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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

AP Practice Exam: Section 1, Part A No Calculator

Slide Duration:

Table of Contents

I. Limits and Derivatives
Overview & Slopes of Curves

42m 8s

Intro
0:00
Overview & Slopes of Curves
0:21
Differential and Integral
0:22
Fundamental Theorem of Calculus
6:36
Differentiation or Taking the Derivative
14:24
What Does the Derivative Mean and How do We Find it?
15:18
Example: f'(x)
19:24
Example: f(x) = sin (x)
29:16
General Procedure for Finding the Derivative of f(x)
37:33
More on Slopes of Curves

50m 53s

Intro
0:00
Slope of the Secant Line along a Curve
0:12
Slope of the Tangent Line to f(x) at a Particlar Point
0:13
Slope of the Secant Line along a Curve
2:59
Instantaneous Slope
6:51
Instantaneous Slope
6:52
Example: Distance, Time, Velocity
13:32
Instantaneous Slope and Average Slope
25:42
Slope & Rate of Change
29:55
Slope & Rate of Change
29:56
Example: Slope = 2
33:16
Example: Slope = 4/3
34:32
Example: Slope = 4 (m/s)
39:12
Example: Density = Mass / Volume
40:33
Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change
47:46
Example Problems for Slopes of Curves

59m 12s

Intro
0:00
Example I: Water Tank
0:13
Part A: Which is the Independent Variable and Which is the Dependent?
2:00
Part B: Average Slope
3:18
Part C: Express These Slopes as Rates-of-Change
9:28
Part D: Instantaneous Slope
14:54
Example II: y = √(x-3)
28:26
Part A: Calculate the Slope of the Secant Line
30:39
Part B: Instantaneous Slope
41:26
Part C: Equation for the Tangent Line
43:59
Example III: Object in the Air
49:37
Part A: Average Velocity
50:37
Part B: Instantaneous Velocity
55:30
Desmos Tutorial

18m 43s

Intro
0:00
Desmos Tutorial
1:42
Desmos Tutorial
1:43
Things You Must Learn To Do on Your Particular Calculator
2:39
Things You Must Learn To Do on Your Particular Calculator
2:40
Example I: y=sin x
4:54
Example II: y=x³ and y = d/(dx) (x³)
9:22
Example III: y = x² {-5 <= x <= 0} and y = cos x {0 < x < 6}
13:15
The Limit of a Function

51m 53s

Intro
0:00
The Limit of a Function
0:14
The Limit of a Function
0:15
Graph: Limit of a Function
12:24
Table of Values
16:02
lim x→a f(x) Does not Say What Happens When x = a
20:05
Example I: f(x) = x²
24:34
Example II: f(x) = 7
27:05
Example III: f(x) = 4.5
30:33
Example IV: f(x) = 1/x
34:03
Example V: f(x) = 1/x²
36:43
The Limit of a Function, Cont.
38:16
Infinity and Negative Infinity
38:17
Does Not Exist
42:45
Summary
46:48
Example Problems for the Limit of a Function

24m 43s

Intro
0:00
Example I: Explain in Words What the Following Symbols Mean
0:10
Example II: Find the Following Limit
5:21
Example III: Use the Graph to Find the Following Limits
7:35
Example IV: Use the Graph to Find the Following Limits
11:48
Example V: Sketch the Graph of a Function that Satisfies the Following Properties
15:25
Example VI: Find the Following Limit
18:44
Example VII: Find the Following Limit
20:06
Calculating Limits Mathematically

53m 48s

Intro
0:00
Plug-in Procedure
0:09
Plug-in Procedure
0:10
Limit Laws
9:14
Limit Law 1
10:05
Limit Law 2
10:54
Limit Law 3
11:28
Limit Law 4
11:54
Limit Law 5
12:24
Limit Law 6
13:14
Limit Law 7
14:38
Plug-in Procedure, Cont.
16:35
Plug-in Procedure, Cont.
16:36
Example I: Calculating Limits Mathematically
20:50
Example II: Calculating Limits Mathematically
27:37
Example III: Calculating Limits Mathematically
31:42
Example IV: Calculating Limits Mathematically
35:36
Example V: Calculating Limits Mathematically
40:58
Limits Theorem
44:45
Limits Theorem 1
44:46
Limits Theorem 2: Squeeze Theorem
46:34
Example VI: Calculating Limits Mathematically
49:26
Example Problems for Calculating Limits Mathematically

21m 22s

Intro
0:00
Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law
0:16
Example II: Evaluate the Following Limit
1:51
Example III: Evaluate the Following Limit
3:36
Example IV: Evaluate the Following Limit
8:56
Example V: Evaluate the Following Limit
11:19
Example VI: Calculating Limits Mathematically
13:19
Example VII: Calculating Limits Mathematically
14:59
Calculating Limits as x Goes to Infinity

50m 1s

Intro
0:00
Limit as x Goes to Infinity
0:14
Limit as x Goes to Infinity
0:15
Let's Look at f(x) = 1 / (x-3)
1:04
Summary
9:34
Example I: Calculating Limits as x Goes to Infinity
12:16
Example II: Calculating Limits as x Goes to Infinity
21:22
Example III: Calculating Limits as x Goes to Infinity
24:10
Example IV: Calculating Limits as x Goes to Infinity
36:00
Example Problems for Limits at Infinity

