INSTRUCTORS Raffi Hovasapian John Zhu

Home » Mathematics » AP Calculus AB » Optimization Problems II

Raffi Hovasapian

Raffi Hovasapian

Optimization Problems II

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 (9)

1 answer

Last reply by: Sarmad Khokhar
Sun Apr 30, 2017 7:35 AM

Post by Sarmad Khokhar on April 30, 2017

In Example we could have also taken the second derivative to check our x value.

1 answer

Last reply by: Professor Hovasapian
Wed Jan 18, 2017 7:55 PM

Post by Rohit Kumar on January 8, 2017

What would the domain be for the last example and how would you find it?

1 answer

Last reply by: Professor Hovasapian
Tue Apr 12, 2016 3:33 PM

Post by Acme Wang on April 12, 2016

In example IV, does the price function denote the price per unit?

0 answers

Post by Gautham Padmakumar on December 5, 2015

Also, you made an error in using the quadratic formula. Its supposed to be
x = 8.01 and x = 6.09

1 answer

Last reply by: Professor Hovasapian
Thu Dec 17, 2015 12:26 AM

Post by Gautham Padmakumar on December 5, 2015

Isn't it a problem to work with different units like that in Example 2? We have both 1.5 km and 7 miles?

Thank you

Optimization Problems II

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: Optimization Problem 0:13
  • Example II: Optimization Problem 17:34
  • Example III: Optimization Problem 35:06
  • Example IV: Revenue, Cost, and Profit 43:22

Transcription: Optimization Problems II

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

Today, we are going to continue our discussion of optimization problems, max/min problems, by doing some more of them.0004

Let us jump right on in.0010

Our first problem says, a piece of wire is 15 m long, it is cut into 2 pieces.0015

One piece is bent into a square and the other into a circle.0020

Where should the wire be cut so that the total enclosed area between the square and the circle is maximized?0026

Let us draw this out to see what we are looking at.0034

We have a piece of wire right here.0038

This is pretty much what we are looking for, some distance x.0042

I’m going to call this x and that is automatically going to make this 15 – x, because the total length was 15.0048

We are going to bend it into a square.0056

We are going to bend it into a circle.0058

Let us go ahead and find out what the areas are.0060

For the square, the perimeter of the square is just going to be x all the way around, if we take.0064

We are going to choose one for the square.0075

It does not really matter which one, I’m just going to go ahead and choose this one, this one piece for the square.0076

If that is the case, if the square, if the total perimeter is x that means that one side is going to be x/4.0082

Therefore, the area of the square is actually going to be x/ 4².0091

We get x²/ 16, that takes care of the area of the square, as for as a formula is concerned.0097

Let us go ahead and do the circle.0105

Our circle, the perimeter of our circle is going to be the rest of this, the 15 – x.0108

The perimeter = 15 - x which I also know is equal to 2 π r.0115

The area of the circle is equal to π r².0123

I take this one over here, if I have 15 - x is equal to 2 π r, I'm trying to find r in terms of x.0130

I get r is equal to 15 - x/ 2 π.0145

Take this r, the 15 - x/ 2 π and I put in there.0153

What I get is area = π × 15 – x/ 2 π².0159

I will go ahead and simplify that.0171

We have got area is equal to π/ 4 π² × 225 - 30x + x².0174

I hope you are checking my arithmetic.0191

Again, I’m notorious for arithmetic mistakes.0193

We get finally area = 1/ 4 π ×, I’m going to go ahead and just switch this around.0197

