INSTRUCTORS Raffi Hovasapian John Zhu

Home » Mathematics » AP Calculus AB » AP Practice Exam: Section 1, Part A No Calculator, cont.

Raffi Hovasapian

Raffi Hovasapian

AP Practice Exam: Section 1, Part A No Calculator, cont.

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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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AP Practice Exam: Section 1, Part A No Calculator, cont.

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
  • 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

Transcription: AP Practice Exam: Section 1, Part A No Calculator, cont.

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

Today's lesson is going to be a continuation of the Section 1 Part A of the AP practice exam.0004

The one with no calculator allowed.0010

Let us continue on.0012

We finished off with question number 14, in the last lesson.0014

Now we are going to start with question number 15.0017

Let me go ahead and go to blue.0020

Question number 15 is asking us to find the average value of a given function over a particular interval.0024

The function at our disposal is 2x + 3³.0031

The particular interval over which we want to take the average value is -3 to -1.0043

The average value is really easy to calculate.0050

It is just, basically, the integral of the function divided by the length of the interval.0054

It is usually written as 1/ b – a.0061

A is the first number, b is the second number, × the integral from a to b of gx dx.0063

It is a simple evaluation of integral.0073

This is going to be 1/ 1 – a - 3 × the integral from -3 to -1 of 2x + 3² dx.0076

I think it was squared, was not it cubed?0092

Yes, squared not cubed.0095

I’m just going to go ahead and multiply this out.0099

= ½ × the integral from -3 to 1 of, this is going to be 4x² + 12x + 9 dx.0102

That is going to equal ½ 4x³/ 3 + 12x²/ 2.0118

That is going to be + 6x² + 9x all evaluated from -3 to -1.0132

When we do the evaluation, we get ½.0141

This is going to end up being.0145

I would not go through all the stuffs, I will just do what it is that we have got.0151

= ½ - 13/3 + 27/3.0159

You are going to end up with a value of 7/3 which is going to be c.0168

That is it, just evaluate the definite integral0173

Here there is no particular technique to use, just straight application of antiderivatives.0177

Number 16, what does number 16 asks us?0184

Excuse me, let me move that over.0189

Number 16 is asking us to evaluate a limit.0195

In this particular case, it is the limit as t goes to 0 of the quotient of tan(π/4) + t – tan(t)/ t.0198

The important thing here is instead of actually evaluating limit, you are not going to evaluate the limit.0226

You are going to recognize that this limit is the definition of derivative for a particular function.0231

This limit is going to be the derivative evaluated at π/4.0238

In this particular case, it is going to be y =, this is the derivative of the tangent function evaluated at π/4.0250

You just want to recognize that is what it is, instead of actually evaluating this limit.0260

Let us write that down.0266

Recognize this as the definition of the derivative.0269

That is all this is, definition of the derivative of the tan(x) at x = π/4.0274

We have y is equal to tan(x).0295

We have y’ of tan(x) which is sec² (x).0298

y’(π/4) = sec² of π/4 which is equal to √2² which is equal to 2.0304

That is choice e.0319

Just recognize that that is the definition of derivative.0322

Let us see, that is number 16, let us go ahead and go to number 17.0328

Let us see what number 17 is asking us.0335

17 is giving us a particular function, a function of t.0337

They want us to find the instantaneous rate of change at t = 0.0342

Nice and straightforward, instantaneous rate of change.0349

We know that an instantaneous rate of change is the derivative.0351

Our derivative is the slope, it is the rate of change.0355

The slope of the secant line is equal to the average rate of change.0362

The slope of the tangent line which is the derivative is the instantaneous rate of change.0365

Let me write number 17, our function of t is equal to 2t³ - 2t + 4 × √t² + 2t + 4.0372

We want to evaluate f’ at 0.0393

Let us go ahead and f’.0401

f’(t), it is going to be product rule.0404

It is going to be a little tedious and long, but that is not too big of a deal.0409

We have got 2t³, this × the derivative of that + that × the derivative of this.0413

-2t + 4 × ½ t² + 2t + 4⁻¹/2 × 2t + 2 + t² + 2t + 4.0420

Instead of running the radical sign, I will go ahead and write it that way.0445

× the derivative of what is inside here which is going to be 6t – 2.0447

Of course, we just plug in t, we do not have to simplify this.0453

We can just plug in our particular 0.0457

When you plug 0 in to t for all of this and evaluate that, you are going to end up with 2 - 4 = -2.0461