36m 31s

Intro
0:00
Example I: Calculating Limits as x Goes to Infinity
0:14
Example II: Calculating Limits as x Goes to Infinity
3:27
Example III: Calculating Limits as x Goes to Infinity
8:11
Example IV: Calculating Limits as x Goes to Infinity
14:20
Example V: Calculating Limits as x Goes to Infinity
20:07
Example VI: Calculating Limits as x Goes to Infinity
23:36
Continuity

53m

Intro
0:00
Definition of Continuity
0:08
Definition of Continuity
0:09
Example: Not Continuous
3:52
Example: Continuous
4:58
Example: Not Continuous
5:52
Procedure for Finding Continuity
9:45
Law of Continuity
13:44
Law of Continuity
13:45
Example I: Determining Continuity on a Graph
15:55
Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous
17:57
Example III: Is the Following Function Continuous at the Given Point?
22:42
Theorem for Composite Functions
25:28
Theorem for Composite Functions
25:29
Example IV: Is cos(x³ + ln x) Continuous at x=π/2?
27:00
Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain?
34:04
Types of Discontinuity
39:18
Removable Discontinuity
39:33
Jump Discontinuity
40:06
Infinite Discontinuity
40:32
Intermediate Value Theorem
40:58
Intermediate Value Theorem: Hypothesis & Conclusion
40:59
Intermediate Value Theorem: Graphically
43:40
Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]
47:46
Derivative I

40m 2s

Intro
0:00
Derivative
0:09
Derivative
0:10
Example I: Find the Derivative of f(x)=x³
2:20
Notations for the Derivative
7:32
Notations for the Derivative
7:33
Derivative & Rate of Change
11:14
Recall the Rate of Change
11:15
Instantaneous Rate of Change
17:04
Graphing f(x) and f'(x)
19:10
Example II: Find the Derivative of x⁴ - x²
24:00
Example III: Find the Derivative of f(x)=√x
30:51
Derivatives II

53m 45s

Intro
0:00
Example I: Find the Derivative of (2+x)/(3-x)
0:18
Derivatives II
9:02
f(x) is Differentiable if f'(x) Exists
9:03
Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other
17:19
Geometrically: Differentiability Means the Graph is Smooth
18:44
Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=0
20:53
Example II: For x > 0
23:53
Example II: For x < 0
25:36
Example II: What is f(0) and What is the lim |x| as x→0?
30:46
Differentiability & Continuity
34:22
Differentiability & Continuity
34:23
How Can a Function Not be Differentiable at a Point?
39:38
How Can a Function Not be Differentiable at a Point?
39:39
Higher Derivatives
41:58
Higher Derivatives
41:59
Derivative Operator
45:12
Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³
49:29
More Example Problems for The Derivative

31m 38s

Intro
0:00
Example I: Sketch f'(x)
0:10
Example II: Sketch f'(x)
2:14
Example III: Find the Derivative of the Following Function sing the Definition
3:49
Example IV: Determine f, f', and f'' on a Graph
12:43
Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value
13:40
Example VI: Distance vs. Time
20:15
Example VII: Displacement, Velocity, and Acceleration
23:56
Example VIII: Graph the Displacement Function
28:20
II. Differentiation
Differentiation of Polynomials & Exponential Functions

47m 35s

Intro
0:00
Differentiation of Polynomials & Exponential Functions
0:15
Derivative of a Function
0:16
Derivative of a Constant
2:35
Power Rule
3:08
If C is a Constant
4:19
Sum Rule
5:22
Exponential Functions
6:26
Example I: Differentiate
7:45
Example II: Differentiate
12:38
Example III: Differentiate
15:13
Example IV: Differentiate
16:20
Example V: Differentiate
19:19
Example VI: Find the Equation of the Tangent Line to a Function at a Given Point
12:18
Example VII: Find the First & Second Derivatives
25:59
Example VIII
27:47
Part A: Find the Velocity & Acceleration Functions as Functions of t
27:48
Part B: Find the Acceleration after 3 Seconds
30:12
Part C: Find the Acceleration when the Velocity is 0
30:53
Part D: Graph the Position, Velocity, & Acceleration Graphs
32:50
Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents
34:53
Example X: Find a Point on a Graph
42:31
The Product, Power & Quotient Rules

47m 25s

Intro
0:00
The Product, Power and Quotient Rules
0:19
Differentiate Functions
0:20
Product Rule
5:30
Quotient Rule
9:15
Power Rule
10:00
Example I: Product Rule
13:48
Example II: Quotient Rule
16:13
Example III: Power Rule
18:28
Example IV: Find dy/dx
19:57
Example V: Find dy/dx
24:53
Example VI: Find dy/dx
28:38
Example VII: Find an Equation for the Tangent to the Curve
34:54
Example VIII: Find d²y/dx²
38:08
Derivatives of the Trigonometric Functions

41m 8s

Intro
0:00
Derivatives of the Trigonometric Functions
0:09
Let's Find the Derivative of f(x) = sin x
0:10
Important Limits to Know
4:59
d/dx (sin x)
6:06
d/dx (cos x)
6:38
d/dx (tan x)
6:50
d/dx (csc x)
7:02
d/dx (sec x)
7:15
d/dx (cot x)
7:27
Example I: Differentiate f(x) = x² - 4 cos x
7:56
Example II: Differentiate f(x) = x⁵ tan x
9:04
Example III: Differentiate f(x) = (cos x) / (3 + sin x)
10:56
Example IV: Differentiate f(x) = e^x / (tan x - sec x)
14:06
Example V: Differentiate f(x) = (csc x - 4) / (cot x)
15:37
Example VI: Find an Equation of the Tangent Line
21:48
Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent?
25:17
Example VIII: Ladder Problem
28:23
Example IX: Evaluate
33:22
Example X: Evaluate
36:38
The Chain Rule