X² - 30x + 225, that takes care of the area of the circle.0205

Here we have the area of the square.0213

Total area, I just add them up.0216

Again, I did this because I want some function of one single variable x,0224

which is why I used this relation to turn the area into a function of x.0229

I have got x²/ 16 + x²/ 4 π - 30x/ 4 π + 225/ 4 π.0235

This is this + this, this gives me our total area as a function of x.0253

Let me put a couple of things together here.0261

I’m going to go ahead and call the area, area sub T.0267

I’m going to go ahead and just put some numbers together.0272

We have 1/16 + 1/ 4 π x² - 30/ 4 π x + 225/ 4 π.0274

I just simplified the equation that I just wrote.0293

This is our equation, now we want to take the derivative, set it equal to 0 to find out where the extreme points are.0295

at’, when we take the derivative of that, we are going to end up with,0307

I’m going to get 1/8 + 1/ 2 π × x - 30/ 4 π, that I set equal to 0.0319

Let us just go ahead and solve this.0336

That is fine, this simplifies this.0350

I get π + 4/ 8 π × x = 30/ 4 π.0352

I get 4 π + 16 x is equal to 240.0363

That gives me x is equal to 240 divided by 4 π + 16, which comes out to around 8.40.0375

When x = 8.40 that gives me an extreme point, a maximum or a minimum.0393

We have a little bit of problem.0401

This is telling me that my wire which is 15 units long, if I cut it at x = 8.40, I’m going to maximize the area.0404

We have a bit of a problem.0423

We have a problem, the x = 8.40, it does not maximize the area.0427

It actually minimizes the area between the circle of the square.0439

It does not maximize the area, it minimizes it.0443

Let us see what is going on here.0458

It minimizes it.0461

When we take the derivative and set it equal to 0, we might have a max, we might have a min.0466

We do not know which one, we have to check and see which one by putting all these values into the original function.0470

All of the x values that work which are all of the critical points and any endpoints that you have,0477

they have to go into the original function to see which one gives you the highest value.0483

That is what we have to check.0488

In this particular case, let us see what is going on here.0490

Let us look at our function.0495

Our function is, our total area, we said was equal to 1/16 + 1/ 4 π × x² - 30/ 4 π x + 225/ 4 π.0506

Our function is quadratic, but more than that, its leading coefficient is positive.0528

What that means, the graph looks like this, not like this.0544

Our x = 8.40 actually minimizes the function.0550

This total area function is quadratic, it hits a minimum at a certain point between 0 and 15.0555

This 8.40 is actually a minimum, it minimizes it.0562

But we want to maximized the area.0567

The question is how then do we find the max?0569

How then do we find the maximum?0573

The answer is check the endpoints.0584

In all of these max/min problem, let me write this down first because we have a hard time doing two things simultaneously.0589

In all of these max/min problems, anytime your domain actually is closed.0596

in other words, anytime you have endpoints that are included in the domain,0601

you not only have checked the internal points, the critical points.0605

When you set the derivative equal to 0 which could be maxes or mins,0610

those x values, you also have to check the left endpoint and the right endpoint.0612

Putting all of these x values into the original function to see which one gives you the highest value or the lowest value.0618

Depending on what you are trying to maximize or minimize, respectively.0624

In this particular case, quadratic function, leading coefficient positive, our 8.4 does not maximize it, it minimizes it.0628

That means that the maximum, the absolute maximum has to happen at one of the endpoints.0636

Let us go ahead and check those now.0640

Our domain is 0 to 15, in other words, we can use all of it for the circle, all of the wire, or all of the wire for the square.0645

That is all that means.0659

If x = 0, then the area of the square is equal to 0/ 4² that = 0.0661

The area of the circle = π × 15 – 0.0679

Go back to the original equation, / 2 π² is equal to 225 π/ 4 π².0689

π cancels, you are left with 225/ 4 π which = 17.90.0703

This is one possibility, an area of 17.90.0710

Now all of the area belongs to the circle but there is no specification that says it has to be split between the square and the circle.0715

It just throws out the problem.0722

In this case, if all the wire is used for the circle, the total area is going to be 17.90.0724

If x is 15, in other words, if we use all of the wire for the square then the area of the square is going to equal 15/ 4².0732

If I done my arithmetic right, this is going to be 14.06.0755

In this case, 17.90 is greater than 14.06.0762

In order to maximize the total area within the constraints of 0 to 15, the total area, use the entire wire for the circle.0773

