INSTRUCTORS Raffi Hovasapian John Zhu

Home » Mathematics » AP Calculus AB » Optimization Problems I

Raffi Hovasapian

Raffi Hovasapian

Optimization Problems I

Slide Duration:

Table of Contents

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
Example VIII: Ladder Problem
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
Example II: Ladder
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
Example I: Shadow
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
Irreducible Quadratic Factors
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
Exam Link
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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Lecture Comments (3)

0 answers

Post by Michael Yang on January 7, 2021

Can I use langrange multipliers to solve the first example too?

1 answer

Last reply by: Professor Hovasapian
Fri Dec 8, 2017 11:38 PM

Post by Maya Balaji on November 11, 2017

Hello Professor. For question 1- I'm not sure why you would check the endpoints of the domain (variable at 0, volume at 0) to see if they are plausible absolute maximums, because technically this domain is not a closed interval. The volume can never be 0, and the length can never be 0- so these would not be included in the domain- so it would not be a closed interval, correct?- and you must only check endpoints if it is a part of a closed interval (please correct me if this isn't true!). Thank you.

Optimization Problems I

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
  • Example I: Find the Dimensions of the Box that Gives the Greatest Volume 1:23
  • Fundamentals of Optimization Problems 18:08
    • Fundamental #1
    • Fundamental #2
    • Fundamental #3
    • Fundamental #4
    • Fundamental #5
    • Fundamental #6
  • 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

Transcription: Optimization Problems I

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

Today, we are going to start talking about optimization and optimization problems,0004

otherwise referred to as maxima and minima with practical application.0010

We have talked about maxima and minima in terms of just functions themselves.0015

Now we are going to apply them to real life situations.0019

There is going to be some quantity that we are going to want to maximize or minimize.0023

In other words, optimize, how to make it the best for our particular situation.0029

The calculus of these problems is actually very simple.0037

Essentially, what you are doing is just taking the first derivative.0040

You are setting it equal to 0 and you are solving.0042

The difficulty with these problems is putting all of this information into an equation.0045

It is the normal problems that people had with word problems, ever since we are introduced to word problems.0054

In any case, let us just jump right on in.0063

What I’m going to do is the first problem, I’m just going to launch right into it so that you get a sense of what it is.0066

I’m going to quickly discuss what is necessary for these problems, and then we are just going to do more.0071

The only way to make sense of them is to do as many problems as possible.0076

This is going to be the first of those lessons.0080

This problem says, if 1400 m² is available to make a box with a square base and no top,0086

find the dimensions of the box that gives the greatest volume.0094

I think I’m going to do this in blue again.0102

Probably, the most important thing to do with all of these optimization problems is draw a picture.0106

Always draw a picture.0112

99% of the time you really need to just draw a picture.0117

Let us see what this is asking.0120

I have got myself a box, let me go ahead and draw a little box here.0122

It is telling me that this box has no top.0133

I want to find the dimensions of the box that gives the greatest volume and also tells me that it has a square base.0137

Therefore, I’m going to call this x, I’m going to call this x.0143

It says nothing about the height, I’m just going to call this h.0146

Find the dimensions of the box that gives the greatest volume.0153

The thing that we are trying to maximize is the volume.0156

All of these problems will always be the same.0159

They are going to ask for some quantity that is maximized or minimized.0161

They are going to give you other information that relates to the problem.0166

The first thing you want to do is find just the general equation for what is being maximized.0170

In this case, it is the volume, greatest volume.0175

What we want to do is, volume, I know here is going to be x² × h.0180

We want to maximize that.0185

When you maximize or minimize something, you are finding the places where the derivative is equal to 0.0189

Once you have an equation, you are going to take the derivative of that equation, set it equal to 0, and solve for x.0195

The problem arises, notice that this is a function of two variables.0200

We cannot do that, this is a single variable calculus.0204

We need to find a way to convert this equation into an equation and just one variable, either h or x.0208

That is going to be our task.0215

This is where the problems tend to get more complicated.0217

Let us see what we can, let us write all of this out.0221

We want to maximize v but it is a function of two variables, mainly x and h.0224

Now we use the other information in the problem to establish a relation between these two variables,0254

so that I can solve for one of those variables.0260

Plug into this one and turn it into a function of one variable, that is essentially all of these problems are like that.0263