24m 56s

Intro
0:00
The Chain Rule
0:13
Recall the Composite Functions
0:14
Derivatives of Composite Functions
1:34
Example I: Identify f(x) and g(x) and Differentiate
6:41
Example II: Identify f(x) and g(x) and Differentiate
9:47
Example III: Differentiate
11:03
Example IV: Differentiate f(x) = -5 / (x² + 3)³
12:15
Example V: Differentiate f(x) = cos(x² + c²)
14:35
Example VI: Differentiate f(x) = cos⁴x +c²
15:41
Example VII: Differentiate
17:03
Example VIII: Differentiate f(x) = sin(tan x²)
19:01
Example IX: Differentiate f(x) = sin(tan² x)
21:02
More Chain Rule Example Problems

25m 32s

Intro
0:00
Example I: Differentiate f(x) = sin(cos(tanx))
0:38
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
2:25
Example III: F(x) = f(g(x)), Find F' (6)
4:22
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
5:35
Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
10:18
Example VI: Differentiate f(x) = sec²(12x)
12:28
Example VII: Differentiate
14:41
Example VIII: Differentiate
19:25
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time
21:13
Implicit Differentiation

52m 31s

Intro
0:00
Implicit Differentiation
0:09
Implicit Differentiation
0:10
Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y
12:15
Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14
19:18
Example III: Find (dy)/(dx) of x³y² + y³x² = 4x
21:43
Example IV: Find (dy)/(dx) of the Following Equation
24:13
Example V: Find (dy)/(dx) of 6sin x cos y = 1
29:00
Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y
31:02
Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x
37:36
Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)
41:03
Example IX: Find (d²y)/(dx²) of x² + y² = 25
44:05
Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)
47:48
III. Applications of the Derivative
Linear Approximations & Differentials

47m 34s

Intro
0:00
Linear Approximations & Differentials
0:09
Linear Approximations & Differentials
0:10
Example I: Linear Approximations & Differentials
11:27
Example II: Linear Approximations & Differentials
20:19
Differentials
30:32
Differentials
30:33
Example III: Linear Approximations & Differentials
34:09
Example IV: Linear Approximations & Differentials
35:57
Example V: Relative Error
38:46
Related Rates

45m 33s

Intro
0:00
Related Rates
0:08
Strategy for Solving Related Rates Problems #1
0:09
Strategy for Solving Related Rates Problems #2
1:46
Strategy for Solving Related Rates Problems #3
2:06
Strategy for Solving Related Rates Problems #4
2:50
Strategy for Solving Related Rates Problems #5
3:38
Example I: Radius of a Balloon
5:15
Example II: Ladder
12:52
Example III: Water Tank
19:08
Example IV: Distance between Two Cars
29:27
Example V: Line-of-Sight
36:20
More Related Rates Examples

37m 17s

Intro
0:00
Example I: Shadow
0:14
Example II: Particle
4:45
Example III: Water Level
10:28
Example IV: Clock
20:47
Example V: Distance between a House and a Plane
29:11
Maximum & Minimum Values of a Function

40m 44s

Intro
0:00
Maximum & Minimum Values of a Function, Part 1
0:23
Absolute Maximum
2:20
Absolute Minimum
2:52
Local Maximum
3:38
Local Minimum
4:26
Maximum & Minimum Values of a Function, Part 2
6:11
Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min
7:18
Function with Local Max & Min but No Absolute Max & Min
8:48
Formal Definitions
10:43
Absolute Maximum
11:18
Absolute Minimum
12:57
Local Maximum
14:37
Local Minimum
16:25
Extreme Value Theorem
18:08
Theorem: f'(c) = 0
24:40
Critical Number (Critical Value)
26:14
Procedure for Finding the Critical Values of f(x)
28:32
Example I: Find the Critical Values of f(x) x + sinx
29:51
Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π]
35:31
Example Problems for Max & Min

40m 44s

Intro
0:00
Example I: Identify Absolute and Local Max & Min on the Following Graph
0:11
Example II: Sketch the Graph of a Continuous Function
3:11
Example III: Sketch the Following Graphs
4:40
Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²
6:13
Example V: Find the Critical Values of f(x) = |2x - 5|
8:42
Example VI: Find the Critical Values
11:42
Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π]
16:57
Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)]
20:08
Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3]
24:39
The Mean Value Theorem

25m 54s

Intro
0:00
Rolle's Theorem
0:08
Rolle's Theorem: If & Then
0:09
Rolle's Theorem: Geometrically
2:06
There May Be More than 1 c Such That f'( c ) = 0
3:30
Example I: Rolle's Theorem
4:58
The Mean Value Theorem
9:12
The Mean Value Theorem: If & Then
9:13
The Mean Value Theorem: Geometrically
11:07
Example II: Mean Value Theorem
13:43
Example III: Mean Value Theorem
21:19
Using Derivatives to Graph Functions, Part I

