  Raffi Hovasapian

Antiderivatives

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 (4) 1 answer Last reply by: Professor HovasapianThu Nov 30, 2017 4:43 AMPost by Magic Fu on November 23, 2017Hello Professor Hovasapian, I had a B on the first semester of AP Calc BC, is it good? Should I drop out to AB? 1 answer Last reply by: Professor HovasapianThu Aug 25, 2016 5:38 PMPost by Isaac Martinez on August 25, 2016Hello Professor Hovasapian,I was wondering how you got 13.432 as an answer for  the second derivative of your example II, Function 3.Thank you,Isaac

### Antiderivatives

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
• Antiderivatives 0:23
• Definition of an Antiderivative
• Antiderivative Theorem
• Function & Antiderivative 12:10
• x^n
• 1/x
• e^x
• cos x
• sin x
• sec² x
• secxtanx
• 1/√(1-x²)
• 1/(1+x²)
• -1/√(1-x²)
• Example I: Find the Most General Antiderivative for the Following Functions 15:07
• Function 1: f(x) = x³ -6x² + 11x - 9
• Function 2: f(x) = 14√(x) - 27 4√x
• Function 3: (fx) = cos x - 14 sinx
• Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )
• Function 5: f(x) = (3e^x) - 2/(1+x²)
• Example II: Given the Following, Find the Original Function f(x) 26:37
• Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 40
• Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 5
• Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.2
• Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 7
• Example III: Falling Object 41:58
• Problem 1: Find an Equation for the Height of the Ball after t Seconds
• Problem 2: How Long Will It Take for the Ball to Strike the Ground?
• Problem 3: What is the Velocity of the Ball as it Hits the Ground?
• Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?

### Transcription: Antiderivatives

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

Today, we are going to start our discussion of anti-derivatives.0005

In a couple of lessons, we are actually going to change the name of that and start calling them integrals.0009

This is the second half of calculus.0013

The first part is differential calculus, now it is integral calculus.0015

As you will see in a minute, it is actually the inverse process.0018

Let us jump right on in.0022

Up to now, we started with functions and we took derivatives of them.0026

Let us go ahead and write this.0032

Up to now, we started with some function f(x) and we found its derivative.0035

We have found its derivative, in other words, we differentiate it.0053

For example, if we had a sin x, we take the derivative of it and we ended up with cos x.0062

If we had, let us say, an x³, we take the derivative of it and we end up with something like 3x².0072

What if we begin with a function then ask is this the derivative of some function?0081

In other words, instead of starting with the function and going to the derivative,0089

let us say that this is a function and this the derivative of some previous function.0093

That is the question we are going to look at here.0099

What if we begin with a function f(x), then ask is f the derivative of some f(x).0100

Can I find this F, can I find this other f?0139

In other words, we are going to be giving you the cos x for the 3x².0147

And we are going to say, how do you recover the sin x?0151

How do you recover the x³, if that original function actually exists?0153

The answer is yes, fortunately.0158

Let us go ahead and start with a definition.0167

You know what, I think I will go ahead and put my definition.0170

The different color, I will go ahead and put in red.0175

The definition F(x) is called an anti-derivative, exactly what it sounds like,0180

it is just the reverse of the derivative, an anti-derivative of f(x).0194

If this f(x) happens to be the derivative of the F(x), we will just mark that with an f’, on a specified interval.0202

Let us go back to blue here, not necessary but what the heck.0225

All we are doing is going backward, that is it.0232

That is all this is.0243

Now the most difficult part is going to be remembering, am I going forward to differentiating0245

or am I going backward and taking the anti-derivative?0250

It is a different set of rules, it is a different set of formulas that you use to find that.0253

That is going to be probably the most difficult thing that you have to do is remember which direction you are going in.0258

If we begin with some f(x), that is the function that is given, we can go ahead and take the derivative which gives us f’.0265

Or we can go this way and take the anti-derivative which gives us our F(x).0276

Some function is going to be the function that is given to you.0284

There are two ways that you can go, depending on what problem that you are trying to solve.0285

That is what makes calculus incredibly beautiful, you start here.0291

As you will see in a few lessons, there is a relationship between derivative and anti-derivative.0296