There is other information in the problem0272

that allows us to establish a relation between x and h, and that is this.0286

They are telling me that I have a total of 1400 m² total, that means the base, the area of the base, and the 4 sides.0311

The base, this side, this side, that side, and that side, let us write that out0323

The area of the base is going to be x².0329

The area of one of these side panels is going to be xh.0334

There are 4 of them, + 4 xh.0337

The sum of those has to be 1400.0342

That is it, we have a second equation, it relates x and h.0346

Let us solve for either x or h and put it in, not a problem at all.0349

What I'm going to do is I'm going to go ahead and solve for h.0355

4 xh = 1400 - x².0358

Therefore, h = 1400 - x²/ 4x.0364

We put this, we put this h into there, and we turn it into a function of one variable x which we can solve.0381

We put this h into v = x² h to get an equation in one variable.0399

In this case, I chose x.0418

Let us go ahead and do that.0425

We have v is equal to x² × h which is 1400 - x²/ 4x.0428

Cancel that, cancel that, multiply through.0444

We end up with 1400 x – x³/ 4.0448

If you want, you can rewrite it as –x³/ 4 + 350x.0458

It is totally up to you how you want to do it.0467

But now I have my equation, I have my v.0469

Now the volume of this box is expressed as a function of a single variable.0479

We know that a function achieves its absolute max or min, in this case, we are talking about a max,0489

I’m just going to leave it as maximum.0515

It achieves its absolute max either at an endpoint of the domain or somewhere in between where the derivative is 0.0518

In other words, a local max/local min.0527

We know that a function achieves its absolute max either at the endpoints of its domain or where f’ is equal to 0.0529

We differentiate this function now.0556

This is the equation of volume, we want to maximize this equation.0562

In order to maximize it, we are going to take the derivative of it, set it equal to 0,0566

and find the places where it either hits a maximum or a minimum.0570

Vx is a function of x is equal to 350x – x³/ 4.0577

V’(x) = 350 - ¾ x².0591

I’m going to set that equal to 0.0598

I have got ¾ x² is equal to 350.0601

When I solve this, I get x² = 1400/3 which gives me x is equal to + or -21.6.0608

We are talking about a distance.0627

Clearly, the negative is not going to be one of the solutions.0628

It is the +21.6 that is going to be the solutions.0632

Let us go over to the next page.0642

First of all, x is a physical length.0644

The -21.6 is not an option.0657

Second, if you rather not think about it physically and have to decide which value that you are going to take, there is another way of doing it.0668

If you prefer a more systematic or analytical approach0680

to excluding a given root or a given possibility, you can do it this way.0702

We said that v(x) is equal to -3/ 4 x³ + 350x.0716

That was the function that we want.0732

That was our original function, -x³.0740

Let me write this again.0747

We said that we had –x³/ 4 + 350x.0752

I will write it this way.0763

I know that when I graph this, I'm looking at this, and this is a cubic function.0765

This is a cubic function and the coefficient of -1/4, the leading coefficient is negative.0772

A normal cubic function begins up here, has two turns and ends down here.0779

This is negative, negative begins up here and ends down here.0790

I already took the derivative and I found that -21.6 and +21.6 are places0804

where it hits a local max or local min because I set the derivative equal to 0.0808

Therefore, I know that -21.6, there is all local min.0814

+21.6, there is a local max.0820

In this particular case, I also know that when x is equal to 0, the function is equal to 0.0823

I know it crosses here.0829

Therefore, I know for a fact that the thing goes like this.0830

Therefore, the maximum is achieved at +21.6.0836

The minimum of the function is achieved at -21.6.0841

We can also use our physical intuition to say that you cannot have, like we did for the first part,0845

like we did for our first consideration, right here.0850

It is a physical length.0853

This is the part of the graph that I'm concerned with.0856

As x gets bigger, there is a certain value of x which happens to be 21.6 where the function –x³/ 4 + 350x is maximized.0859