That is our answer and that is choice b.0471

Nice and straightforward.0476

Let us try number 18.0482

Let us see, what is number 18 asking us to do?0487

It wants us to calculate, it wants us to do ddx of 11 ⁺cos(x).0491

We know that the derivative of some constant e ⁺u is equal to e ⁺u × ln(a) × du.0507

If we have 11 ⁺cos(x) and we subject it to the differential operator, in other words, take the derivative of it.0527

It is a fancy term for that.0535

We subject it to differentiation.0537

We get the derivative of 11 ⁺cos(x) is equal to 11 ⁺cos(x) × natlog of 11 × the derivative of the u.0539

Because u is a function of x, it is not just x.0552

The derivative of cos(x) is - sin(x).0556

That is it, basically, you just end up with -sin x 11 cos(x) ln(11) which is d.0561

Again, if you want to separate them out, so you can actually not get confused as to which is which with the symbols,0576

just put some parentheses around it, not a problem.0581

Great, nice and straightforward.0584

Those are the ones that we love.0588

Number 19, let us see what we have got here.0591

Number 19, we have a solid which is generated by rotating a particular region that is enclosed by these graphs and these lines.0596

We are rotating about the x axis and they want to know which one of these integrals0610

actually gives you the volume of the particular solid that is generated.0615

Let us see what we got.0621

We have y = √x and we have x = 1, x = 2, and y = 1.0622

Let us go ahead and draw this out.0632

Something like this, we know what the √x function looks like.0637

It looks something like that.0643

X = 1, let us just put it here.0646

x = 2, let us go ahead and put it here.0650

y = 1, that is this line right here.0654

This is our origin, y = 1.0663

I will just go ahead and put it right there.0665

It is the region that is bounded by all of these.0671

Bounded by x = 1, x = 2, the function y = √x, and the line y is 1.0673

This is the region that we are looking for.0688

That is the region that we are going to rotate around the x axis.0689

Basically, we want to come down here.0694

Now we have that one, we are going to have this.0697

This is the cell that is going to be generated.0705

We are going to have this solid, we want to find the volume of that solid or the integral representing that volume.0708

I think what I’m going to do here is I’m going to go ahead and use washers.0717

I'm going to integrate from 1 to 2.0721

I’m going to integrate in the horizontal direction.0724

I’m definitely going to be using dx.0726

My upper limit and lower limits of integration, my lower to upper is actually I’m going to go from 1 to 2.0730

Essentially, what I'm doing is I’m taking a little bit of a washer here.0739

It looks like this.0744

It is going to be something like this.0754

This is the region.0756

We are going to take the area of that washer and we are going to add up all the areas along the x axis.0759

Let us go ahead and call this radius the outer radius.0774

We will call this one the inner radius.0778

The volume is going to be the integral from a to b of an area element × dx.0781

We said we are going to integrate from 1 to 2, that is 1 to 2.0792

The area of this region is going to be π × the outer radius² - π × inner radius².0797

In other words, the big circle - the inner circle.0805

That is just π × the outer radius² - the inner radius² × dx.0808

Now we just need to know what the outer and inner radiuses are, as functions of x.0818

This is going to equal, I’m going to pull the π out of the integral sign, it is 1 to 20822

The outer radius is, if this is my x value, the outer radius is √x.0830

The inner radius is 1.0840

It is going to be the outer radius, it is going to be √x²,0842

that is the outer radius² - the inner radius which is from here to here, from here to here, which is 1² dx.0853

We have π × the integral 1 to 2 of x - 1 dx.0864

That is our answer, it did not ask us to evaluate it.0876

This is going to be choice d.0879

In this particular case, we decided to use washers.0883

I’m going to stick with red, I guess it is kind of nice.0888

This is going to be question number 20.0891

There is no calculator involved in this particular section, the first section.0895

The first section is no calculator, the second section is there are going to be calculator.0898

When you do the free response questions, one of the sections is going to be no calculator,0903

the other one is going to be with calculator.0908

Let us see what 20 is asking us.0912

It is asking is to calculate a limit.0915

Let us see what we have got.0918

They want us to evaluate the limit as x goes to 0 of 4x/ sin(3x) + x/ cos(3x).0919

The limit of the sum is the sum of the limits.0939

Let me actually separate this out.0942

This is equal to the limit as x approaches 0 of 4x/ sin(3x) + the limit as x approaches 0 of x/ cos(3x).0944