They call it the fundamental theorem of calculus, it actually connects the two,0300

which as we said are inverse processes.0304

Let us note the following.0308

If we had some f(x) which is equal to sin x + 6 and we had some g(x),0320

let us say this is sin x + 4, these are not the same function.0329

If you put an x value in, you are going to get two different y values.0335

These are not the same function, very important to know that.0338

These are not the same function but you can probably see where this is going.0343

But f’(x), f’(sin x) + 6 is equal to cos(x).0357

Not f, I’m talking about g and g‘(x).0367

When I take the derivative of the sin x + 4, it is also going to equal cos(x).0370

These are the same derivative.0378

You get two different functions that end up going to the same derivative.0382

That could be a bit of an issue, but it is not, fortunately.0387

If we begin with h(x) = cos x, if we begin with the derivative and ask for its anti-derivative, which one do I choose?0393

Do is say that sin x + 6 is the anti-derivative?0426

Do I say sin x + 4 is the anti-derivative?0428

Is it sin x + 6?0436

Is it sin x + 4?0444

Or how about just sin x + c, where c can be any constant?0447

Because we know that the derivative of a constant = 0.0465

I does not matter what that number is, it can be sin x + π, because the derivative of π is 0.0470

Here is our theorem, let us go ahead and mark this in red.0477

Our theorem says, if f(x) is an anti-derivative of f(x) on an interval that we will just call I,0483

then f(x) + c, any other constant is the general solution.0514

It is the general solution to this anti-differentiation problem.0531

In other words, if I'm given cos(x) and if I take the anti-derivative of that,0551

I know that the anti-derivative is going to be sin x + something.0555

What is it that I’m going to choose for that something?0560

In general, if we are just speaking about general solutions and taking anti-derivatives, you are just going to write cos x + c.0562

There is going to be other information in the problem that allows you to find what c is.0569

sin x + 6 or sin x +, or sin x + 48, those are specific solutions, particular solutions to a particular problem where certain data is given to you.0576

When we speak of the general situation, we put the anti-derivative and you just stick a c right after it,0589

to make sure that it is formally correct.0594

Are you going to forget the c, yes you are going to forget the c.0597

I still forget the c, after all these years.0601

Do not worry about it but this is the general solution.0603

anti-derivative, without any other information, just add the constant to it.0605

I will say, technically, you must always include the c.0616

Try your best to remember it.0628

Once again, extra information will allow you to find what c is.0631

We will see that when we start doing some of the example problems.0648

To find what c is, extra information will allow you to find what c is, in a particular case.0652

Thus, giving you what we call the specific solution or often called the particular solution.0677

Let us do some examples.0690

Before we do the examples, I’m going to give you a list of the anti-derivatives that we actually already know.0703

Some anti-derivative formulas we already know.0711

I think I will go ahead and do this in red.0730

Here we have the function and here I'm going to put the anti-derivative.0732

Once again, we have to make sure that we know what direction we are going in.0745

If I have x ⁺n, the way to find the anti-derivative of x ⁺n is you take x ⁺n + 1/ n + 1.0749

In other words, if this were the function that we are given,0761

you know that what you do is to take the exponent, you bring it down here.0764

You subtract one from the exponent.0769

Differentiation is this way.0770

What we are saying is, if you are starting with a function anti-differentiation, this is the formula that you use.0773

Again, you will see in just a minute what we mean.0779

Right now, I’m just going to write down some formulas here.0782

1/x, the anti-derivative of that is the natlog of the absolute value of x e ⁺x.0786

The anti-derivative is e ⁺x.0795

If I have cos(x), I know the anti-derivative of that is sin(x).0798

If I’m given sin(x), the derivative is cos(x).0803

If I’m given cos(x), the anti-derivative is sin(x).0806

Direction is very important here.0810

Do not worry, you will make a thousand of mistakes, as far as direction is concerned.0812

You will be asked to get an anti-derivative and you end up differentiating.0817

That is just the process that we go through, do not worry about it.0819

One more page here, let us write our function and let us write our anti-derivative.0825

If I have a sin x and I want the anti-derivative, it is actually going to be a -cos x.0841