They gave us the greatest volume.0871

You want to use all the resources at your disposal, if you are dealing with a function.0873

You know what a cubic function looks like, where the negative over the leading coefficient is negative, it looks like this.0877

This tells you systematically, analytically, that -21.6 is not your solution.0885

Not to mention the fact that it physically makes no sense.0891

There are many things that you want to consider.0894

You do not just want to do the calculus.0896

Whatever you get, you want to stop and think about if the calculus makes sense.0899

Does your -21.6, does your +21.6 actually makes sense?0904

It does, based on other things that you need to consider.0909

Let us see, where are we, we are not done yet.0916

Let us go ahead.0922

We know that x = 21.6, that is the dimension of our base.0928

For h, h is equal to 1400 - x²/ 4x which is equal to 1400 - 21.6²/ 4 × 21.6.0934

When we do the calculation, we get xh = 10.8.0957

There you go, our box is 21.6 by 21.6 by 10.8.0963

Our unit happens to be in centimeters.0975

There you go, that is it, nice and simple.0978

Let us go ahead and actually show you the particular graph.0983

This is the graph of the function, volume function.0987

This is volume = 1400x – x³/ 4.0992

21.6 is right about there, that is our maximum point.1006

This was the function that we wanted to maximize.1011

In this particular case , we have a certain restriction on the domain.1014

This right here, that is the particular domain of this function.1019

The smallest that x can be is 0, no length.1026

The biggest that x can be is whatever that happens to be, when you set this equal to 0.1030

It turns out that x is equal to about 37.4, that is the other root of this equation.1037

That is the other 0 of that equation so that give us a natural domain.1044

In other words, if x = 0, there is no box.1048

If x = 37.4, there is no box.1052

Between 0 and 37.4, for a value of x, which is the base of the box, x by x, the volume goes up and comes down.1055

There is some x value that maximizes the volume.1067

That x is the 21.6 that we found, local maximum of this function.1070

Again, you can use the graph to help you out to find your domain, to restrict your domain, whatever it is that you need.1077

Let us talk about this a little bit.1087

All optimization problems are fundamentally the same.1089

There is a quantity that is asked to be maximized or minimized.1116

It might be an area, might be a volume.1143

It might be a distance, it might be an angle, whatever it is.1145

There are some quantity that is maximized or minimized.1150

Two, your task is to find a general equation for that quantity, for this quantity.1153

Number 3, if the equation that you get in part 2, if the equation is a function of more than one variable,1172

you use other information in the problem + any other mathematical manipulation you need1197

to find a relation between or among the variables.1236

I say among because you might end up with a general equation that has 3 or 4 variables.1250

And you have to find the relationship among all 3 or 4, not just between the two.1255

Part 4, you use the relations above among the variables1263

to express the desired quantity as a function of one variable, if possible.1285

Again, there might be situations where, we will do when we come up with them, not a problem.1308

I know the thing that you might want to do, this is a little looser but it is always a good idea to do this, if you need to.1317

A lot of this will come up with more experiences in solving these kind of problems.1323

You want to find the domain of the equation.1328

The reason you want to find the domain is,1335

Remember, what we are find here is absolute maximum of a function.1339

The absolute maximum of a function can happen within the domain, at places where it is a local max or min.1344

That is where you set the function, the derivative of a function equal to 0.1349

But you also have to consider the endpoints.1352

If you know the domain, if a domain is a closed interval, like it was in the first problem, 0 and 37.4,1355

you are still going to check those points to see if the value of the function that you get is going to be greater.1363

Because we want to find the absolute maximum.1370

Let us say there were two points in an interval, in the domain.1374

Let us go back to the first problem.1380

You had, 0 you have a 21.6, and you have a 37.4.1381

The 21.6 is the answer but you still have to technically check the 0 and the 37.4.1386

Put those values of x into the original equation.1392

You are going to get 0 for the value of the function.1395

When you put 21.6 in, you are actually going to get a number that is the biggest one among the three.1399

You remember when we were doing absolute maxes and absolute mins,1406

we have to check the values at the endpoints to see if maybe f of those values was actually bigger than what it is at a local max or min.1409