This one right here, when I put 0 in, I get 0 sin(3x), sin(3) × 0, sin(0) is 0.0962

What I ended up with is 0/0.0968

This is an indeterminate form.0971

We have something great for this, L’Hospitals rule.0973

I just take the derivative of that or the derivative of that, and I take the limit again.0978

This limit is actually equal to the limit as x approaches 0.0983

The derivative of 4x is equal to 4.0989

The derivative of sin of 3x is 3 cos of 3x.0992

Now when I take x to 0, the cos(3) × 0 is cos(0).1000

The cos(0) is 1.1004

The limit of 4/3, this is actually equal to 4/3.1006

This first limit right here, that is equal to 4/3.1013

This one, when I put 0 in, I end up with 0/1 which is 0.1017

My final answer is 4/ 3 + 0 = 4/3.1025

My answer is b.1032

Again, just a straight application of l’Hospitals rule, any of the indeterminate forms.1034

If you have 0/0, if you have infinity/ infinity, if you have infinity × 0, things like that.1038

Let us try number 21.1049

What is 21 asking us?1053

For 21, it says that y is greater than 0 and it is telling us that dy dx is equal to 3x² + 4x/ y.1058

They are telling us that the point 1√10 is on the graph.1080

What is 0y?1090

In other words, find the y value.1096

They are telling us the y is greater than 0.1102

They are telling us that the derivative is equal to 3x² + 4x/ y.1104

They are telling us that the graph itself contains the point 1 √10.1109

They we want to know what the y value is, when x = 0 of the function.1112

Let us what we have got.1121

We know y’, that is dy dx.1124

y’ is equal to 3x² + 4x/ y.1126

I'm going to go backwards.1137

This is essentially an implicit differentiation.1139

Somebody who has done implicit differentiation and they come up with this.1143

Just go backwards.1146

What you end up with is yy’, just multiply through by y = 3x² + 4x.1148

What you will end up, when you rearrange this, you get -3x² - 4x + y, y’ is equal to 0.1159

Essentially, now you are just going to integrate this function one at a time.1173

The integral of -3x² is x³.1177

We are just going backwards to what y is equal.1181

This is -2x², the integral of -4x is -2x² because the derivative of -2x² is -4x.1184

Here, this derivative is ½ y², the antiderivative.1193

Because if I take ½ y², take the derivative of it, implicitly, it is going to be 2 × ½ y × y’.1200

The 2 and ½ cancel, you are left with y and y’.1210

That is equal to some constant c.1214

We do not know what c is.1219

Let us go ahead and work it out.1223

We figured out what our original function is, in implicit form.1225

That is this thing.1229

We also know that we have an x and y value.1230

We know that 1√10 is on this graph.1234

This is –x³, sorry.1240

-1³ - 2 × 1² + 1/2 y which is √10² is equal to c.1245

Here what we end up with is c is equal to 2.1263

If c is equal to 2, let us plug that back in.1270

Now we know that –x³ - 2x² + ½ y² is equal to 2.1274

We want to know what y is, when x is 0, just plug this in.1286

I get 0 - 0 + ½ y² is equal to 2.1291

y² is equal to 4.1301

y is equal to + or -2.1304

But they said early on, y is greater than 0 so that means that y = 2.1307

I took the implicit derivative formula and I worked my way backwards integrating to the original function.1316

I hope that made sense, this choice was a.1322

Let us see, that was number 21.1329

We are almost done with this section.1332

We are actually moving along pretty well.1334

This is 22, what is 22 asking us to do?1335

It looks like it is asking us to evaluate,1340

Excuse me, let me turn the page here.1343

I have difficulty turning these things.1345

It wants us to evaluate an integral.1348

Let us evaluate the integral from 1 to 2 of 1/ 4 - t², all under the radical.1350

Again, this is just a question of recognizing antiderivatives and doing a little manipulation.1362

I’m going to rewrite this as.1368

I think I want to go back to blue, I like it better.1370

The integral from 1 to 2, I’m going to factor out a √4 in the bottom, 1 - t²/ 4.1373

Let me write this a little bit better.1388

It is going to be 1/ √4 × √1 - t²/ 4.1393

I hope that makes sense that this is this.1402

I factored out a √4 under the radical sign.1404

I just pulled it out of the radical sign, because a radical × a radical is equal to a radical.1408

All I’m doing is a little bit of mathematical manipulation.1414

This 1/ √4 comes out as ½ × the integral 1 to 2, 1/ 1 -, t²/ 4 is the same as t/2² under the radical dt.1416

