find the Volume of the Following Region
0:12
Volumes by Cylindrical Shells: Integrating Along x
14:12
Volumes by Cylindrical Shells: Integrating Along y
14:40
Volumes by Cylindrical Shells Formulas
16:22
Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid
18:33
Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid
25:57
Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid
31:38
Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid
38:44
Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid
44:03
The Average Value of a Function

32m 13s

Intro
0:00
The Average Value of a Function
0:07
Average Value of f(x)
0:08
What if The Domain of f(x) is Not Finite?
2:23
Let's Calculate Average Value for f(x) = x² [2,5]
4:46
Mean Value Theorem for Integrate
9:25
Example I: Find the Average Value of the Given Function Over the Given Interval
14:06
Example II: Find the Average Value of the Given Function Over the Given Interval
18:25
Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]
24:04
Example IV: Find the Average Density of a Rod
27:47
Section 6: Techniques of Integration
Integration by Parts

50m 32s

Intro
0:00
Integration by Parts
0:08
The Product Rule for Differentiation
0:09
Integrating Both Sides Retains the Equality
0:52
Differential Notation
2:24
Example I: ∫ x cos x dx
5:41
Example II: ∫ x² sin(2x)dx
12:01
Example III: ∫ (e^x) cos x dx
18:19
Example IV: ∫ (sin^-1) (x) dx
23:42
Example V: ∫₁⁵ (lnx)² dx
28:25
Summary
32:31
Tabular Integration
35:08
Case 1
35:52
Example: ∫x³sinx dx
36:39
Case 2
40:28
Example: ∫e^(2x) sin 3x
41:14
Trigonometric Integrals I

24m 50s

Intro
0:00
Example I: ∫ sin³ (x) dx
1:36
Example II: ∫ cos⁵(x)sin²(x)dx
4:36
Example III: ∫ sin⁴(x)dx
9:23
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx
15:59
#1: Power of sin is Odd
16:00
#2: Power of cos is Odd
16:41
#3: Powers of Both sin and cos are Odd
16:55
#4: Powers of Both sin and cos are Even
17:10
Example IV: ∫ tan⁴ (x) sec⁴ (x) dx
17:34
Example V: ∫ sec⁹(x) tan³(x) dx
20:55
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx
23:31
#1: Power of sec is Odd
23:32
#2: Power of tan is Odd
24:04
#3: Powers of sec is Odd and/or Power of tan is Even
24:18
Trigonometric Integrals II

22m 12s

Intro
0:00
Trigonometric Integrals II
0:09
Recall: ∫tanx dx
0:10
Let's Find ∫secx dx
3:23
Example I: ∫ tan⁵ (x) dx
6:23
Example II: ∫ sec⁵ (x) dx
11:41
Summary: How to Deal with Integrals of Different Types
19:04
Identities to Deal with Integrals of Different Types
19:05
Example III: ∫cos(5x)sin(9x)dx
19:57
More Example Problems for Trigonometric Integrals

17m 22s

Intro
0:00
Example I: ∫sin²(x)cos⁷(x)dx
0:14
Example II: ∫x sin²(x) dx
3:56
Example III: ∫csc⁴ (x/5)dx
8:39
Example IV: ∫( (1-tan²x)/(sec²x) ) dx
11:17
Example V: ∫ 1 / (sinx-1) dx
13:19
Integration by Partial Fractions I

55m 12s

Intro
0:00
Integration by Partial Fractions I
0:11
Recall the Idea of Finding a Common Denominator
0:12
Decomposing a Rational Function to Its Partial Fractions
4:10
2 Types of Rational Function: Improper & Proper
5:16
Improper Rational Function
7:26
Improper Rational Function
7:27
Proper Rational Function
11:16
Proper Rational Function & Partial Fractions
11:17
Linear Factors
14:04
15:02
Case 1: G(x) is a Product of Distinct Linear Factors
17:10
Example I: Integration by Partial Fractions
20:33
Case 2: D(x) is a Product of Linear Factors
40:58
Example II: Integration by Partial Fractions
44:41
Integration by Partial Fractions II

42m 57s

Intro
0:00
Case 3: D(x) Contains Irreducible Factors
0:09
Example I: Integration by Partial Fractions
5:19
Example II: Integration by Partial Fractions
16:22
Case 4: D(x) has Repeated Irreducible Quadratic Factors
27:30
Example III: Integration by Partial Fractions
30:19
Section 7: Differential Equations
Introduction to Differential Equations

