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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Example Problems for the Fundamental Theorem

Slide Duration:

Table of Contents

I. 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
II. 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
III. 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
IV. 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
V. 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
VI. 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
VII. 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
VIII. 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 (2)

1 answer

Last reply by: Professor Hovasapian
Mon Feb 8, 2016 1:56 AM

Post by nathan lau on January 31, 2016

at 13:30, isn't it 2pi/3, not just pi/3?

Example Problems for the Fundamental Theorem

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 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)
    • Where Does g(x) Achieve Its Absolute Max on [0,8]
    • On What Intervals is g(x) Concave Up/Down?
    • Sketch a Graph of g(x)

Transcription: Example Problems for the Fundamental Theorem

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

Today, we are going to be doing some example problems for the fundamental theorem of calculus that we just covered.0004

These are going to be extraordinarily straightforward and simple, almost surprisingly.0010

Our first example is find the derivative of the following function.0017

The function is g(x) and it is defined as the integral from 1 to x, whatever x might be.0021

5/ t³ + 9 dt, nice and simple.0027

Since the upper limit is x, we would literally just put x where we have the t.0032

Our final answer is just 5/ x³ + 9, and we are done.0039

That is the beauty of the fundamental theorem.0046

When a function is defined as an integral or the upper integral is a variable,0048

this particular variable right here xx, we just literally put that in there.0054

Let me correct this, a little bit of a notational issue here.0064

g’ because we are looking for the derivative.0071

g’(x), that is a little more clear, I think.0075

We have 5/ x³ + 9.0079

Literally, when we take the derivative of this, we are just getting rid of this integral sign0084

because differentiation and integration or anti-differentiation are inverse processes.0090

That is the whole idea.0095

That is it, nice, straightforward, and simple.0097

Find the derivative of the following function.0102

g(x), upper variable is x, we do the same thing.0105

Therefore, the derivative of this g’(x), all we have to do is we put an x where we see a t.0110

We get x² sin³ x, that is it, very nice.0117

As long as this upper limit of integration is the variable that is inside the function defining it,0128

we can just put it straight into the function itself.0144

Let us try this one, find the derivative of the following function.0150

g(x) = the integral from 1 to x⁴ sin(t) dt.0155

Now this is x, this time, we have not x but we have a function of x.0161

The only thing we do know is we do the same thing.0168

We are just going to be putting in the x here.0171

g’(x) is equal to sin(x).0173

But because this is not x but it is a function of x, by the chain rule,0177

we are just going to multiply by the derivative of this function.0181

Let us write that out.0186

When the upper limit is not just x but a function of x, some more complicated function of x, we do the following.0188

This is the long process, the easy part is just multiply by the derivative of the upper limit, whatever the function happens to be.0220

Essentially, what you are doing is you are letting x⁴, the upper limit integration equals some letter u.0227

Essentially, you are just doing a substitution.0239

And then, du dx is going to equal 4x³.0240

And then, you rewrite this g(x) = the integral from 1 to u of sin(t) dt.0250

Then, when you do this, when you put this into here, in other words, when you do g’(x) is equal to the sin(u),0263

because u is a function of x, by the chain rule, the derivative is sin(u) du dx.0277

You are multiplying by du dx.0285

What you end up getting is g’(x) is equal to sin(x)⁴ × 4x³.0292

This part is exactly the same, you are just putting the upper limit into this variable to get the sin x⁴.0303

And then, you are just multiplying by the derivative of that because now it is a function of x, it is not just x itself.0311

I hope that makes sense.0321

Essentially, what you are doing is you are doing the same thing as you did before.0322

If this were just an x, you would put it in and it would be sin(x).0325

The derivative of x is just 1, you were technically doing du dx.0331

The derivative of x is just 1, which is why it just ends up staying that way.0336

But if it is a function, just multiply by its derivative.0342

That is your final answer, right there, very straightforward.0345

Find the derivative of the following functions, same thing.0356

Now π/3, the upper limit is cos(x).0359

We are just going to put this into the t's and then multiply everything by the derivative of cos(x).0363

g’(x), we would literally just read it off.0372

It is going to be 3 × cos(x) + 7 × cos² (x) × the derivative of that function cos(x) which is –sin(x).0375

We are done, that is all.0391

Just do what the first fundamental theorem of calculus says which is just put the upper limit into those things.0399

Then, multiply by ddx of the upper limit.0415

That is all, moving along nicely.0431

Evaluate the following integral.0435

We want to evaluate a definite integral.0437

We have a lower limit, an upper limit, they are both numbers.0440

We are going to find the antiderivative, the integral.0443

And then, we are going to put in 5, we are going to put in 1,0446

and we are going to subtract the value of what we get from 1, from the value of the upper limit 5.0450

Let us go ahead and do that.0456

The integral of this, let us go ahead and actually multiply this out first because we have x³ - 4x².0457