Again, the problems will help make more sense of this.1420

And then, once you have all of this information, you find the absolute max or min.1426

You find the absolute max or min.1434

If your domain is not a closed interval, that does not matter.1439

All you need to do is look for the local maxes and mins.1443

That is where you are going to pick one of those to maximize or minimize, whichever is it that you are trying to do.1446

Again, if you have a closed interval, you have to check the end points of the domain.1452

Most important, draw a picture always.1458

Always draw a picture.1471

Let us do some more examples here.1475

Demonstrate that of all rectangles with a given perimeter, the one with the largest area is a square.1477

Pick a random rectangle.1487

I’m going to call this x, I’m going to call this y.1491

In short, demonstrate that the one with the largest area is a square.1496

In short, we must show that y is equal to x, that it is a square.1502

Of all rectangles with a given perimeter.1520

The perimeter, that equals 2x + 2y, and they say of a given perimeter, some constant 5, 10, 20, 30, 86.6, whatever.1523

I'm just going to say c, c stands for a constant.1536

One of the largest area is a square.1540

The general equation for area is xy.1544

The largest area, that is the one they want us to maximize right here.1548

Largest area means maximize this, maximize this.1554

It is a function of two variables.1570

It is a function of two variables, I need a relationship between those two variables x and y,1573

in order for me to turn this into a function of one variable.1577

I have a relationship, that is my relationship right there.1582

I’m going to solve for y and plug it into this equation right over here.1585

I’m going to write 2y = c - 2x.1589

I have y = c - 2x/ 2.1595

I’m going to put this into here.1603

I get the area = c/2 – x.1608

I get the area = cx/2 – x².1622

This is my function, that is the function.1628

Now it is a function, I’m trying to maximize it.1634

I now have the area expressed as a function of one variable, x and x².1636

It is taken into account the perimeter.1641

The c, that is where that comes in.1643

Now I have to differentiate.1646

A’ is equal to c/2 - 2x, I set that equal to 0.1648

When I solve for this, I get 2x = c/2 which implies that x = c/4.1656

I found what x has to be.1673

Let us find y.1678

We said that y is equal to c/2 – x, that is equal to c/2 – c/4.1686

C/2 – c/4, it is equal to c/4.1697

Y does equal x which equals c/4.1705

I have demonstrated that, in order to maximize an area of a given rectangle.1710

I have maximized it by finding the derivative of the function of the area.1716

Found the value of x, it turns out that it has to be a square.1721

For a fixed perimeter, the sides have to be the perimeter divided by 4.1725

That is it, square.1730

I have demonstrated what is it that I set out to demonstrate.1733

Let us see here.1740

Notice that I did not explicitly specify a domain.1745

Let us tighten this up a little bit and talk about the domain.1773

Let us tight this up and discuss domain.1781

For a rectangle with a given perimeter c, the domain 0 to c/2.1790

The domain is what the x value can be.1824

If x = 0, if I take the endpoint x = 0.1826

Then, 2 × 0 + 2y is equal to c.1834

2y is equal to c, y = c/2.1847

The area is equal to x × y, that is equal to 0 × c/2, the area is 0.1856

That is this endpoint.1872

If x = c/2, then the perimeter 2 × c/2 + 2y which is equal to c, we get c + 2y = c.1879

We get 2y = 0, we get y = 0.1900

The area equals xy which equals c/2 × 0.1904

Again, the area = 0, our domain is this.1909

X cannot go past c/2 because we already set that the perimeter has to be c.1917

If you have a rectangle where this is c/2 and this is c/2, basically what you have is just a line because there is no y.1926

X, our domain, has to be between 0 and c/2.1943

When we check the endpoints, we got a value of 0.1947

C/4 is in the domain and it happens to be the local max.1952

When you put c/4 into this, you are actually going to get an area that is a number.1957