25m 54s

Intro
0:00
Using Derivatives to Graph Functions, Part I
0:12
Increasing/ Decreasing Test
0:13
Example I: Find the Intervals Over Which the Function is Increasing & Decreasing
3:26
Example II: Find the Local Maxima & Minima of the Function
19:18
Example III: Find the Local Maxima & Minima of the Function
31:39
Using Derivatives to Graph Functions, Part II

44m 58s

Intro
0:00
Using Derivatives to Graph Functions, Part II
0:13
Concave Up & Concave Down
0:14
What Does This Mean in Terms of the Derivative?
6:14
Point of Inflection
8:52
Example I: Graph the Function
13:18
Example II: Function x⁴ - 5x²
19:03
Intervals of Increase & Decrease
19:04
Local Maxes and Mins
25:01
Intervals of Concavity & X-Values for the Points of Inflection
29:18
Intervals of Concavity & Y-Values for the Points of Inflection
34:18
Graphing the Function
40:52
Example Problems I

49m 19s

Intro
0:00
Example I: Intervals, Local Maxes & Mins
0:26
Example II: Intervals, Local Maxes & Mins
5:05
Example III: Intervals, Local Maxes & Mins, and Inflection Points
13:40
Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
23:02
Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity
34:36
Example Problems III

59m 1s

Intro
0:00
Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
0:11
Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
21:24
Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D
37:56
Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes
46:19
L'Hospital's Rule

30m 9s

Intro
0:00
L'Hospital's Rule
0:19
Indeterminate Forms
0:20
L'Hospital's Rule
3:38
Example I: Evaluate the Following Limit Using L'Hospital's Rule
8:50
Example II: Evaluate the Following Limit Using L'Hospital's Rule
10:30
Indeterminate Products
11:54
Indeterminate Products
11:55
Example III: L'Hospital's Rule & Indeterminate Products
13:57
Indeterminate Differences
17:00
Indeterminate Differences
17:01
Example IV: L'Hospital's Rule & Indeterminate Differences
18:57
Indeterminate Powers
22:20
Indeterminate Powers
22:21
Example V: L'Hospital's Rule & Indeterminate Powers
25:13
Example Problems for L'Hospital's Rule

38m 14s

Intro
0:00
Example I: Evaluate the Following Limit
0:17
Example II: Evaluate the Following Limit
2:45
Example III: Evaluate the Following Limit
6:54
Example IV: Evaluate the Following Limit
8:43
Example V: Evaluate the Following Limit
11:01
Example VI: Evaluate the Following Limit
14:48
Example VII: Evaluate the Following Limit
17:49
Example VIII: Evaluate the Following Limit
20:37
Example IX: Evaluate the Following Limit
25:16
Example X: Evaluate the Following Limit
32:44
Optimization Problems I

49m 59s

Intro
0:00
Example I: Find the Dimensions of the Box that Gives the Greatest Volume
1:23
Fundamentals of Optimization Problems
18:08
Fundamental #1
18:33
Fundamental #2
19:09
Fundamental #3
19:19
Fundamental #4
20:59
Fundamental #5
21:55
Fundamental #6
23:44
Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square
24:36
Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0)
35:13
Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R
43:10
Optimization Problems II

55m 10s

Intro
0:00
Example I: Optimization Problem
0:13
Example II: Optimization Problem
17:34
Example III: Optimization Problem
35:06
Example IV: Revenue, Cost, and Profit
43:22
Newton's Method

30m 22s

Intro
0:00
Newton's Method
0:45
Newton's Method
0:46
Example I: Find x2 and x3
13:18
Example II: Use Newton's Method to Approximate
15:48
Example III: Find the Root of the Following Equation to 6 Decimal Places
19:57
Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point
23:11
IV. Integrals
Antiderivatives

55m 26s

Intro
0:00
Antiderivatives
0:23
Definition of an Antiderivative
0:24
Antiderivative Theorem
7:58
Function & Antiderivative
12:10
x^n
12:30
1/x
13:00
e^x
13:08
cos x
13:18
sin x
14:01
sec² x
14:11
secxtanx
14:18
1/√(1-x²)
14:26
1/(1+x²)
14:36
-1/√(1-x²)
14:45
Example I: Find the Most General Antiderivative for the Following Functions
15:07
Function 1: f(x) = x³ -6x² + 11x - 9
15:42
Function 2: f(x) = 14√(x) - 27 4√x
19:12
Function 3: (fx) = cos x - 14 sinx
20:53
Function 4: f(x) = (x⁵+2√x )/( x^(4/3) )
22:10
Function 5: f(x) = (3e^x) - 2/(1+x²)
25:42
Example II: Given the Following, Find the Original Function f(x)
26:37
Function 1: f'(x) = 5x³ - 14x + 24, f(2) = 40
27:55
Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 5
30:34
Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.2
32:54
Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 7
37:54
Example III: Falling Object
41:58
Problem 1: Find an Equation for the Height of the Ball after t Seconds
42:48
Problem 2: How Long Will It Take for the Ball to Strike the Ground?
48:30
Problem 3: What is the Velocity of the Ball as it Hits the Ground?
49:52
Problem 4: Initial Velocity of 6 m/s, How Long Does It Take to Reach the Ground?
50:46
The Area Under a Curve