This is going to be the integral from 1 to 5 of x⁶ – 4x × that is going to be 4x⁴, then another 4x⁴.0468

It is -8x⁴ + 16x², I just multiply this out, dx.0484

And that is going to equal x⁷/7 - 8x⁵/5 + 16x³/3.0495

I'm going to evaluate this from 1 to 5.0512

In other words, I’m going to put 5 in for all of these and get a number, then I'm going to subtract by 1.0514

Putting them into all the x here.0520

When I put 5 into this, I get 78,000, 5⁷ is 78,125/7.0522

When I put 5 into here, multiply by 8 and divide by 5.0535

I get - 2500/5, I hope you are going to be confirming my arithmetic.0539

I’m notorious for arithmetic mistakes.0544

+ 2000/3 that takes care of the 5.0546

I’m going to subtract by putting 1 in.0553

This is going to be 1/7 - 8/5 + 16/3.0556

Find the antiderivative of the integrand, and then evaluate.0565

Put the upper number into this, you are going to get a value.0570

Put the lower number into the x values, you are going to get this and then you subtract.0573

The rest is just arithmetic.0577

I'm just going to go ahead and leave it like this.0579

You can go ahead and do the arithmetic, if you want, to get a single answer.0581

Evaluate the following integral, once again, we are using the second fundamental theorem of calculus.0588

The integrals of f from 2 to 8 is going to be the f(8) - f(2), F(8) – F(2), that is what the second fundamental theorem says.0594

Remember, it is this one, if from a to b of f(x) dx is just equal to f(b) – f(a), where F is the antiderivative of f.0611

This is going to be the integral from 2 to 8, I’m going to separate these out.0629

I’m going to write this as x²/3, it is going to be x/ x²/3 - 3/ x²/3.0638

x/ x²/3 dx - the integral from 2 to 8 of 3/ x²/3 dx.0646

That is going to equal, the integral from 2 to 8 of x¹/3 dx - the integral from 2 to 8 of 3 × x⁻²/3 dx.0662

I can go ahead and integrate these, not a problem.0678

1/3 + 1 is 4/3, I get x⁴/3/ 4/3 - 3 × x – 2/3 + 1 is 1/3/ 1/3.0681

That is going to equal ¾ x⁴/3 - 9x¹/3.0700

I'm going to evaluate that from 2 to 8.0712

When I put 8 in for here, I'm going to get 12.0716

When I put 8 into here, it is going to be -18, that is the first one.0721

-, when I put 2 into here, I end up with 1.89.0727

And then, - when I put 2 into here, I end up with 11.34.0734

There is your answer.0740

Once you have the antiderivative, you put the upper limit into the x value to evaluate, that gives you the first term.0742

You put the lower limit into the x, that gives you the second term.0753

You subtract the second term from the first term, f(b) - f(a).0758

Nice and straightforward.0765

Another good example, the integral from π/6 to 2π/ 3 of cosθ dθ.0772

This is going to equal where the integral of cosθ dθ is going to be sin(θ).0780

We are going to evaluate sin(θ) from π/6 to π/3.0786

Not a problem, that is going to be the sin(π/3) - sin(π/6).0796

We are just putting in upper and lower limit into the actual variable of integration.0809

The sin(π/3) is √3/2 – sin(π/6) which is ½.0815

The integral from 1 to 3 of 5/ x² + 1.0835

You should recognize that the integral of 1/ x² + 1 is equal to the inv tan.0841

Remember, the integral or the antiderivative, whatever you want to call it,0847

of 1 + x² dx that was equal to the inv tan(x).0852

Therefore, this is, that is fine, I will go ahead and separate it out.0860

This is just a constant, it is 5 integral from 1 to 3, 1/ x² + 1 dx = 5 × inv tan(x), evaluated from 1 to 3.0866

That is it, that is all it is.0885

This is just 5 × inv tan(3) – inv tan(1).0887

That is all, you can go ahead and multiply the 5 through.0901

You can do 5 or you can evaluate this from here to here.0904

This - this and then multiply the constant through.0911

The order actually does not matter.0914

I’m just somebody who tends to like to see the constant on the outside, personal choice.0916

g(x) is equal to the integral from 0 to x of f(t) dt.0928

f being the function whose graph is given below.0934

This graph right here, this is a function of f.0937

Answer the following questions.0943

We take a look at this graph, it looks like some sort of a modified sin or cos graph, definitely periodic.0945

We are taking a look real quickly, we notice that we have about a 3. something, then we have a 6.2 something.0952

It turns out that these are actually just π/2, π.0961

I’m going to go ahead and label them that way.0970

I left the labels just as is, but it is nice to recognize the numbers for what it is that they are.0972

Especially, since you are talking about a periodic function, then more than likely the graphs that you are going to be dealing with0980

are going to pass through the major points, π/2, π, π/6, π/3, π/4, things like that.0985