That is it, that is all that is happening here.0799

Basically, what we have done is we found this x = 8.40.0803

But given the fact that our equation was actually a parabola that opens upward, at 8.40 minimizes it.0811

If I take any x value between 0 and 15, my total area is actually going to end up being minimized not maximized.0819

I have to check the endpoints.0828

The endpoints tells me that, if I choose the circle to use the entire wire,0830

that is going to give me the total maximum area, 17.90/ 14.06.0837

If you were to put in the values, 8.40, and calculate the area each,0843

which I probably should have done but you can do it yourself, use x = 8.40.0850

Calculate the area of the square, the area of the circle, and add them up.0854

You will find that it is actually minimized.0858

That is the 17.90, when x = 0, that is the real answer.0860

Let us go ahead and show you what this looks like here, pictorially.0865

We said that our area total, our equation area total was 1/16 + 1/ 4 π x² - 30/ 4 π x + 225/ 4 π.0870

This was our equation for total area, this is the graph for that.0896

It is minimized at this 8.4, this was our 8.40.0901

I actually could did do it here.0911

The area here, the total area is going to be 7.88.0914

At 15, at this point right here, when x = 15, the area is going to end up being 14.06.0920

That is what the y value is, the y value is the total area because this is our area function, the total area.0929

Here, 0, 17.90, this is where the absolute maximum occurs on this domain, from 0 to 15.0935

Total area is 17.9 at 0, total area is 7.88 at 8.40, and total area is 14.06, when x = 15.0949

If you did not catch the quadratic nature of the function and the positive leading coefficient, coefficient is actually not a problem.0960

Just check the endpoints and the critical point x = 8.40.1004

You put all of those points into the original function.1024

The one that gives you the lowest number, in this case because we are trying to minimize, that is the one you pick.1027

If we were trying to maximize it, we are trying to maximize the area,1031

that is what you pick, depending on what the problem is asking for.1037

The lesson actually of this is closed intervals, you always have to check the endpoints.1042

That is all that is going on.1046

Let us go onto the next problem here and see what that is.1051

It seems a little long, do not worry about it, it is actually pretty straight forward.1055

We will draw a nice picture and see what is going on.1059

A gas company is on the north shore of a river that is a 1.5 km wide.1062

It has storage tanks on the south bank of the river, 7 miles east of a point directly across the river from the company.1067

They want to run a pipeline from the company to the storage tanks by first heading east from the company over land,1076

to a point p on the north shore, then, going under the water to the storage tanks on the other bank.1081

It costs $350,000/km, to run a pipe over land.1088

$600,000/km, to do so under water.1092

Where should the point p be, in order to minimize the cost of the pipe line.1096

Let us draw what is going on here.1100

I have a north shore of the riverbank, I have a south shore of the riverbank.1103

They tell me that the company, the north shore of the riverbank,1114

here is my company, at c.1118

River that is 1.5 km wide, this is 1.5 km wide.1122

Storage tanks on the south bank of the river 7 miles east of the point directly across the river.1129

Across the river and 7 miles east.1135

The storage tanks are right here, we will call this s or t, whichever you want.1139

They tell me that this distance right here is 7.1146

They want to run a pipe line from the company to the storage tanks,1152

by first heading east from the company over land to a point p.1155

They want to go this way.1159

They want to head out this way.1163

This is our point p, to the point p in the north shore, then going underwater to the storage tanks to the other bank.1172

Then, they want to go under the water there.1179

This is the cost, minimize that cost.1185

The cost function is as follows.1190

I’m going to call this distance x and I’m going to call this distance y.1198

$350,000/km, the cost is going to be 350,000 x for x km + 600,000 y under the water.1203

That is it, we want to maximize this.1219

Notice that it is a function of two variables.1228

More than likely, we were going to try to find some relation between the two variables, substitute into either one of them.1229

Find an equation in one variable for the cost, take the derivative, so on, and so forth.1236