51m 3s

Intro
0:00
The Area Under a Curve
0:13
Approximate Using Rectangles
0:14
Let's Do This Again, Using 4 Different Rectangles
9:40
Approximate with Rectangles
16:10
Left Endpoint
18:08
Right Endpoint
25:34
Left Endpoint vs. Right Endpoint
30:58
Number of Rectangles
34:08
True Area
37:36
True Area
37:37
Sigma Notation & Limits
43:32
When You Have to Explicitly Solve Something
47:56
Example Problems for Area Under a Curve

33m 7s

Intro
0:00
Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve
0:10
Example II: Using 5 Rectangles, Approximate the Area Under the Curve
11:32
Example III: Find the True Area by Evaluating the Limit Expression
16:07
Example IV: Find the True Area by Evaluating the Limit Expression
24:52
The Definite Integral

43m 19s

Intro
0:00
The Definite Integral
0:08
Definition to Find the Area of a Curve
0:09
Definition of the Definite Integral
4:08
Symbol for Definite Integral
8:45
Regions Below the x-axis
15:18
Associating Definite Integral to a Function
19:38
Integrable Function
27:20
Evaluating the Definite Integral
29:26
Evaluating the Definite Integral
29:27
Properties of the Definite Integral
35:24
Properties of the Definite Integral
35:25
Example Problems for The Definite Integral

32m 14s

Intro
0:00
Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
0:11
Example II: Express the Following Limit as a Definite Integral
5:28
Example III: Evaluate the Following Definite Integral Using the Definition
6:28
Example IV: Evaluate the Following Integral Using the Definition
17:06
Example V: Evaluate the Following Definite Integral by Using Areas
25:41
Example VI: Definite Integral
30:36
The Fundamental Theorem of Calculus

24m 17s

Intro
0:00
The Fundamental Theorem of Calculus
0:17
Evaluating an Integral
0:18
Lim as x → ∞
12:19
Taking the Derivative
14:06
Differentiation & Integration are Inverse Processes
15:04
1st Fundamental Theorem of Calculus
20:08
1st Fundamental Theorem of Calculus
20:09
2nd Fundamental Theorem of Calculus
22:30
2nd Fundamental Theorem of Calculus
22:31
Example Problems for the Fundamental Theorem

25m 21s

Intro
0:00
Example I: Find the Derivative of the Following Function
0:17
Example II: Find the Derivative of the Following Function
1:40
Example III: Find the Derivative of the Following Function
2:32
Example IV: Find the Derivative of the Following Function
5:55
Example V: Evaluate the Following Integral
7:13
Example VI: Evaluate the Following Integral
9:46
Example VII: Evaluate the Following Integral
12:49
Example VIII: Evaluate the Following Integral
13:53
Example IX: Evaluate the Following Graph
15:24
Local Maxs and Mins for g(x)
15:25
Where Does g(x) Achieve Its Absolute Max on [0,8]
20:54
On What Intervals is g(x) Concave Up/Down?
22:20
Sketch a Graph of g(x)
24:34
More Example Problems, Including Net Change Applications

34m 22s

Intro
0:00
Example I: Evaluate the Following Indefinite Integral
0:10
Example II: Evaluate the Following Definite Integral
0:59
Example III: Evaluate the Following Integral
2:59
Example IV: Velocity Function
7:46
Part A: Net Displacement
7:47
Part B: Total Distance Travelled
13:15
Example V: Linear Density Function
20:56
Example VI: Acceleration Function
25:10
Part A: Velocity Function at Time t
25:11
Part B: Total Distance Travelled During the Time Interval
28:38
Solving Integrals by Substitution

27m 20s

Intro
0:00
Table of Integrals
0:35
Example I: Evaluate the Following Indefinite Integral
2:02
Example II: Evaluate the Following Indefinite Integral
7:27
Example IIII: Evaluate the Following Indefinite Integral
10:57
Example IV: Evaluate the Following Indefinite Integral
12:33
Example V: Evaluate the Following
14:28
Example VI: Evaluate the Following
16:00
Example VII: Evaluate the Following
19:01
Example VIII: Evaluate the Following
21:49
Example IX: Evaluate the Following
24:34
V. Applications of Integration
Areas Between Curves

34m 56s

Intro
0:00
Areas Between Two Curves: Function of x
0:08
Graph 1: Area Between f(x) & g(x)
0:09
Graph 2: Area Between f(x) & g(x)
4:07
Is It Possible to Write as a Single Integral?
8:20
Area Between the Curves on [a,b]
9:24
Absolute Value
10:32
Formula for Areas Between Two Curves: Top Function - Bottom Function
17:03
Areas Between Curves: Function of y
17:49
What if We are Given Functions of y?
17:50
Formula for Areas Between Two Curves: Right Function - Left Function
21:48
Finding a & b
22:32
Example Problems for Areas Between Curves

42m 55s

Intro
0:00
Instructions for the Example Problems
0:10
Example I: y = 7x - x² and y=x
0:37
Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=2
6:25
Example III: y=(1/x), y=(1/x³), and x=4
12:25
Example IV: 15-2x² and y=x²-5
15:52
Example V: x=(1/8)y³ and x=6-y²
20:20
Example VI: y=cos x, y=sin(2x), [0,π/2]
24:34
Example VII: y=2x², y=10x², 7x+2y=10
29:51
Example VIII: Velocity vs. Time
33:23
Part A: At 2.187 Minutes, Which care is Further Ahead?
33:24
Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
36:32
Part C: At 4 Minutes Which Car is Ahead?
37:11
Part D: At What Time Will the Cars be Side by Side?
37:50
Volumes I: Slices