Because if I’m given –cos x, the derivative of that is sin x.0847

Sec² x, the anti-derivative is the tan(x).0854

If I'm given sec x tan x, the anti-derivative sec x.0859

If I'm given 1/ 1 - x², all under the radical,0867

the anti-derivative of that is the inv sin(x) 1/ 1 + x², that anti-derivative is the inv tan(x).0872

One last one, -1/ 1 - x², all under the radical and that is going to be the inv cos(x), that is the anti-derivative.0886

If I were given the inv cos, the derivative of that would be -1/ √1 - x².0896

Now let us go ahead and jump right into the examples.0904

Find the most general anti-derivative for the following functions.0909

General means just add c to your answer, that is all that means.0912

1, 2, 3, 4, 5, let us go ahead and jump right on in.0917

Let me go ahead and make sure that I have everything here.0924

I have copied the functions, 14 x⁵, 4/3, and 3 ⁺x.0928

Let us start with number 1, I think I will go back to blue here, I hope you do not mind.0939

Number 1, our f(x) was x³ – 6x² + 11x – 9.0947

We said that the formula for the anti-derivative, when you are given some x ⁺n,0962

the anti-derivative of that is x ⁺n + 1/ n + 1.0968

Add 1 to the exponent , divide by the number that you get which is now the new exponent, very simple.0972

Therefore, our f(x), our anti-derivative is going to be x⁴/ 4.0979

How much easier can this possibly be?0987

-6, the constant, x³/ 3, we will simplify it, just a minute.0992

+ 11 x² because this is 1, add 1 to it and divide by that same number, -9x.1001

This is x⁰, x⁰, add 1 to the exponent, it becomes 1, divide by 1, it becomes 9x.1009

Now we can simplify, divide where we need to.1017

This is perfectly valid, you do not have to take the 6 and the 3, and divide it.1020

You can stop there, if you want to.1023

It just depends on what you teacher is going to be asking for.1025

We have x⁴/ 4, 6/3 is 2.1028

It is going to be -2x³, it is going to be +11 x²/ 2 – 9x + c.1035

I will go ahead and put that c here.1045

Again, we are going to add that c because it is the most general solution.1046

There you go, that is your anti-derivative.1051

You can always double check by differentiating your F.1055

The anti-derivative that you got, just differentiate and see if you get the original function.1066

That corroborates the fact that you have done it right, by differentiating f(x).1070

We have f(x) right here, therefore, f’, let us see what happens when I take the derivative of that.1085

It is going to be 4x³/ 4 - 6x² + 22x/ 2 – 9.1091

Sure enough, f’(x) is equal to x³ - 6x² + 11x – 9, which is exactly what the original = f(x).1111

F’(x) = f(x), that is what our theorem said, that is all we are doing.1128

We just have to remember which direction we are going in.1133

If we are taking the derivative of x³, it is going to be 3x², the original function.1137

If we are taking anti-derivative, it is going to be x⁴/ 4.1144

Direction is all that matters.1148

Let us go to function number 2, we had f(x) is equal to 14 × √x - 27 × 4√x.1151

We are going to write this with rational exponents.1167

This is going to be equal to 14 × x ^½ - 27 × x¹/4.1170

When we have an x ⁺n, when we take our anti-derivative, our formula is x ⁺n + 1/ n + 1.1180

That is it, you just subjected to the same thing.1188

It does not matter whether the exponent is rational or not.1191

This is going to be 14 × x, ½ + 1 is 3/2 divided by that number 3/2 – 27 × x.1194

¼ + 1 is 5/4 divided by 5/4.1206

We have to have our + c.1213

Therefore, we end up with, our final anti-derivative is going to be 14/ 3/2, that is going to be 28/3 × x³/2 - 4 × 27.1217

That is going to be 108 divided by 5 × x 5⁴ + c.1234

That is your most general anti-derivative.1244

If you took the derivative of this, you would get the original back.1248

Example number 3, we have f(x) = cos(x) - 14 × sin(x).1254

We are doing anti-derivative.1267

We go back to that list where we have the function and its anti derivative, which is also in your book or anywhere on the web.1270

You can just look at table of anti-derivatives also called table of integrals.1278

The anti-derivative of cos x was sin x - 14 which is the constant.1284

The anti-derivative of sin x was -cos x + c.1296

When we simplify, we get sin x + 14 cos(x) + c.1305

Once again, if you want to go ahead and check, the derivative of sin x is cos x.1318