Let us go ahead and mark our domain to double check, to see whether we are dealing with open or closed endpoints.1243

X can be 0, in other words, I can run it just straight from c all the way under the water, to the storage tanks.1254

It can certainly be 0, or I can go all the way to 7 over land, and then, cut straight across under the river.1262

Our domain is 0 and 7, closed.1270

We have to check those endpoints.1273

We will also check the endpoints.1278

Let us talk about what it is that we are going to do with this figure and how we are going to make sense of this.1289

I’m going to go ahead and draw a little dotted line straight across.1294

I’m going to call that c.1301

This angle here, I’m going to call this angle θ.1305

This is my p, this is my x, this is my y.1311

Just in case, I’m going to go ahead and I got a right triangle right there.1316

Perfect, now I’m going to go to the next page and redraw this, just this figure, without anything else.1323

Let me go ahead and draw the figure over here.1334

I have got my company, I got my point p.1338

I got the storage tanks over here.1348

I think I will call this c, I think I will call this s, does not really matter.1350

This was c, this was x, this was y, this was our angle θ.1358

I’m going to go ahead draw this regular triangle here, something like that.1364

This was 1.5, that was the width of the river, and this was 7.1368

Let me see if I can come up with some relationship between x and y.1374

I’m going to go ahead and take c.1379

C is easy to find, that is just Pythagorean theorem, this is a right triangle.1382

C is nothing more than 1.5² + 7², under the radical.1386

It is going to be 51.25 which is going to be 7.16 km, that is c.1395

Now θ, this θ over here, that is actually going to be fixed.1406

X is going to change, that is what I’m trying to find.1413

How far do I have to go this way?1417

But from here to here is a fixed line.1421

If I move along this, it is a fixed line, θ stays fixed.1424

This θ is the same as that, that angle is the same.1428

If I want to find θ, θ is just the inv tan(1.5) divided by 7, which ends up being 12.1°.1433

We can go ahead and use the law of cosines.1447

Using the law of cosines, we get the following.1459

We get y² is equal to x² + 7.16² - 2 × 7.16 × x × cos(12.1).1466

That is the law of cosines, I have established a relationship between y, x, c, and θ.1489

It is right there.1496

Ultimately, what I have got is a relationship between y and x,1499

which is what I wanted so that I can put it back into my cost equation.1503

I end up with y² = x² + 51.25 - 14x.1509

Therefore, y is equal to this x² + 51.25 - 14x, all under the radical.1520

We now have a relation between x and y, cost function.1539

Therefore, the cost function which is now going to be a function of x,1553

is going to equal 350,000x + we said 600,000 × y.1558

Y is this, x² +, let me go ahead and write it, -14x + 51.25.1567

That is it, that is my cost function.1582

Let me go ahead, let us see, should I do it here, should I do it there?1585

Let us go ahead and stick with what I have got.1594

Now we take c’(x), let me go to the next page.1596

I have got, let me rewrite c(x).1606

C(x) = 350,000x + 600,000 × x² - 14x + 51.25.1611

Our c’(x) is going to equal 350,000 + 600,000 × ½ x² - 14x + 51.25 ^- ½1634

× the derivative of what is inside, which is going to be 2x – 14.1655

I get c’(x) is equal to 350,000 + 600,000 divided by 2, I will just leave that alone,1664

put this on top, bring this down to the bottom.1674

I have got 300,000 × 2x - 14/ x² - 14x + 51.25.1677

Of course, the rest of this is just algebra, it is not a problem.1697

I will go ahead and go through it all.1699

Common denominator, let me get, 350,000 × x² - 14x + 51.25 + 600,000x.1702

I have distributed this, -4,200,000/ I knew this is c’(x) = this/ x² - 14x + 51.25,1721

all of this is going to equal 0, which means the numerator = 0.1744

Therefore, we are going to have 350,000 × √x² - 14x + 51.25 is equal to 4,200,000 - 600,000x.1749