46m 37s

Intro
0:00
Introduction to Differential Equations
0:09
Overview
0:10
Differential Equations Involving Derivatives of y(x)
2:08
Differential Equations Involving Derivatives of y(x) and Function of y(x)
3:23
Equations for an Unknown Number
6:28
What are These Differential Equations Saying?
10:30
Verifying that a Function is a Solution of the Differential Equation
13:00
Verifying that a Function is a Solution of the Differential Equation
13:01
Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation
17:20
General Solution
22:00
Particular Solution
24:36
Initial Value Problem
27:42
Example I: Verify that a Family of Functions is a Solution of the Differential Equation
32:24
Example II: For What Values of K Does the Function Satisfy the Differential Equation
36:07
Example III: Verify the Solution and Solve the Initial Value Problem
39:47
Separation of Variables

28m 8s

Intro
0:00
Separation of Variables
0:28
Separation of Variables
0:29
Example I: Solve the Following g Initial Value Problem
8:29
Example II: Solve the Following g Initial Value Problem
13:46
Example III: Find an Equation of the Curve
18:48
Population Growth: The Standard & Logistic Equations

51m 7s

Intro
0:00
Standard Growth Model
0:30
Definition of the Standard/Natural Growth Model
0:31
Initial Conditions
8:00
The General Solution
9:16
Example I: Standard Growth Model
10:45
Logistic Growth Model
18:33
Logistic Growth Model
18:34
Solving the Initial Value Problem
25:21
What Happens When t → ∞
36:42
Example II: Solve the Following g Initial Value Problem
41:50
Relative Growth Rate
46:56
Relative Growth Rate
46:57
Relative Growth Rate Version for the Standard model
49:04
Slope Fields

24m 37s

Intro
0:00
Slope Fields
0:35
Slope Fields
0:36
Graphing the Slope Fields, Part 1
11:12
Graphing the Slope Fields, Part 2
15:37
Graphing the Slope Fields, Part 3
17:25
Steps to Solving Slope Field Problems
20:24
Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y
22:38
Section 8: AP Practic Exam
AP Practice Exam: Section 1, Part A No Calculator

45m 29s

Intro
0:00
0:10
Problem #1
1:26
Problem #2
2:52
Problem #3
4:42
Problem #4
7:03
Problem #5
10:01
Problem #6
13:49
Problem #7
15:16
Problem #8
19:06
Problem #9
23:10
Problem #10
28:10
Problem #11
31:30
Problem #12
33:53
Problem #13
37:45
Problem #14
41:17
AP Practice Exam: Section 1, Part A No Calculator, cont.

41m 55s

Intro
0:00
Problem #15
0:22
Problem #16
3:10
Problem #17
5:30
Problem #18
8:03
Problem #19
9:53
Problem #20
14:51
Problem #21
17:30
Problem #22
22:12
Problem #23
25:48
Problem #24
29:57
Problem #25
33:35
Problem #26
35:57
Problem #27
37:57
Problem #28
40:04
AP Practice Exam: Section I, Part B Calculator Allowed

58m 47s

Intro
0:00
Problem #1
1:22
Problem #2
4:55
Problem #3
10:49
Problem #4
13:05
Problem #5
14:54
Problem #6
17:25
Problem #7
18:39
Problem #8
20:27
Problem #9
26:48
Problem #10
28:23
Problem #11
34:03
Problem #12
36:25
Problem #13
39:52
Problem #14
43:12
Problem #15
47:18
Problem #16
50:41
Problem #17
56:38
AP Practice Exam: Section II, Part A Calculator Allowed

25m 40s

Intro
0:00
Problem #1: Part A
1:14
Problem #1: Part B
4:46
Problem #1: Part C
8:00
Problem #2: Part A
12:24
Problem #2: Part B
16:51
Problem #2: Part C
17:17
Problem #3: Part A
18:16
Problem #3: Part B
19:54
Problem #3: Part C
21:44
Problem #3: Part D
22:57
AP Practice Exam: Section II, Part B No Calculator