The derivative of 14 cos x is -14 sin x.1323

Let us see what we have got, number 4.1336

We have x⁵, let me write down f(x).1341

F(x) = x⁵ + 2 × √x/ x⁴/3.1349

Let us write with rational exponents here.1364

We have x⁵ + 2 × x ^½/ x⁴/3.1366

I'm going to go ahead and separate this out.1378

It is going to be x⁵/ x⁴/3 + 2x ^½ / x⁴/3.1380

This is going to be x⁵/ x⁴/3 + 2x ^½/ x⁴/3.1386

I do not like writing my fractions that way, sorry about that.1398

I’m going to write it as it is supposed to be written, x⁴/3.1402

We get this is equal to x ⁺15/3 - 4/3 + 2 × x³/6 – 8/6.1406

Our f(x) is actually equal to x ⁺11/3 + 2 × x⁻⁵/6.1430

We can go ahead and take the anti-derivative.1445

I have just simplified that and made it such that there was an x to some exponent, so that I can use my formula.1447

I hope that make sense.1457

I cannot do anything with this, I have to convert it to something where I have x ⁺n and x ⁺n.1458

Now I can apply the formula.1463

The anti-derivative of f(x), now it is equal to, it is going to be x ⁺11/3 + 1 / 11/3 + 1 + 2 × x⁻⁵/6 + 1 all divided by -5/6 + 1 + c.1465

We have f(x) = x ⁺14/3/ 14/3 + 2x⁻⁵/6 + 6/6 is x¹/6 divided by 1/6 + c.1490

Our final answer is going to be 3/14 x ⁺14/3 + 12x¹/6 + c.1511

Our final answer, slightly longer not a problem.1533

It was only because of the simplification that we have to do.1536

Our number 5, we have our f(x) is equal to 3e ⁺x - 2/1 + x².1542

Really simple, this we can just read off.1555

The anti-derivative of e ⁺x is e ⁺x.1558

This stays 3e ⁺x.1561

Hopefully, we recognize that 1/1 + x², the anti-derivative of that is the inv tan.1564

Sorry about that, it is -2 × inv tan(x).1574

Of course, we add our c to give us our most general anti-derivative.1580

There you go, that takes care of that.1585

Hopefully, those examples help.1589

Again, it is all based on the basic formulas.1590

Let us do example number 2, given the following, find the original function f(x).1595

This time, they have given us extra information.1600

They have not only given us the f’, we are going to find the anti-derivative which is the f.1603

They have given it to us as f’, we just need to find f.1614

They also gave us other information, they said that the original f at 2 is equal to 40.1618

This extra information now is going to allow us to find what c is, in a particular case.1623

We are going to find the general solution.1629

And then, we are going to use this extra information to find the particular constant.1630

Let us get started here.1637

I think I have the wrong number here.1646

F(2) = 40, let me double check and make sure that my numbers are correct here.1652

I think I ended up actually using a different number when I solve this.1661

I had f(2) = 47, sorry about that, slight little correction.1665

Number 1, we have that f’(x) is equal to 5x³ - 14x + 24.1671

They tell us that f(2) is equal to 47.1684

F(x), notice that if I’m using prime notation, f is the anti-derivative of f’.1691

This just becomes 5x⁴/4 - 14x²/ 2 + 24x + c.1701

x ⁺n, just add 1 to the exponent, put that new exponent also in the denominator.1716

Let us go ahead and simplify a little bit.1722

f(x) = 5/4 x⁴ - 7x² + 24x + c.1724

This is our f, they tell me f(2) is equal to 47.1740

I put 2 wherever I have an x, I set it equal to 47, and I solve for c.1744

They tell me that f(2) which is 5/4⁴ - 7 × 2² + 24 × 2 + c.1752

They are telling you that all of that actually = 47.1765

I hope that I have done my arithmetic correctly.1775

We have got 20 - 28 + 48 + c = 47 and that gives me a final c = 7.1777

I’m hoping that you will confirm.1790

Now that I have c which is equal to 7, I can go ahead and put it back in to my equation that I have got.1793

My anti-derivative, my specific, my particular solution is going to be 5/4 x⁴ - 7x² + 24x + 7.1801