Go ahead and divide, I end up with x² - 14x + 51.25, all under the radical.1773

It is equal to 12 - 1.7x, square both sides.1783

X² - 14x + 51.25 = 144 - 40.8x + 2.89x².1789

Rearrange, I end up with 1.89 x² - 26.8x + 92.75.1806

This is my c’(x), then, when I solve this which is going to be 0.1823

When we solve this quadratic, we get x = 5.929 km and x = 8.068.1840

The x = 5.929, that is actually the answer.1870

Notice the 8.068 is outside of the domain.1875

We will talk a little bit more about that in just a minute.1879

We are going to use the 5.929.1882

We now put x = 0, x = 5.929, and x = 7.1887

0 and 7 are the endpoints, this was the critical point.1902

You put these into the cost function.1906

We said that the cost function, let me rewrite it, in case we need it.1917

It was 350,000x + 600,000 × √x² - 14x + 51.25.1920

When we take c(0), we end up with 4.3 × 10 ^$6, $4.3 million.1943

When we take c(5.929), we end up with 3.1 × 10 ^$6.1955

When we take c(7), we end up with 3.4 × 10 ^$6.1969

Clearly, this one minimizes the cost, minimum cost.1978

Therefore, we want x to be 5.929 km, that is what is going on.1987

Let us go ahead and take a look at what this looks like.1998

Our red, this was our original function, this was our cost function.2005

It is going to end up minimizing someplace.2009

But clearly, our domain is 0 to 7.2013

We are only concerned with from here to about here.2015

But this is the overall function, if we need it.2017

The blue, this was the c’, just to show you where it is.2020

That is it, the 5.929, that is this number right here.2028

As you can see, that is where it actually minimizes that.2033

Also, you should notice that at least here, it only passes through 0 once.2038

This c’ that we got, we ended up with a quadratic.2047

Whatever it was that we got, it only passes through 0 once.2050

This other root, this 8.068 that we got, that was actually a false root that showed up because we ended up squaring a radical.2053

When you do that, you tend to sometimes introduce roots that do not exist.2063

You can think about it that way.2067

You can think of it as, it is outside the domain so we can ignore that.2069

You can think of it as a false root.2074

If you go ahead and graph it, you can see that it does not touch 0 again at 8.068.2076

Clearly, only the 5.929 is the answer, whatever you need.2081

You just have to be vigilant.2086

It is not just about putting the numbers in, take the derivative, whatever they are, check the answer.2089

You want to still stand back and make sure things make some sort of physical sense.2093

You want to take a look at the function, think about the function, use every resource at your disposal.2098

Let us go ahead and go to the next problem here.2106

A long pipe is being carried down the hallway that is 10 ft wide.2108

At the end of the hallway, a right angle turned to the right must be made into the hallway that is 7 ft wide.2112

What is the longest pipe that can make that turn.2117

Let us draw this out, see what we are looking at here.2121

We have got a hallway, it is going to be some hallway this way and it is going to be this way.2124

The initial hallway that we are going down is 10 ft wide, that is this one.2136

And then, we are going to make a right turn that must near the hallway that is 7 ft wide.2143

This is 7, we are carrying this long pipe.2150

The only way this pipe is going to make it is like that.2153

When we turn it, it has to just barely touch the wall, in order to actually make this turn.2163

That is our pipe.2173

What is the longest pipe that can make that turn?2177

Let us see what we have got.2182

I think we are going to break this up.2184

You stare at this a little bit, we try different things.2187

We try to see what is going to happen.2191

Based on this, I decided to call this l1 length 1 and I will call this length 2.2194