31m 20s

Intro
0:00
Problem #4: Part A
1:35
Problem #4: Part B
5:54
Problem #4: Part C
8:50
Problem #4: Part D
9:40
Problem #5: Part A
11:26
Problem #5: Part B
13:11
Problem #5: Part C
15:07
Problem #5: Part D
19:57
Problem #6: Part A
22:01
Problem #6: Part B
25:34
Problem #6: Part C
28:54
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• ## Transcription 2 answers Last reply by: Professor HovasapianSat May 22, 2021 6:01 PMPost by Leonardo Luo on May 22, 2021On Graph number 1, would the point (-0.5, 5.69) also be a local maximum? 1 answer Last reply by: Professor HovasapianFri Apr 7, 2017 9:57 PMPost by Daniel Persaud on February 24, 2017For the critical value question how did you get x = pi. should it not be -1 1 answer Last reply by: Professor HovasapianThu Apr 7, 2016 2:09 AMPost by Zhe Tian on April 2, 2016For the first graph, wouldn't there be a local minimum at x=0? 1 answer Last reply by: Professor HovasapianThu Dec 3, 2015 12:55 AMPost by Gautham Padmakumar on November 28, 2015You made a small writing typo error in example 2 you wrote down cos x = 1/4 where its actually cos x = -1/4 but the values for x are right anyways!

### Maximum & Minimum Values of a Function

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 0:00
• Maximum & Minimum Values of a Function, Part 1 0:23
• Absolute Maximum
• Absolute Minimum
• Local Maximum
• Local Minimum
• Maximum & Minimum Values of a Function, Part 2 6:11
• Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
• Function with Local Max & Min but No Absolute Max & Min
• Formal Definitions 10:43
• Absolute Maximum
• Absolute Minimum
• Local Maximum
• Local Minimum
• Extreme Value Theorem
• Theorem: f'(c) = 0
• Critical Number (Critical Value)
• Procedure for Finding the Critical Values of f(x)
• Example I: Find the Critical Values of f(x) x + sinx 29:51
• Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π] 35:31

### Transcription: Maximum & Minimum Values of a Function

Hello, and welcome back to www.educator.com, welcome back to AP Calculus.0000

Today, we are going to talk about the maximum and minimum values of a function.0005

A function on a given domain, it is going to achieve different types of max and min.0009

We are going to talk about an absolute max, an absolute min, local max, and local min.0015

Let us jump right on in.0020

Let us start off by just looking at this graph.0024

Let me go ahead and tell you what this graph is.0026

It is what the function is actually, that this graph represents.0030

Let us go to black here.0041

Let us call it f(x).0043

Here f(x) is equal to 3x⁴ - 14x³ + 15x².0044

The domain is restricted on this one.0056

The domain happens to be -0.5.0061

X runs from -0.5 and it is less than or equal to 3.5.0065

We know that a function is complete, when you actually specify its domain.0073

Generally, for the most part, we do not talk about domains but domains really are important.0078

In this case, even though this function is defined over the entire real line, we are going to restrict its domain here.0082

Notice that this is less than or equal to.0090

The endpoints do matter, so it is defined.0092

We have a point there and a point there.0096

Like I said, we are going to be talking about an absolute max, an absolute min, a local max, and local min.0100

I’m going to go through a few of these graphs and talk about it informally.0106

And then, I will go ahead and give formal definitions for what they are.0110

The formal definitions are there just for you, they are in your book.0113

Our discussion right now in the first couple of minutes is going to make clear exactly what these things are.0117

The absolute max of a function on a given domain is exactly what it sounds like,0123

it is the highest value that f(x) takes on that domain.0126

In this particular one, this point right here, which happens to be 3.5 and 33.69.0131

This is the absolute max.0141

On the domain, it is the highest value that f(x) actually achieves.0147

We know that if did not restrict the domain, it would go up into infinity.0151

In that case, there is no absolute max, there is no upper limit that we can say.0157

But here, we can because we have restricted the domain.0161

Let us talk about the absolute minimum.0165

The absolute minimum, the lowest value that a function actually takes on its domain happens to be over here.0168

This point is 2.5 and -7.81.0174

This is an absolute min, it is the lowest value that it takes on its domain.0181

Let us talk about something called a local max and local min.0189

A local max and a local min, that is where the function achieves a local max and local min at a point x in the domain.0192

Such that, if you take some little interval around that particular x, that f(x) is bigger than every other number.0202

Or the f(x) is smaller than every other number.0213

In this case, this right here, this point which happens to be the point 1,4, this is a local max.0216