I found my constant and I have a particular solution, a specific solution, that is all I'm doing.1818

Do the anti-derivative and then use the information that is given to you to find the rest.1825

Number 2, we have an f’(x) is equal to 3 × sin(x) + sec² (x).1834

They are telling me that the f(π/6) happens to equal 5.1847

If this is f’, I take the anti-derivative.1856

This is going to be my f without the prime symbol.1858

It is going to be -3 × cos(x) because the anti-derivative of sin x is -cos(x).1861

The anti-derivative of sec² is tan(x).1868

This is going to be + c.1874

Now I use my information, f(π/6) is equal to -3 × cos(π/6) + tan(π/6) + c.1875

They are telling me that all of that is equal to 5.1892

Here we have cos(π/6) is going to be √3/2, -3 √3/2.1896

Tan(π/6) is going to be 1/ √3 + c is equal to 5.1905

Therefore, my c is going to equal, I’m not going to solve for of them, I’m just going to write it out straight.1913

It is going to be 5 – 1/ √3 + 3 √3/2, that is my c.1919

Therefore, I stick my c there and I get f(x) is equal to -3 × cos(x).1928

I will make my o's a little closer here, × cos(x) + tan(x) + whatever c I got which is 5 – 1/ √3 + 3 √3/ 2.1941

There you go, nice and simple.1957

Your teacher can tell you about the extent to which they want this simplified, put together, however they want to see it.1961

Number 3, let us see what we have got here.1970

Number 3, this one involves taking the anti-derivative twice.1977

They are telling me that f”(x) is equal to 8x - cos x.1982

They gave me two bits of information.1990

They are giving me f(1.5) is equal to 12.7.1992

They are telling me that f’(1.5) is equal to 4.2.1998

I have to take two anti-derivatives.2006

Therefore, I'm going to have two initial conditions.2009

One is going to be for f, one is going to be for f’.2011

I’m going to take the anti-derivative once, find f’.2014

Use this information, the f’(1.5) = 4.2, to find that constant.2017

I’m going to take the anti-derivative again, we will call it integration later, it is not a problem.2023

We are going to take the anti-derivative again, of the f’ to get our original function f.2030

We are going to use this first bit of information to find that constant.2034

Each step has a constant in it.2039

From f”, we are going to take the anti-derivative which means find f’.2042

This is going to be 8x²/ 2 - the anti-derivative of cos x which is sin x.2049

I will call this constant 1.2057

This is f’(x) = simplify a little bit, we have got 4x² – sin x + the constant of 1.2061

Now I use this information right here, the f(1) f’.2072

F' of 1.5 is equal to 4 × 1.5² – sin(1.5) + c1, they are telling me that it = 4.2.2076

I will write it all, that is not a problem.2096

When I solve this, I get 9 - 0.997 + c1 = 4.2.2097

I get that my c1 is equal to -3.803.2106

I found my first c1, that is the one that I’m going to plug in to here.2113

Therefore, my f’(x) is going to equal 4x² – sin x - 3.803.2119

Now that I have my f’, I want my original function f.2134

I’m going to take the anti-derivative again.2136

F(x) = 4x³/ 3 + cos(x) because the anti-derivative of sin x is -cos(x).2140

It is going to be -3.803x.2154

This is x⁰, it becomes x¹/1, and then now, + c2, always add that constant.2158

I know, I always forget.2165

I think the only reason that I actually remember is because I'm doing a lesson now, I’m trying hard to remember this, to put that c there.2169

Now we use this bit of information.2177

They are telling me that f(1.5) which is equal to 4/3 × 1.5³ + cos(1.5) - 3.803 × 1.5 + c2.2180

They are telling me that it = 12.7.2200

When I do that, I have got f(1.5) is equal to, it is going to be 4.5 + 0.0707 - 5.303 + c2 = 12.7.2207

When I solve, I get 13.432, that is my c2.2228

I put it back to my original and I end up with f(x) = 4/3 x³ + cos x - 3.803x + 13.432.2236

This is my particular solution to this particular anti-differentiation problem, given those two initial conditions.2256