Now I do a u substitution, u = t/2, du = ½ dt, dt = 2 du.1439

Therefore, this integral is equal to ½ × the integral from 1 to 2 of 1/ 1 - u² × 2 du which is equal to the integral from 1 to 2.1456

This is 2/2 which turns into 1, 1/ 1 - u² du.1485

I recognize this as the inv sin(u), from 1 to 2.1494

I plug u back in as the inv sin of t/2, from 1 to 2 = the inv sin of, when I put 2 in there,1503

it is going to be the inv sin of 1 - the inverse sin of ½.1516

The inverse sin of 1 is π/2, the inverse sign of 1/2 is π/6.1523

I'm left with π/3 which is choice a.1532

That is it, just a little bit of manipulation to turn it into an antiderivative that we recognize.1538

In this case, the inverse sin function.1543

That is 22, let us try number 23.1548

23 is asking us to evaluate an integral.1558

Again, this time it looks like an indefinite integral.1562

We have the integral e ⁺2x × √e ⁺x + 1 dx.1566

This is going to be a little strange.1580

I’m going to start with u = e ⁺x + 1.1582

Du = e ⁺x dx.1590

I’m also going to write dx is equal to du/ e ⁺x.1596

What this gives me is, when I plug these into that, I get the integral of e ⁺2x × √e ⁺x + 1, that is just u and dx.1604

dx is equal to du/ e ⁺x.1621

This is equal to e ⁺2x divided by e ⁺x.1628

I get the integral of e ⁺x × that u du.1634

I have the u, I have the du, I just need to take care what this e ⁺x is.1642

Let me write that down.1652

I need to get everything in terms of u, one variable.1655

I have taken care of that, I just need to take care of e ⁺x.1659

We said that u is equal to e ⁺x + 1.1664

U - 1 = e ⁺x which means x is equal to natlog of u – 1.1670

Therefore, this integral now becomes the integral,1684

I do not really need to do that.1694

Should I leave it as u⁻¹?1697

I do not need to do this part, that is not even necessary.1703

Again, I have got e ⁺x, now it is u⁻¹.1710

Now I have the integral of u⁻¹ × √u × du.1713

Perfect, this is great, this is equal to the integral of, this distribute.1720

This says u³/2 - u ^½.1725

du = 2/5 u⁵/2 - 2/3 u³/2 + c.1732

When we substitute back in what u is, we get that this integral is equal to 2/5 × e ⁺x + 1⁵/2 - 2/31754

× e ⁺x + 1, because u was e ⁺x + 1³/2 + c.1772

This is choice e.1779

That is it, nice application.1780

We set the mess around with that e ⁺x.1783

We can work one function at a time.1785

Whenever you are doing the u substitution, everything has to be changed into u.1787

You may not be able do it in one step, you may have to take that extra step.1791

Number 24, let us see what does number 24 asks us.1797

Let us come over here.1806

Number 24, we are given an acceleration and we are given an initial position of velocity.1809

What is the position of the particle at another time?1818

I’m given acceleration a ⁺t is equal to 12t + 4.1824

They are telling me that my initial position x(0) is equal to 2.1831

They are telling me that my velocity at 1 is equal to 5.1835

They want to know what is my position at time t = 2.1841

We know that if you are given that position graph, the first derivative is velocity, the second derivative is acceleration.1850

If you are given an acceleration, integrate once, you get velocity.1855

Integrate twice, you get position.1859

v(t) or velocity of t = the integral of a(t) dt which is equal to the integral of 12t + 4 dt, which is equal to 6t² + 4t + c.1862

That gives us our velocity, we have to find c.1891

We have an initial value.1893

V(1) which is equal to putting 1 in here.1899

You get 6 + 4 + c.1902

They tell me that v(1) is actually equal to 5.1906

I get c is equal to -5.1909

My velocity function v(t), I just put this -5 back into here.1912

I get 6t² + 4t – 5.1918

For f(t), f(t) is equal to the integral of v(t) which is equal to the integral of 6t² + 4t - 5 dt is equal to 2t³, not²,1924

2t³ + 2t² - 5t, + this time I will just call it d.1945

They are telling me that x(0) is equal to 2.1956

X(0) which is equal to 0 + 0 - 0 + d is equal to 2.1958

I get d is equal to 2.1968

Therefore, my function, my x(t) is equal to 2t³ + 2t² - 5t + 2.1976

Therefore, my function at t = 2, just plug the 2 in, is equal to 2 × 2³ + 2 × 2² - 5 × 2 + 2.1987