I ended up calling this θ, this is also θ.2203

The total length is equal to l1 + l2.2216

Let me do it over here.2230

The cos(θ) is equal to 10/l1.2235

Therefore, l1 is equal to 10/ cos θ.2244

Over here, sin(θ) is equal to 7/l2.2252

Therefore, l2 is equal to 7/ sin(θ).2261

Therefore, I put these into here.2269

Therefore, l is equal to 10/ cos(θ) + 7/ sin(θ).2272

Now at least I have a function l, a function of only one variable, θ.2287

Our domain in this case, θ is going to be 0 - 90°.2294

We do have 0 and we have 90°.2299

I’m going to say 0 to π/2.2306

Let us stick with radian measure.2310

That is that, great.2313

L is just function of θ.2315

Now I’m going to take dl dθ or l’.2317

What I get is the following.2322

Dl dθ is equal to 10 sin θ/ cos² θ.2326

This is quotient rule, or if you want to bring this and write this is 10 cos⁻¹, however you want to do it.2338

This × the derivative of that - that × the derivative of this/ this².2347

That is 10 sin θ cos² θ.2354

And then this one, this × the derivative of that 0 - that × the derivative of this/ this².2356

You are going to get -7 cos θ/ sin² θ.2365

We are going to set this derivative equal to 0.2373

Now we have this, now we just need to solve this equation.2379

I have got a common denominator there.2383

Actually, let me go ahead and rewrite it, it is not a problem.2389

I have got dl dθ is equal to 10 sin θ/ cos² θ - 7 cos θ/ sin² θ, we want that equal to 0.2395

How do we solve this?2414

Let us do a little common denominator here.2416

I will do a common denominator, I'm going to get to 10 sin³ θ - 7 cos³ θ/ cos² θ sin² θ that = 0.2420

It is the numerator that equal 0, I have got 10 sin³ θ – 7 cos³ θ, that is going to be equal to 0.2446

I have got 10 sin³ θ = 7 cos³ θ, switch things around.2464

Sin³ θ/ cos³ θ = 7/10.2475

This is 10³ θ = 7/10.2483

When I take this, I get 10 θ = 0.8879.2490

When I take the inverse, I get θ is equal to 41. 6°, that is one of the critical points.2502

Now I have got l1 which is equal to 10/ cos θ which is equal to 10/ cos(41.6) is equal to 13.37 ft.2513

That gives me the length of l1.2535

L2 was equal to 7/ sin θ which is 7/ sin(41.6), which ends up being 15.06 ft.2540

Our total l is going to be 28.13 ft.2558

That is it, it is that simple.2567

When you put in the 0 and the 90, I should have done it, I apologize.2569

You are going to get numbers that are not going to give you an ultimate length.2582

The longest that it can be is going to be the 28.13 ft.2587

For this particular one, I will let you take care of the 0 and the π/2 in for θ.2593

Let us go to our final example here, revenue, cost, and profit.2601

Show that the marginal revenue = marginal cost, when profit is maximized.2606

B, if the cost function is this function and the price function is this function, what level of production will maximize profit?2611

Level of production means how many units sold, how many units should I make, when all of them are sold, in other words, x?2622

Let us talk about this a little bit.2631

Let us talk about some variables here.2633

We will talk generally about what these things are.2640

X is going to be the number of units sold, or the number of units produced, the number of units.2642

Our price function, the price function, we will generally use a p(x), that is this one.2656

Our revenue function, revenue function is just your revenue, how much money you are actually bringing in?2671

The amount of money that you are bringing in is going to be the price of one unit × the number of units that you sell.2680

It is going to be x × p(x), the number of units × the price, that is the revenue function.2686

Cost function, your cost function is how much it costs you to make each unit?2696

In other words, if I'm making toasters and it cost me $10.00 to make one toaster, let us say I end up selling it for $15.00.2704

That is that is difference, there is a price and there is a cost that you actually incur for making this thing.2712

The cost function, we will usually just call it c(x), whatever that happens to be.2718

In this case, our cost function is this.2723

If I make 500 units, I put 500 in for x, that gives me the cost that I pay,2725

that I have to incur as the manufacturer, in order to make 500 units of this thing.2734

Hopefully, I make that cost back + any more, that is going to be my profit.2739

There you go, your profit function P(x), it is equal to your revenue function,2744

how much money you bring in total - your cost function, how much you actually spend.2754