It is a local max because if I move away from the point 1, if I move a little bit this way or a little bit this way,0228

notice the function is lower than this, the function is lower than this.0235

For a nice small little region around the point, if at that point the function achieves the maximum value that it can,0240

in that little bit, it is a maximum locally speaking.0248

Locally meaning some little neighborhood around that point.0252

Here this is an absolute max, it is also called the global max.0255

Absolute min also called the global min.0259

Overall, what is the biggest?0261

Locally, that is this.0263

This is a local max.0265

Interestingly enough, this point right here also happens to be a local min.0267

This point 2.5, if I take a little neighborhood around 2.5,0273

within that little neighborhood locally around the 2.5, this is the lowest point.0279

Because if I move to the right, the function is bigger.0284

If I move to the left, the y value of the function is bigger.0288

This also happens to be a local min.0291

That can happen, a local min can be an absolute min.0294

A local max can be an absolute max.0298

You can have more than one local min and local max.0301

You can have only one absolute max and absolute min.0305

You might have no absolute max, no absolute min, but you have a bunch of local max and min.0308

You might have no local max and min but you might have an absolute max and absolute min.0314

These all kind of combinations.0318

In this particular case, we have an absolute maximum that it achieves.0321

We have an absolute minimum, also happens to be local minimum.0327

We have a local maximum.0332

This end point over here, it does not really matter.0333

It is the y value is someplace in between.0336

That is it, that is what is going on with absolute max, absolutely min, local max, and local min.0338

We also speak about the point at which the function achieves its absolute max, local max.0346

In this case, this function achieves an absolute max at x = 3.5.0352

It achieves an absolute min at a local min at x = 2.5.0357

It achieves a local max at 1.0361

The values themselves are the y values, 33.694 and -7.81.0364

Let us look at another function here.0372

Once again, a graph can have all or none of these things.0379

Let me go ahead and write that.0381

I think I will use blue.0384

A graph can have all, some, or none of absolute max, absolute min, and the local max and min.0390

In this particular case, let me see, what function I have got here, it looks like the x² function.0415

Here we are looking at the function y = x².0422

Once again, we have restricted its domain.0426

2x being greater than or equal to 0.0429

This point is absolutely included and this just goes off to positive infinity.0433

In this particular case, is there a highest point on this domain?0439

No, because this goes up into infinity, we cannot say that there is an absolute max.0444

There is no absolute max.0449

Is there an absolute min?0454

Yes, there is, this is the lowest point overall on this domain.0455

It is lowest y value.0460

This point 0,0 is the absolute min.0462

Are there local max or min?0467

No, there are not.0470

There is no local max, there is no local min.0472

You might think yourself, could this not be considered a local minimum?0478

No, a local minimum requires that at a point, in this particular case 0,0,0482

that there is actually an interval to the left and to the right of it, which satisfies the conditions.0487

In other words, yes, if I move to the right, it is true.0494

The function gets higher in value but it is not defined to the left.0497

It is not defined to the left but for a local, I need something that is defined around.0501

An interval around it has to surround that thing.0507

The local min always looks like a little valley.0511

A local max always looks like a crest of the hill.0514

That is it, that is local max and local min.0518

In this case, absolute min and no absolute max, no local max, no local min.0522

Let us take a look at another function here.0528

This particular function right here, we have not restricted the domain at all.0541

This goes off to positive infinity, this goes down to negative infinity.0545

It looks like some sort of a cubic function.0549

I actually did not write down what function this is but that is not a problem.0552

We are here to identify graphically absolute max and min, and local max and min.0555

In this particular case, there is no absolute max and there is no absolute min.0560

However, we do have a local min and we do have a local max.0572

Yes, there is a local minimum and it looks like it achieves that minimum at x = 2.0577

There is a local maximum and it looks like it achieves that maximum at x = -2.0583

The maximum values and the minimum values happen to be the y values.0591

Whatever that happens to be, it looks like somewhere around 16, something like that.0596

The same thing around here.0601

That is it, local min, local max, no absolute max, no absolute min.0603

Let us go ahead and give some formal definitions to these concepts, because you are often going to see the formal definitions.0616

We have talked about these things informally, geometrically, let us give them some algebraic identity.0623

Formal definitions, we will let f(x) be a function.0650

We will let d be its domain.0672

Let us define what we mean by absolute max.0680

The absolute max also called the global max.0685

If there is a number c that is in the domain such that0692

the value of f at c is actually bigger than or equal to the value of f(x) for every single x in the domain.0710