34m 15s

Intro
0:00
Volumes I: Slices
0:18
Rotate the Graph of y=√x about the x-axis
0:19
How can I use Integration to Find the Volume?
3:16
Slice the Solid Like a Loaf of Bread
5:06
Volumes Definition
8:56
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
12:18
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
19:05
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation
25:28
Volumes II: Volumes by Washers

51m 43s

Intro
0:00
Volumes II: Volumes by Washers
0:11
Rotating Region Bounded by y=x³ & y=x around the x-axis
0:12
Equation for Volumes by Washer
11:14
Process for Solving Volumes by Washer
13:40
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
15:58
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
25:07
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
34:20
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
44:05
Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

Intro
0:00
Solids That Are Not Solids-of-Revolution
0:11
Cross-Section Area Review
0:12
Cross-Sections That Are Not Solids-of-Revolution
7:36
Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H
10:54
Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles
20:39
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H
29:27
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64
36:47
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis
46:13
Volumes IV: Volumes By Cylindrical Shells

50m 2s

Intro
0:00
Volumes by Cylindrical Shells
0:11
Find the Volume of the Following Region
0:12
Volumes by Cylindrical Shells: Integrating Along x
14:12
Volumes by Cylindrical Shells: Integrating Along y
14:40
Volumes by Cylindrical Shells Formulas
16:22
Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid
18:33
Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid
25:57
Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid
31:38
Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid
38:44
Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid
44:03
The Average Value of a Function

32m 13s

Intro
0:00
The Average Value of a Function
0:07
Average Value of f(x)
0:08
What if The Domain of f(x) is Not Finite?
2:23
Let's Calculate Average Value for f(x) = x² [2,5]
4:46
Mean Value Theorem for Integrate
9:25
Example I: Find the Average Value of the Given Function Over the Given Interval
14:06
Example II: Find the Average Value of the Given Function Over the Given Interval
18:25
Example III: Find the Number A Such that the Average Value of the Function f(x) = -4x² + 8x + 4 Equals 2 Over the Interval [-1,A]
24:04
Example IV: Find the Average Density of a Rod
27:47
VI. Techniques of Integration
Integration by Parts

50m 32s

Intro
0:00
Integration by Parts
0:08
The Product Rule for Differentiation
0:09
Integrating Both Sides Retains the Equality
0:52
Differential Notation
2:24
Example I: ∫ x cos x dx
5:41
Example II: ∫ x² sin(2x)dx
12:01
Example III: ∫ (e^x) cos x dx
18:19
Example IV: ∫ (sin^-1) (x) dx
23:42
Example V: ∫₁⁵ (lnx)² dx
28:25
Summary
32:31
Tabular Integration
35:08
Case 1
35:52
Example: ∫x³sinx dx
36:39
Case 2
40:28
Example: ∫e^(2x) sin 3x
41:14
Trigonometric Integrals I

24m 50s

Intro
0:00
Example I: ∫ sin³ (x) dx
1:36
Example II: ∫ cos⁵(x)sin²(x)dx
4:36
Example III: ∫ sin⁴(x)dx
9:23
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx
15:59
#1: Power of sin is Odd
16:00
#2: Power of cos is Odd
16:41
#3: Powers of Both sin and cos are Odd
16:55
#4: Powers of Both sin and cos are Even
17:10
Example IV: ∫ tan⁴ (x) sec⁴ (x) dx
17:34
Example V: ∫ sec⁹(x) tan³(x) dx
20:55
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx
23:31
#1: Power of sec is Odd
23:32
#2: Power of tan is Odd
24:04
#3: Powers of sec is Odd and/or Power of tan is Even
24:18
Trigonometric Integrals II

22m 12s

Intro
0:00
Trigonometric Integrals II
0:09
Recall: ∫tanx dx
0:10
Let's Find ∫secx dx
3:23
Example I: ∫ tan⁵ (x) dx
6:23
Example II: ∫ sec⁵ (x) dx
11:41
Summary: How to Deal with Integrals of Different Types
19:04
Identities to Deal with Integrals of Different Types
19:05
Example III: ∫cos(5x)sin(9x)dx
19:57
More Example Problems for Trigonometric Integrals

17m 22s

Intro
0:00
Example I: ∫sin²(x)cos⁷(x)dx
0:14
Example II: ∫x sin²(x) dx
3:56
Example III: ∫csc⁴ (x/5)dx
8:39
Example IV: ∫( (1-tan²x)/(sec²x) ) dx
11:17
Example V: ∫ 1 / (sinx-1) dx
13:19
Integration by Partial Fractions I

55m 12s

Intro
0:00
Integration by Partial Fractions I
0:11
Recall the Idea of Finding a Common Denominator
0:12
Decomposing a Rational Function to Its Partial Fractions
4:10
2 Types of Rational Function: Improper & Proper
5:16
Improper Rational Function
7:26
Improper Rational Function
7:27
Proper Rational Function
11:16
Proper Rational Function & Partial Fractions
11:17
Linear Factors
14:04
Irreducible Quadratic Factors
15:02
Case 1: G(x) is a Product of Distinct Linear Factors
17:10
Example I: Integration by Partial Fractions
20:33
Case 2: D(x) is a Product of Linear Factors
40:58
Example II: Integration by Partial Fractions
44:41
Integration by Partial Fractions II