We know that that number is the maximum, precisely because of how we did it.1973

We took the derivative, we set it equal to 0, and that is what happened.1978

Let us go ahead and actually take a look at this.1984

In both cases, let me actually draw it out.1987

In both cases that we just did for the endpoints, the area was equal to 0.1993

Between 0 and c/2, there is a number such that a is maximized.2005

That number was c/4.2024

What did we say our function was, our a’, our a?2029

We said that our area function of x was equal to -x² + cx/2.2034

This is a quadratic function where the leading coefficient is negative.2041

I know that the graph goes like this.2047

I know that there are some point where I’m going to hit a maximum, that is what is going on here.2049

This is my 0, this is my c/2.2055

Let us see what this actually looks like.2058

I have entered the function cx/2 - x².2061

I have taken a particular value of c = 15.2064

I end up with this graph.2068

Notice 0, c/2, 15/2 is 7.5, that 7.5.2069

This right here, this is c/2, this is c/4.2077

That is why I hit my max.2080

This is the function that I’m maximizing.2082

It happens to be the quadratic function where the leading coefficient is negative.2086

Therefore, I know that this is the shape.2090

If I do not know it, let me use a graphical utility to help me out.2092

If I need a graphical utility to help me get the domain, that is fine.2095

I do not necessarily need this, I already know that if my perimeter is c, the most that any one side can be is c/2.2099

Therefore, my domain is 0 to c/2, hope that makes sense.2107

Let us see what we have got here.2115

What is our next one?2119

Find the points on the ellipse 9x² + y² = 9, farthest away from the point 1,0.2120

Let us go ahead and draw this out.2130

I got myself an ellipse.2134

I have got 9x² + y² = 9 x²/ 1² + y²/ 3² is equal to 1.2141

I have got, this is 1, this is 1, this is 1, 2, 3, 1, 2, 3.2158

I have an ellipse that looks like this.2166

Find the points on the ellipse farthest away from the point 1,0.2172

Here is my point 1,0, I need to find the points on the ellipse that are the farthest away from this.2176

Just eyeballing it, I’m guessing it is somewhere around here.2181

We will try to maximize this distance right here.2187

We want to maximize the distance from the point 1,0 to some random point xy on the ellipse, that satisfies this equation.2194

We know we are going to have two answers.2220

We already know that.2222

This is going to be xy1, xy2.2225

Probably you are going to have the same value of x, different values of y.2228

We want to maximize the distance, the distance formula.2233

The distance formula = x2 - x1² + y2 - y1², all under the radical sign.2239

Let me put it in.2252

I have x - 1² - y - 0².2253

This is going to give me x - 1² + y², all under the radical.2270

Let us move on to the next one.2282

I have got d is equal to, I expand the x - 1².2285

I get x² - 2x + 1 + y², all under the radical.2290

I know that 9x² + y² = 9.2300

Therefore, y² = 9 - x².2305

I put that into here, I find my d is equal to x² - 2x + 1 + 9 - 9x².2310

Therefore, I get d = -8x² - 2x + 10.2326

This is my distance function expressed as a single variable x.2337

This is what I want to maximize.2341

Maximize this, we maximize it, we take the derivative and set it equal to 0.2346

D’(x) that is going to equal ½ of -8x² - 2x + 10⁻¹/2 × the derivative of what is inside which is -16x -2.2354

D’(x), when I rearrange this, I get -8x – 1/ √-8x² – 2x + 10.2376

I set that equal to 0.2393

What I get is -8x - 1 = 0.2396

When I solve this, I get 8x = -1, x = -1/8.2400

I have found my x, my x value is -1/8.2410

Now I need to find my y so that I can find what the two points are.2414

I know that the function was 9x² + y² = 9.2424

I’m going to go 9 × -1/8² + y² = 9.2431

I get 9/64 + y² = 9, that gives me y² = 9 - 9/64.2441

I get y² = 567/64.2457

Then, I get y = + or -567/64 that equals + or -2.97.2465

Therefore, I have -1/8 - 2.97, that is one point, I have -1/8 and 0.97.2486

These two points are the points that are on the ellipse, farthest away from the point 1,0.2501

Once again, I have an ellipse, this is 1,0.2510

The points are here and they are here.2516

Those are the points that are farthest away from that.2518

This is the function that we have to maximize.2526

This graph, this is not the ellipse.2528

This is the function we have to maximize.2531

This is the -8x² - 2x + 10, under the radical.2535

This is the function that we maximized.2542

It happens to hit a maximum at -1/8.2545

Be careful, this is not the ellipse, this is the function that you end up deriving, that you needed to maximize.2554