Then, f achieves its absolute max at c.0726

The value f(c) is the absolute maximum value.0741

Once again, if there is a number c that has to be in the domain,0753

such that f(c) is bigger than f(x) for all the other x in the domain, then f achieves its absolute max at c and f(c) is the absolute max.0757

That is it, it is the largest y value that the function takes in the domain.0769

This just happens to be the formal definition.0774

Let us give a definition for absolute min which is also called a global min.0778

You can imagine, it is going to be exact same thing except this inequality is going to be reversed.0784

Global min, if there is … everything else is the same.0791

Such that floats that f(c) is actually less than or equal to f(x), for all x in d.0802

Then, f achieves its absolute min at c and f(c) is that absolute min.0824

Nothing strange, completely intuitive.0847

You know what is going.0849

But again, it is very important.0850

To start the formal definitions in mathematics, very important because you want to be very precise0853

We want to be able to take intuitive notions and put them into some symbolic form.0861

Let us go ahead and define what we mean by local max and local min.0868

You know what, I think I’m going to go all these wonderful colors to choose from.0872

I’m going to go to black for this one.0877

Local max also called the relative max.0883

The definition is, if there is a c in the domain such that f(c) is greater than or equal to f(x),0893

for all x not in d, for all x in some open interval around c.0914

Remember what we said, once you have a point c, we have to take some interval around this point c.0927

If I go to the right of c and to the left of c, that the function drops, that is that.0949

Now it is not over everything, it is just locally speaking, a little bit.0955

If there is a c and d such that f and c is greater than or equal f(x) for all x in some open interval around c,0960

then f achieves a local max at c and f(c) is that local max.0970

The definition for local min which is also called a relative min.0989

Everything is exactly the same.0998

I’m just going to do if … the defining condition is f(c) is less than or equal to f(x) for all x in some open interval around c.0999

Then, f achieves its local min at c and f(c) is that local min.1013

Nothing strange, let me go back to blue.1038

The absolute max value and the absolute min values are also called the extreme values.1045

We are going to list a important theorem called the extreme value theorem.1075

We have the extreme value theorem.1090

If f is continuous on a closed interval ab,1106

then f achieves both an absolute max and an absolute min on ab.1126

Very important, the two hypotheses of this theorem that, if f is continuous,1152

f has to be continuous and it has to be a closed interval.1158

If those two hypotheses are satisfied, then the conclusion is that f has an absolute max and an absolute min on that closed interval.1163

It might be inside the interval, in other words it might be a local max or local min were achieved its highest.1176

Or it might actually be at the endpoints because the closed interval, the endpoint are part of the domain.1182

If it is continuous, if it is closed interval, then both absolute max and absolute min are achieved.1188

If one or both of the hypotheses are not satisfied, you cannot conclude that it has a max or a min.1195

May or may not, but you cannot conclude it.1201

If a function is not continuous on its domain, if the interval is not closed, all bets are off.1204

Let us go ahead and take a look at a couple of examples of that.1211

I will do this in red.1214

We have got something like that.1216

Let us go something like that.1221

Here is a and here is d, this is a closed interval.1230

The function is continuous.1234

Therefore, it achieves its absolute max and absolute min somewhere on this interval, based on the extreme value theorem.1236

In this particular case, here is your absolute min,1245

I’m reversing everything today.1249

In this particular case for this function, they also happen to be local max and local min but that does not matter.1259

Continuous function, closed interval, it achieves an absolute max and it achieves an absolute min.1265

Let us look at another graph.1272

This is a, this is b, continuous function, closed interval.1280

We have the absolute min, we have the absolute max, on that closed interval always.1286

Let me draw a little circle, something like that.1311

This is a, this is b, this is a closed interval.1317

However, the function is not continuous.1322

Therefore, it does not achieve both an absolute max and an absolute min.1325

Here I see some absolute max but there is no absolute min.1332

Because in this particular case, this is an open circle.1336

It gets smaller and smaller but we do not know how small it actually gets.1340

If there is no absolute smallest value of y on this.1348

There is not because it is discontinuous there.1353

It does not satisfy the hypotheses, so it does not apply.1356

We will do one more.1366

We will take the function f(x) = 1/x, your standard hyperbola.1370

Here there is no max, there is no absolute max, and there is no absolute min.1379