42m 57s

Intro
0:00
Case 3: D(x) Contains Irreducible Factors
0:09
Example I: Integration by Partial Fractions
5:19
Example II: Integration by Partial Fractions
16:22
Case 4: D(x) has Repeated Irreducible Quadratic Factors
27:30
Example III: Integration by Partial Fractions
30:19
VII. Differential Equations
Introduction to Differential Equations

46m 37s

Intro
0:00
Introduction to Differential Equations
0:09
Overview
0:10
Differential Equations Involving Derivatives of y(x)
2:08
Differential Equations Involving Derivatives of y(x) and Function of y(x)
3:23
Equations for an Unknown Number
6:28
What are These Differential Equations Saying?
10:30
Verifying that a Function is a Solution of the Differential Equation
13:00
Verifying that a Function is a Solution of the Differential Equation
13:01
Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation
17:20
General Solution
22:00
Particular Solution
24:36
Initial Value Problem
27:42
Example I: Verify that a Family of Functions is a Solution of the Differential Equation
32:24
Example II: For What Values of K Does the Function Satisfy the Differential Equation
36:07
Example III: Verify the Solution and Solve the Initial Value Problem
39:47
Separation of Variables

28m 8s

Intro
0:00
Separation of Variables
0:28
Separation of Variables
0:29
Example I: Solve the Following g Initial Value Problem
8:29
Example II: Solve the Following g Initial Value Problem
13:46
Example III: Find an Equation of the Curve
18:48
Population Growth: The Standard & Logistic Equations

51m 7s

Intro
0:00
Standard Growth Model
0:30
Definition of the Standard/Natural Growth Model
0:31
Initial Conditions
8:00
The General Solution
9:16
Example I: Standard Growth Model
10:45
Logistic Growth Model
18:33
Logistic Growth Model
18:34
Solving the Initial Value Problem
25:21
What Happens When t → ∞
36:42
Example II: Solve the Following g Initial Value Problem
41:50
Relative Growth Rate
46:56
Relative Growth Rate
46:57
Relative Growth Rate Version for the Standard model
49:04
Slope Fields

24m 37s

Intro
0:00
Slope Fields
0:35
Slope Fields
0:36
Graphing the Slope Fields, Part 1
11:12
Graphing the Slope Fields, Part 2
15:37
Graphing the Slope Fields, Part 3
17:25
Steps to Solving Slope Field Problems
20:24
Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y
22:38
VIII. AP Practic Exam
AP Practice Exam: Section 1, Part A No Calculator

45m 29s

Intro
0:00
Exam Link
0:10
Problem #1
1:26
Problem #2
2:52
Problem #3
4:42
Problem #4
7:03
Problem #5
10:01
Problem #6
13:49
Problem #7
15:16
Problem #8
19:06
Problem #9
23:10
Problem #10
28:10
Problem #11
31:30
Problem #12
33:53
Problem #13
37:45
Problem #14
41:17
AP Practice Exam: Section 1, Part A No Calculator, cont.

41m 55s

Intro
0:00
Problem #15
0:22
Problem #16
3:10
Problem #17
5:30
Problem #18
8:03
Problem #19
9:53
Problem #20
14:51
Problem #21
17:30
Problem #22
22:12
Problem #23
25:48
Problem #24
29:57
Problem #25
33:35
Problem #26
35:57
Problem #27
37:57
Problem #28
40:04
AP Practice Exam: Section I, Part B Calculator Allowed

58m 47s

Intro
0:00
Problem #1
1:22
Problem #2
4:55
Problem #3
10:49
Problem #4
13:05
Problem #5
14:54
Problem #6
17:25
Problem #7
18:39
Problem #8
20:27
Problem #9
26:48
Problem #10
28:23
Problem #11
34:03
Problem #12
36:25
Problem #13
39:52
Problem #14
43:12
Problem #15
47:18
Problem #16
50:41
Problem #17
56:38
AP Practice Exam: Section II, Part A Calculator Allowed

25m 40s

Intro
0:00
Problem #1: Part A
1:14
Problem #1: Part B
4:46
Problem #1: Part C
8:00
Problem #2: Part A
12:24
Problem #2: Part B
16:51
Problem #2: Part C
17:17
Problem #3: Part A
18:16
Problem #3: Part B
19:54
Problem #3: Part C
21:44
Problem #3: Part D
22:57
AP Practice Exam: Section II, Part B No Calculator

31m 20s

Intro
0:00
Problem #4: Part A
1:35
Problem #4: Part B
5:54
Problem #4: Part C
8:50
Problem #4: Part D
9:40
Problem #5: Part A
11:26
Problem #5: Part B
13:11
Problem #5: Part C
15:07
Problem #5: Part D
19:57
Problem #6: Part A
22:01
Problem #6: Part B
25:34
Problem #6: Part C
28:54
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Lecture Comments (7)

0 answers

Post by Magic Fu on July 7 at 12:54:44 AM

Thank You so much!!!!
My textbook sucks, it has zero practice problems and examples.
Lucky, I found your AP Calc AB videos, and I got 5 on my Calc BC and Calc AB exams.


(Also, I took AP Chemistry exam last year, but I ended up with a 4, which I am still very salty about that).