Let us try another one of those.2270

This time we have, sorry about that, this is a double prime.2274

f”(x) is equal to 5/ √x.2283

We have f(2) is equal to 15 and we also have f’(2) is equal to 7.2289

Two initial conditions.2299

Let us write this in a way that we can manipulate.2301

f”(x) is equal to 5 × x⁻¹/2.2307

I take the anti-derivative so this is now going to become f’(x) and2314

this is going to be 5 × x⁻¹/2 + 1/ -1/2 + 1 which = 5x ^½/ ½, which is equal to 10x ^½ + c1.2318

There you go, that is my f’.2342

They tell me that f’(2) which is going to be 10 ^½ + c1 is equal to 7.2348

Therefore, my c1 is going to equal -7.14, when I do the calculation.2364

Therefore, I put this 7.14 into there and I get my f’, my specific solution f’(x) is equal to 10 x ^½ - 7.14.2371

f(x), I will do my f(x), I take the anti-derivative of this.2392

This is going to be 10x ½ + 1 is 3/2/ 3/2 - 7.14 × x + c2.2398

Let me see what I have got here, let me go to the next page.2416

I have got, when I simplify this, I have got f(x) = 20/3 x³/2 - 7.14 × x + c2.2429

They are telling me that f(2) which is equal to 20/3 × 2³/2 - 7.14 × 2 + c2.2445

They are telling me that that = 15.2459

When I solve for this, I get c2 is equal to 10.42.2461

I have my final f(x), my original function is 20/3 x³/2 - 7.14 × x + 10.42.2468

I think I did that right, I hope I did that right.2491

That is it, just anti-derivative, anti-derivative.2501

With each anti-derivative that you take, you want to go ahead and make sure to put the c2504

and then use the other information for wherever you are to find that c, and then take the next step.2509

Let us do a practical problem here.2518

A steel ball was dropped from rest from a tower 500 ft high, answer the following questions.2521

Take the acceleration of gravity to be 9.8 m/s2.2527

The first thing we want to do is find an equation for the height of the ball, after t seconds.2533

After I have dropped it, how long will it take for the ball to hit the ground?2537

What is the velocity of the ball as it hits the ground?2543

Part 4, if the stone is not dropped from rest, but if the stone is actually thrown downward with an initial velocity of 6 m/s,2546

how long does it take to reach the ground?2555

Let us see what we have got.2561

We have this tower, let us go ahead and draw this out.2568

This is the ground level, I’m just going to make this tower like that.2571

They tell us that this is 500 ft high.2575

I'm going to go ahead and take that as ground 0.2580

This is the 500, right.2584

Number 1 wanted the equation for the height of the ball after t seconds.2590

After a certain number of seconds, the ball is going to be like right there.2610

The height is going to be 500 - the distance that it actually traveled.2614

What I'm going to do is I'm going to find an equation for the distance that it actually traveled, and then take 500 – that.2621

That will give us our equation.2627

We will let s(t) be the distance function, how long it travels?2630

Be the distance function also called the position function.2638

You remember we called the position before.2644

That is the actual function that I’m looking for.2647

I’m looking for s(t).2649

S’(t), I know that the derivative of the position function is my velocity function.2651

It is my velocity and I know that if I take the derivative again, in other words, s”(t),2661

that is my acceleration, that is my acceleration function.2670

What do we know, we know that the acceleration of gravity is 9.8.2682

It is just a constant, that is it, you are just dropping it from rest.2686

The only force that is acting on this is the acceleration of gravity.2689

Therefore, our acceleration function s”(t) is actually just equal to 9.8, it is a constant.2693

Therefore, s’(t), when I take the first anti-derivative, that is just going to equal 9.8 t + c1.2704

What do I know, I know that s’(t) which is the velocity,2720

let us try it again, I know that s’ is the velocity.2731

I know that the velocity at time 0 which is the s’ at time 0 was starting from rest.2736

I’m just dropping it, it is 0.2742

S’(0) is 0, let us plug it in here.2746

That means that s’(0) is equal to 9.8 × 0 + c1.2749

I know that that = 0, that implies that c1 is actually equal to 0.2762

When I plug that in to my original equation, to my s’, I get s’(t) = 9.8 t.2768

I found an equation for the velocity at time t, it is 9.8 t.2777

Now I'm looking for s(t), now I’m going to take the anti-derivative of that.2786

Now I have got s’(t) = 9.8.2794

Therefore, s(t) is going to equal 9.8 t²/ 2 + c2.2797

What else do I know, now I need to find c2.2812

I know that s(t) or s(0), in other words the position at time 0 is 0.2817

I take that as my 0 position, that also = 0.2828

I’m going to put that in here, s(0) = 4.9 × 0² + c2 is equal to 0, that implies that c2 is equal to 0.2833