The reason is it is continuous but there is no closed interval.1385

I have not specified a closed interval that has well defined endpoints.1391

This is going to keep climbing and climbing.1397

This is going to keep dropping and dropping.1399

You might think yourself, wait a minute, in this particular case, cannot I just say that 0 is an absolute min?1403

No, what 0 is a lower bound.1408

In other words, the function will never drop below 0.1414

But I cannot say that there is a smallest number that is still bigger than 0, that this function will hit.1418

It is going to keep getting smaller and smaller, heading towards 0.1425

In this case, like that one, this number is a lower bound on this function.1429

In other words, it will never be lower than that.1434

But that does not mean that that is in absolute minimum because it does not achieve its minimum.1437

Because for every number I find that is small, that is close to this lower bound, like close to 0 over here,1442

I can find another number smaller than that closer to 0.1448

That is the whole idea of this infinite process.1452

There is a very big difference.1455

A lower bound or upper bound is not the same as absolute max and absolute min.1457

Absolute max and absolute min, they have to belong to the domain.1461

The x values at which the absolute max and absolute min are achieved, they have to be part of the domain.1468

Let us move on, one more theorem here.1480

Let us go ahead and leave it in red.1483

If f(x) has a local max or a local min at c in the domain of the function, then, f’ at c is equal to 0.1488

All that means is the following.1522

We already know what local max and local min look like.1525

Local min is a valley, local max is a crest.1527

If I have some function like this, this is a local max and this is a local min.1531

We will call this c1, we will call this c2, whatever the x value happens to be.1539

This says that at the local max and at the local min, the slope is 0.1544

The derivative f’ at c is 0, the derivative is 0, the slope is 0.1551

We can see it geometrically.1557

We are going to have a positive slope, from your perspective, if you are moving from negative to positive.1558

Positive slope, it is going to hit 0 and it is going to go down like this.1563

That tells us that we have a crest, a local max.1567

Then this one, local min.1571

That is it, that is all this theorem says.1573

Let us go ahead and give the definition.1577

Definition, something called a critical number or a critical value.1583

The number c that is in the domain such that, f’ at = 0 or f’ at c does not exist.1605

A critical number or a critical value of the function, it is a number in the domain such that it is a number c in the domain,1631

such that f’ at that number is either equal to 0 or f’ at c does not exist.1636

In this particular case, these values, f’ of these values is definitely 0, it is a horizontal slope.1645

These are critical values.1652

An example of one where it does not exist is the absolute value function.1656

Absolute value function goes that way and it goes that way.1660

It is continuous there at 0 but is not differentiable there.1667

Because it is not differentiable there, that is a critical value of the absolute value function.1675

F’ at c = 0 or f’ at c does not exist.1683

If it is not defined there, that is not considered a critical value.1687

It has to actually be defined there.1690

It has to be in the domain, that is important.1692

If it is not part of the domain, then all bets are off.1697

Let me write this a little bit better.1710

Let us do it in red.1713

The procedure for finding the critical values, very simple.1718

The procedure for finding the critical values of a function f(x).1723

Find the derivative f’(x) into = 0.1735

Set f’(x) equal to 0 and solve for all values of x that satisfy this equation.1751

This equation, the only other thing that you have to watch out for is place on the domain1771

with the function is not differentiable.1778

Other than that, find the derivative, set the derivative equal to 0, and you are done.1782

Let us go ahead and actually do an example of this.1789

I’m going to call this example 1.1792

Example 1, find the critical values of f(x) = x + sin(x).1797

We know that f(x) = x + sin(x).1820

F’(x) is equal to 1 + cos(x).1827

We take the derivative and we set it equal to 0.1833

1 + cos(x) is equal to 0 and we solve.1836

Cos(x) = -1.1841

Therefore, x = π.1846

Let us just stick to a particular domain, let us go from 0 to 2π.1852

We know that it repeats over and over again but that is fine.1857

We will stick to 0 and 2 π.1860

In this particular case, in this domain, the critical values are x = π.1862

That is a place where the derivative is equal to 0.1869

Are there any places where this function is actually not differentiable on this domain?1872

No, the cosign function is discontinuous everywhere and it is differentiable everywhere.1876

I do not have to worry about that other part of the definition of critical value.1883

I just have to worry about taking the derivative and setting it equal to 0, and solving.1887