We needed to maximize the distance.2563

It actually gives me the x value.2567

Once I have the x value, I put it back into the original equation for the ellipse to find out where the y values are for the ellipse.2570

There are a lot to keep track of.2579

My best advice with all of math and science is go slowly, that is all.2582

Let us do one last example here.2590

Find the dimensions of the rectangle of the largest area.2592

We are going to be maximizing area, know that already.2595

That can be inscribed in a circle of a given radius r.2598

Let us draw it out.2602

We have a circle and we are going to try to inscribe some random rectangle in it.2604

Probably, not going be the best drawing in the world, sorry about that.2612

It tells me that the radius is r.2614

We are going to maximize area.2620

I’m going to call this x, and I’m going to call this side y, of the rectangle.2622

Area is equal to x × y.2627

We have our general equation, we want to maximize this.2631

I have two variables, I need to find the function of one variable.2639

I have to find the relationship between x and y.2642

I have a relationship between x and y.2645

If I draw this little triangle here, this side is y divided by 2 and this side is x/2.2648

Therefore, I have by the Pythagorean theorem, x/ 2² + y/ 2² = r².2663

I have got x²/ 4 + y²/ 4 = r² which gives me x² + y² = 4r²,2674

which gives me y² = 4r² - x², which gives me a y equal to √4r² - x².2688

This is what I plug into here, to this.2701

Therefore, I get an area which is equal to x × 4r² - x².2706

Now I have a function of one variable.2716

Take the derivative and set it equal to 0.2719

A’(x) is equal to this × the derivative of that, x × ½ of 4r² – x²⁻¹/2 × the derivative of what is inside.2722

4r² is just a constant at 0.2737

It is only -2x + that × the derivative of that.2740

We get just 4r² - x² × 1.2745

I rearranged this to get, 2 and 2 cancel, -x².2752

I get -x²/ 4r² - x², under the radical, +√4r² - x².2762

Then, I find myself a nice common denominator.2777

I end up with a’(x) is equal to -x² + 4r².2779

I hope the algebra is not giving you guys any grief.2788

I just found the common denominator, over √4r² - x².2792

This is the derivative we said is equal to 0.2797

When we set it equal to 0, I have the top -x² - x².2801

I end up with a’(x) =, this is 0 so the denominator goes away.2806

I’m left with -2x² + 4r² = 0.2812

2x² = 4r², x² = 2r².2820

Therefore, x is equal to r√2.2829

Again, I take the positive because I’m talking about a distance here.2836

X = r√2.2842

We know what y is, we said that y is equal to √4r² - x² which is equal to 4r² - r√2²,2844

all under the radical, which is equal to 4r² - 2r², all under the radical.2858

That equals √2r² which is equal to r√2.2869

Y is also equal to r√2.2878

Again, I will just say y is equal to x.2891

In other words, the rectangle of largest area that you can describe in a circle is a square,2899

where the sides of the square are equal to the radius of the circle × √2.2908

That is what we have found.2914

Let us go ahead and show you what it looks like.2919

I have the function, the area function that I try to maximize.2922

√4r² - x².2926

I picked a particular value of r radius of the circle happens to equal 2.2931

This is the function.2935

Again, x is the physical distance, really, our domain is here and here.2938

I set the function equal to 0 to find the end points.2946

When I check the endpoints, when I put the endpoints into the area function, I'm going to get an area of 0.2949

0 does not work, 0 does not work.2955

However, there is a point someplace here.2957

What we found is r√2.2960

When I put r√2 into the function for area, I end up getting the largest area.2964

This graph confirms it.2974

The maximum of this graph, the maximum of the area function happens at √2.2975

It happens where the derivative of this function = 0.2981

I hope that helped.2987

Do not worry about it, in the next lesson we are going to be continuing to do more optimization problems,2988

more complicated optimization problems.2993

Thank you so much for joining us here at www.educator.com.2995

We will see you next time, bye.2998

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