In this case, let me mark these off.0992

3.14, this is π.1003

This over here looks like it is π/2.1010

Over here, this looks like 2π and this looks like 3π/ 2, 4π/ 2.1017

It looks like we have got 5π/ 2.1028

At least that way we have some numbers that we are accustomed to seeing,1030

instead of just referring them as 3.1, 1.5, things like that.1033

Give the values of x for which g(x) achieves its local maxes and mins.1040

You have to be very careful here, this is a graph of f.1047

g is the integral of f from 0 to x.1052

Therefore, g is actually the area under the graph.1059

The graph actually falls below the x axis, the area is going to become negative.1066

Give the values of x for which g(x) achieves its local maxes and mins.1075

As we start moving from 0 x, moving forward g(x), as x gets bigger and bigger,1080

this value is just the area under the graph itself.1091

It is going to hit a high point, when I hit π/2.1095

And then, at π/2, because the graph itself falls below the axis, as I keep going,1101

as x keeps getting bigger and bigger, the integral itself is going to be negative.1108

The area is going to hit a maximum and then the integral is going to start to go negative.1115

The area is going to diminish.1120

At this point, the area is going to go down.1123

At some point, it is going to hit 0.1127

The area is going to rise to a certain number, that is up to here.1130

We are going to start subtracting from the area, at some point it is going to hit 0.1134

Over here, it is going to hit a minimum because now this area under the x axis is a lot bigger than the area above the x axis.1138

The integral from 0 to π is going to be a negative number.1147

It is actually going to hit a maximum.1154

One of the maximums that it is going to hit is right there.1157

And then, it is going to hit a minimum right there.1161

After that, it is going to climb again, the area under the graph.1164

That is what we are doing, g(x) is the area of f, the area under the graph of f because this is the graph of f.1169

The area is going to get bigger again.1178

At this point, it is going to hit another maximum.1180

And then at this point, the area is going to be negative.1185

We are going to hit another minimum, 2π.1187

We are going to increase the area again.1191

It is going to pass this and it is going to hit another maximum.1194

That is what is going to happen.1198

Give the values of x for which g(x) achieves its local maxes and mins.1199

It is going to achieve its local maxes and mins where the area under the graph hits a maximum and a minimum.1205

It is a maximum at π/2, a minimum at π.1215

Maximum at 3π/ 2, a local minimum at 2π, local max and min at 5π/ 2.1218

We have π/2, π, 3π/ 2, 2π, and 5π/ 2.1225

The reason is because g(x) is the net area under the graph of f.1241

This red is the graph of f.1248

Where does g(x) achieves its absolute max, between 0 and 8, this point right here.1255

Let us go ahead and draw this out.1266

Basically, what we are going to have is this.1267

Let me do this in black.1270

g(x) is going to look like this, it is going to rise.1272

It is going to hit a maximum here and it is going to hit a minimum.1277

It is going to start rising again.1282

It is going to hit a maximum and it is going to hit a minimum, and it is going to hit a maximum.1286

It is going to start coming down to there.1296

The area keeps getting bigger and bigger.1300

The absolute value, the area keeps getting bigger and bigger.1304

Our highest point is going to be this point right here, which is going to be our 5π/ 2.1307

Again, because g(x), that is what we want, where does g(x) achieve its absolute max.1316

g(x) is the net area under the graph.1322

Positive, if it is above the x axis, negative if it is below.1326

This is the graph of g.1330

On what intervals is g(x) concave up or concave down?1335

For concave up or concave down, we want to see the sin(g”).1341

g’ is equal to f.1356

By the fundamental theorem of calculus, if I take the derivative of this, I get rid of this.1360

I just have f(x), this is g’ that mean g“ is equal to f’.1364

f’ is just the derivative of the red graph.1373

The derivative is the slope.1379

It is going to be concave up where the slope is positive.1383

It is going to be concave down where the slope is negative.1387

If the slope is positive, that is f’.1392

f’ happens to be g”.1395

Therefore, we have from here to here, is concave up.1400

From here to here, it is concave down.1415

From here to here, concave up.1424

From here to here, concave down.1429

From here to here, concave up.1433

And from here to 8, we got concave down which is exactly what we see.1437

Concave up graph, this is g that we are looking at.1443

Now it is concave down, it starts to be concave up again up to here.1445

From here to here, it is concave down, it matches.1450

I hope that makes sense.1457

You are given a graph f, if g is defined as the integral from 0 to x of f.1460

It is the net area under that particular graph.1467

I hope that made sense.1472

Let us take a look at what it actually looks like.1475

The original function f was this one right here.1478

This one, this is the actual g(x).1484

This was our f(x).1488

It is a maximum here, it is a minimum here.1494

Local, hits a local max here, it is right there.1502

It is a local min here, it is right there.1507

Hits a local max here, that is right there.1511

It is exactly what we said before.1515

Thank you so much for joining us here at www.educator.com.1518

We will see you next time, bye.1520

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