Therefore, my position function s(t) is equal to 4.9 t².2849

That means after a certain number of seconds, t seconds, I have actually traveled 4.9 × t² ft, m, whatever the length is.2857

Therefore, that means that is this distance.2869

After a certain number of seconds, I traveled this distance.2876

Therefore, my height above the ground is going to be 500 - this distance,2879

my height function is going to be 500 - 4.9 t².2884

Again, this is only based from the fact that I chose this as my 0.2892

You could have chosen this as your 0, just a different frame of reference.2896

I hope that make sense.2902

Let us go back to blue here.2910

How long before the ball strikes the ground, in other words, how many seconds go by before it?2915

How long before the ball strikes the ground?2930

Let us see, we came up with our s(t) which was going to be 4.9 t².2937

That was our position function.2948

We are falling 500 ft, our tower was this one.2953

We need to find out, we are going to set s(t) which is 4.9 t², we are going to set it to 500.2959

In other words, how many seconds does it take to go 500 ft?2969

When I solve this, I get t = 10.10 s.2973

Nice and simple, I have my position function.2982

How long does it take to go the 500 ft?2986

Number 3, velocity, as the ball hits the ground.2990

We said that our velocity function which was our first anti-derivative, s’(t), we said that that = 9.8 × t.3008

After 10.10 s which is when the ball is hitting the ground, I get the velocity at 10.10 s = 9.8 × 10.10 s.3019

It is going to be 99 m/s.3033

That is it, very straight forward.3038

Let us do the last one, number 4, if our initial velocity is 6 m/s downward, what is the velocity of the ball as it hits the ground?3044

Let us do this again, let us start from the beginning?3083

We have s”(t) which is our acceleration function, we know that that = 9.8.3085

When I take the anti-derivative of that, that is going to give my velocity function.3092

That is what I’m interested in.3095

Again, I have s’(t) which is my velocity function, that is going to equal 9.8 t + c1.3096

Standard anti-differentiation but now it is slightly different.3109

Now my initial velocity, in other words, my s’(0) which is my v(0) is now 6, it is not 0.3113

My velocity at time 0 which is 9.8 × 0 for t + c1 is equal to 6 m/s.3126

This implies that my c1 is actually equal to 6, that goes in here.3137

Therefore, my s’(t) function is actually equal to 9.8 t + 6.3143

When I find my s(t), I take my anti-derivative again,3156

I’m going to get my 9.8 t²/ 2 + 60 + c(2) which is equal to 4.9 t² + 60 + c/2.3160

I have a different function now and I also know that s(0) is still 0.3182

My 0 point is my starting point.3189

Therefore, I have got s(0) is equal to 4.9 × 0² + 6 × 0 + c2 = 0, which implies that our c2 is equal to 0.3191

Therefore, I get s(t) is now equal to 4.9 t² + 6t, that is my equation.3207

I need to find out how many seconds it takes, now that I have thrown it with an initial velocity which introduces the second term,3222

which was not there before, now I need to set this equal to, s(t) = 4.9 t² + 60.3227

I need to set that equal to 500, when I do that, I get t is equal to 9.51 s.3242

Exactly, what I expect, I threw it down with initial velocity instead of dropping from it rest.3251

It is going to take less time for it to get to the ground.3256

It took 10.1 seconds, now it is only taking 9.51 seconds.3258

This 9.51 is now, what I actually am going to put into my velocity function.3263

I knew velocity function which includes this extra term for the initial velocity.3275

S’(t) which is my velocity function is equal to 9.8 t + 6.3286

Therefore, the velocity of 9.51 = 9.8 × 9.51 + 6.3295

I get my velocity 1.51 = 99.2 m/s.3306

Not a lot faster but certainly faster.3314

There you go, that takes care of anti-derivatives.3320

Thank you so much for joining us here at www.educator.com.3323

We will see you next time, bye.3325

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