Let us list the procedure for finding.1896

Do not worry about it, as far as this example is concerned,1902

this particular lesson is just the presentation of the material with a quick example.1905

The following lesson is going to be many examples of what it is that we are doing.1910

There are going to be plenty examples, I promise you.1916

Now we are going to talk about the procedure for finding the absolute max and absolute min.1919

Remember, we have that extreme value theorem.1924

We said that, if a function is continuous on a closed interval, that it achieves its max and min, absolute max and min.1925

How do we find that absolute max and min?1934

Here is how you do it.1935

Procedure for finding the absolute max and absolute min of f(x) on ab, the closed interval.1936

One, find the critical values of f(x), that is what we just did, that procedure.1955

Evaluate the original function, evaluate f(x) at each critical value.1970

Third step, we want to evaluate f(x), the original function, that is going to be the hardest part,1988

especially now that we get into this max and min.1995

We are going to be talking more about derivative, setting them equal to 0, using it to graph.1996

You are going to be dealing with functions, their derivatives, the first derivative and the second derivative.2002

You are going to find certain values and you are going to be plugging them back in.2007

Which do you plug it back in?2010

You do have to be careful.2012

It is an easy procedure but just make sure the values that you find, you are plugging back into the right function.2013

When we find the critical values, we are going to be using f’(x), setting it equal to 0.2020

When we find those values, we are going to be actually to be putting them back into f(x) not f’(x).2025

The third part is evaluate f(x) at each end point.2032

In other words, you want to evaluate f(a) and f(b).2045

Now you have a list of values.2057

You have a list of values f(x) at each critical value.2061

You have f(a) and you have f(b).2064

Of these tabulated values for f(x), the greatest one, the greatest is the absolute max and the smallest is the absolute min.2072

That is it, nice and simple.2099

Let us do an example.2102

Let me skip this graph.2130

Now example 2, what are the absolutely max and absolute min of f(x) is equal to x + 4 sin x on 0 to 2 π.2132

Let us go ahead and take f’.2171

F’(x) is equal to 1 + 4 × cos(x).2173

We set that equal to 0 because our first step is to find the critical values of this function.2180

Critical values, we take the derivative and we set it equal to 0.2184

I have got cos(x) is equal to 1/4 that means that x is equal to the inverse cos of 1/4 or 0.25.2189

On the interval, from 0 to 2 π, I find that x is equal to 1.82 rad or 104.5°, if you prefer degrees.2194

Or we have 4.46 rad, I’m sorry this is going to be -1/4, 255.5°.2218

These are our critical values.2233

We want to evaluate f at those critical values, that is our second step.2237

F(1.82) is equal to 5.70.2242

In other words, I put these back into the original function to evaluate it.2249

F(4.46) is equal to 0.59.2256

I’m going to evaluate at the endpoints, 0 and 2 π.2264

F(0) is equal to 0 and f(2 π) = 2 π which = 6.28.2271

I have 5.7 and 0.59, 0 and 6.28.2283

The absolute max is the biggest number among these.2289

The absolute max is equal to 6.28.2294

The absolute max happens at x = 2 π.2300

The absolute min value, that is going to be the smallest number here is 0, and that happens at x = 0.2309

That is all, find the critical values, evaluate the function of the critical values.2322

Evaluate the function at the endpoints, pick the biggest and pick the smallest.2327

You are done, let us see what this looks like.2330

Here is the function.2336

This right here is your f(x) and I decided to go ahead and draw the derivative on there too.2339

This is f’.2346

This f(x) here, this is the x + 4 sin x.2352

This f’(x), this is equal to 1 + 4 × cos(x).2359

You know it achieves its maximum value at whatever it happened to be which was 2 π, I think, and its minimum at 0.2369

There you go, that is it.2377

Over the domain, you are done.2378

2 π would put you on 6.28, somewhere around here.2383

Sure enough, that is a the highest value because we are looking at that.2394

This is going to be the lowest value, that is all.2403

We have the critical values where the derivative equal 0.2407

Those were here and here.2410

In other words, whatever that was, the 1.82 and I think the 4.46, those are local max and min.2414

Local max and local min but they are not absolute max and absolute min.2420

We have to include the endpoint.2424

In this particular case, it is the end points at which this function achieves its absolute values, its extreme values.2427

Thank you so much for joining us here at www.educator.com.2436

We will see you next time so that we can do some example problems with maximum and minimum values.2438

Take care, bye.2444

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