You end up with 16, e is our choice.2003

Very nice and straightforward.2008

What is number 25 asking us?2017

They ask us to determine an integral, combination of integrals here.2020

We have the integral from 0 to π/2 of the sin(3x) dx + the integral from 0 to π/6.2026

Is that correct, yes, π/6 of the cos(3x) dx.2040

Just straight evaluation of an integral.2048

Sin(3x) dx, this is going to be a u substitution.2052

Hopefully, you have done enough of these to recognize that,2055

when you are taking the trig of some constant × x, that constant comes out as fraction.2058

This is going to equal, the integral of sin x - cos x.2064

It is going to be – 1/3 of cos(3x), evaluated from 0 to π/2 + the integral of cos.2068

It is going to be + 1/3 × sin(3x), evaluated from 0 to π/6.2078

You are going to end up this, plug in these, this – that, this – that.2088

You are going to end up with 0 - a - 1/3 + 1/3 – 0.2092

You are going to end up with 2/3.2112

This is choice c.2115

That is it, straight application.2117

All you have to worry about is that.2119

The integral of sin of a(x) is equal to -1/ a cos of a(x).2121

The integral of cos of a(x) is equal to 1/a × sin of a(x).2133

When we do a u substitution, it will work out.2146

It will make itself clear.2149

Let us see number 25.2155

Let us see number 26, we are almost there.2157

Number 26, it is asking us to determine the derivative of this particular function at π/2.2161

We got f(x) is equal to cos(2x) - 2².2167

They want us to find f’ at π/2.2179

F’(x), this is just an application, a very careful application of the chain rule.2185

Easy, just you got to be careful, that is all.2189

We are going to get 3 × cos(2x) - 2² × the derivative of what is inside which is - sin(2x) – 22193

× the derivative of what is inside, × 2 = -6 × cos(2x) - 2² × sin(2x) – 2.2212

Now you just basically just plug π/2 in.2236

F’(π/2), here is -6 × cos(2) × π/2 - 2² × sin(2) × π/2 – 2.2238

I think if I'm not mistaken, the choice is c.2271

Number 27, we have come to the end of this first part.2278

What is number 27 asks you to do?2283

We are going to compute the derivative of something that is given to us, in terms of an integral.2290

f(x) is defined as the integral from 0 to x² of the natlog of t² + 1 dt.2296

We have learned from the fundamental theorem of calculus that whenever f(x) is defined as an integral from some constant to x,2312

really all you do is you are taking the derivative of an integral.2319

Since, the derivative of the integrals are inverse processes, all you are doing is getting rid of the integral sign.2323

Your answer would be have been ln(t)² + 1 or ln(x)² + 1.2327

However, this upper limit is not x, it is x².2332

Really, all you have to do is, remember, f’(x), everything is exactly the same.2337

It is going to be the ln of,2345

You are going to put this into here.2354

It is going to be x⁴ + 1.2357

You just have to remember, because this is an x, you have to multiply by the derivative of this thing.2361

The derivative of x is just 1. It would be just 1, normally.2368

But the derivative of x² is 2x.2371

It is literally that simple.2375

This is choice e, the only difference is that they put the x and the 2x in front.2380

Once again, here, just plug in this into here, that will give you the function.2386

But remember that, if this is not x, then you need to just multiply by the derivative of this thing.2394

It looks like we have one more, I apologize, there are 28 questions not 27.2403

Number 28, here it wants us to evaluate the derivative with respect to x of the ln of 2 - cos x.2412

That is one parenthesis, that is two parentheses, and a bracket.2429

This is just an application of chain rule, it is not a not a big deal.2432

f’(x) which is ddx is equal to the derivative of natlog is 1/ the argument.2436

This is the argument, it is going to be 1/ derivative of ln(u), it is 1/ u du dx.2449

1/ ln(2) - cos x × the derivative of the argument ×, the derivative of ln(2) - cos x is 1/ 2 - cos x2457

× the derivative of the argument which is the derivative of -cos x is sin x.2484

That is it, it is choice c.2493

Just have to watch the differentiation.2495

That takes care of Section 1, Part A, no calculator allowed.2499

Next lesson, we will start with section 1 part B, where the calculator is allowed on the multiple choice sections.2505

Thank you so much for joining us here at www.educator.com.2512

We will see you next time, bye.2514

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