I hope that make sense.2762

You bring in $1,000,000 but if you end up spending $300,000 to make those things that you just sold, you lose that money.2763

Your profit ends up being $700,000, it is that simple.2773

Revenue – cost, profit = revenue – cost.2776

When we speak of marginal in the world of money and finance, as far as mathematics is concerned,2782

marginal just means take the derivative.2791

Marginal cost c’, marginal profit p’, what is p’?2802

It is marginal, it is r’ – c’.2808

It is marginal revenue - marginal cost.2811

Marginal just means it is a derivative, it is a rate of change.2815

Let us do part a, very simple actually.2821

Part A says, show that the marginal revenue = marginal cost, when profit is maximized.2825

The profit function, we already know what that is.2831

The profit function is equal to the revenue function, the money that I bring in - the cost function, the money that I spend.2833

Profit is maximized when p’ = 0.2842

Profit is maximized when p’(x) = 0.2855

P’ is nothing more than r’ – c’.2867

P’ is just r’ – c’, the derivative is linear.2873

R’ – c’ is equal to 0, I just solved this equation.2877

R’ = c’, marginal revenue = marginal cost.2884

That takes care of part A, very simple.2891

Let us go ahead and deal with part B.2896

A derivative is a rate of change, r’(x) marginal revenue is the rate at which the revenue changes per unit sold.2903

How fast is my revenue increasing or decreasing, if I sell one more unit?2946

C’(x) marginal cost is the rate at which cost changes per unit produced.2954

In other words, how fast is the cost that I'm incurring changing if I make one more unit?2982

I asked my people in my company, if I make one more unit, how much more is it going to cost me?2992

That is c’, it is the derivative of the cost function.2999

Let us do B, our profit is equal to our revenue - our cost.3004

Revenue is how much you bring in.3016

It is how many you sell × the price that you are selling it at.3018

That = x × the price function pp, P profit, p price, -c(x) = x × 1500.3022

1500 - 5x, I think was the price function, - the cost function, - the whole cost function.3042

You have to put them in brackets, [14,000 + 400x - 1.3 x² + 0.0035 x³].3050

Therefore, our profit function is equal to 1500x - 5 x² - 14,000 - 400x + 1.3 x² - 0.0035 x³.3071

Simplify, we end up with -0.0035 x³ - 3.7 x² + 1100x - 14,000.3101

I take the derivative, try to maximize profit.3121

What value of x, what production level will allow me to maximize my profit?3125

My profit function is equal to my revenue – cost, I have that function.3129

Simplify that function, now I take the derivative of that function, -0.0105 x² - 7.4 x + 1100.3133

I set the derivative equal to 0.3154

When I solve this quadratic equation, I end up with x = 126.1, x = -830.9.3173

Clearly, I cannot make a negative amount of things.3188

Producing 126 units maximizes profit.3197

Let us take a look at what this looks like.3212

This right here, this is our p(x).3218

Remember, our p(x) was a cubic equation.3223

This is not a quadratic equation, it is actually cubic.3227

I have just taken this piece of it so that you see the piece of it matters.3228

Obviously, the minimum I can make is 0 things.3232

I'm not going to make 0 things, I'm going to make more than that.3238

This, as I make more and more, this was my profit function.3242

At some point, I'm going to hit a maximum.3248

This right here, that is p’.3253

It is p’(x), that is the one that we set to 0.3256

Notice it crosses 0 at 126.3259

When I make 126 items, my profit is maximized and the maximum is something like that, whatever that number is.3266

600,000, somewhere near 600,000.3277

That is what is going on here.3279

I have zoomed in on this, this is not a quadratic function.3282

It is the high point of a cubic function.3285

This function actually goes down and comes back up the other end.3288

I’m not concerned about this part, the negative part.3293

I’m concerned because I have to make at least one unit.3295

Clearly, maximizes at x = 126.3301

Thank you so much for joining us here at www.educator.com, we will see you next time, bye.3308

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