0 answers

Post by R K on April 23, 2017

For number 7, since it is u substitution, shouldn't we change the bounds?

1 answer

Last reply by: Professor Hovasapian
Mon Jul 25, 2016 7:19 PM

Post by Peter Ke on July 23, 2016

For Problem 5, shouldn't the derivative of-4y^2 be -8y and not -8yy' because you bring down the 2 and multiply it by -4 which is -8 and subtract the exponent by 1?

2 answers

Last reply by: R K
Sun Apr 23, 2017 11:41 PM

Post by nathan lau on May 5, 2016

for question number 9 I keep getting letter b do you know what I could be doing wrong?

AP Practice Exam: Section 1, Part A No Calculator

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
  • 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

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

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

Today, we are going to start on the AP practice exam.0004

What I would like to do, the practice exam that we are going to use is going to be a 2006 version.0009

You can find it on, I will write the link here.0015

The link is also going to be down at the bottom of the page in the quick notes.0019

www.online.math.uh.edu/apcalculus/exams.0024

When you enter that or when you click on the link down below,0044

the page will come up and it is going to give you a bunch of different versions.0048

The version that we are going to use is going to be version 5.0051

Just click on version 5 for part A, the section on part A.0054

We are also going to be doing when we get to, it is going to be version 5 for the part B.0068

For the written part, the free response questions, we are going to go with version 2.0073

You can either pull it up, head on your screen for you.0079

If you want, you can print it out, whichever you prefer.0082

Let us jump right on in.0086

Question number 1, we have got a couple of functions here.0088

We have f is equal to -2x + 1 and our g(x), I will write it as g = -x/ x² + 1.0096

In this particular case, we are to find f of g(1).0109

Nice and straightforward composition of functions.0120

First thing we do is we find the g(1).0123

g(1) is equal to -1/ 1² + 1 = -½.0125

f of g(1) is equal to f(-½).0138

We have -2 × -1/2 + 1.0145

This is 1 + 1 is equal to 2.0152

In this particular case, our answer is c.0156

Question number 2, I think I will do this on the next page.0164

You know what, I think I’m going to work in blue.0170

Question number 2, we have f(x) is equal to -x² – 4√x and we want f’ at 4.0173

Take the derivative, plug in 4, and see what it is that you actually get, the slope of the tangent line.0194

The slope of the tangent line is the value of the derivative at that point.0205

f(x) is that, f’(x), what do we get for f’(x)?0215

The derivative of this is -2x and this is going to be -4 × 1/2 × x⁻¹/2 = -2x - 2/ √x.0222

f’ at 4 is -2 × 4 - 2/ √4.0245

This is equal to -8 – 1, we have -9.0255

Therefore, in this particular case, the answer is going to be c.0261

Let us see what we have got, let me move this a little bit over here.0270

Question number 3, question number 3 is asking us to evaluate,0278

they want the limit as x goes to infinity of 2x³ + 4x/ -2x⁵ + x² – 2.0286

To evaluate this particular limit, a couple ways that we can do it.0309

One of the systematic way, whenever you are dealing with a rational function is to divide the top and bottom,0314

the numerator and the denominator by the highest degree in the denominator.0322

Basically, divide everything on the top by x⁵ and divide everything in the bottom by x⁵.0326

And then, take the limit as x goes to infinity.0331

Let us go ahead and do that.0334

Let us go ahead and bring this over here.0336

This is going to be dealing with a functional.0338

I do not want to keep writing limit symbol over and over again.0340

When I divide the top, it is going to be 2x³/ x⁵ - 4x/ x⁵.0343

On the bottom, I’m going to get -2x⁵/ x⁵ + x²/ x⁵ - 2/ x⁵.0353

This becomes 2/ x² + 4/ x⁴/ -2 + 1/ x³, I think -2/ x⁵.0365

Now we take the limit as x goes to infinity of this function.0385

As x goes to infinity, this goes to 0 because this becomes really big, therefore, this becomes really small, it goes to 0.0394

This goes to 0, this stays -2, goes to 0, goes to 0.0403

What you are left with is 0/ -2 which is equal to 0.0409

In this particular case, the answer will be a.0414

That is it, nice and straightforward limit.0418

Nothing particularly difficult about it.0421

Question number 4, let us see what question number 4 is asking us.0425

We have a particular function and it is bounded by certain numbers, certain functions.0434

It wants you to express the integral that gives you the area of the particular region that is bounded.0443

We have f(x), in this particular case is equal to x³.0454

Let us go ahead and we know what x³ looks like.0462

x³ looks like that, let us see, from -1 to 0.0470

They say that region 1, x goes from -1 to 0.0480

From -1 to 0, we are bounded by,0490

We have got, this is 1, this is -1.0508

We have this region and we have this region.0518

From -1 to 0, we call this our region 1.0523

From 0 to 1, this is our region 2.0527

Our total area is going to be the area of region 1 + the area of region 2.0534

This is fully symmetric here about the origin.0542

I can basically just take the integral of one of them and multiply it by 2.0545

I can just say it is 2 × the area of region 2.0549

It is equal to 2 × the integral from 0 to 1 of the upper function - the lower function.0555

It is going to be 1 - x³ dx.0565

As far as our choices are concerned, 0 to 1,2, just pull the constant in there, 1 - x³ dx.0573

It looks like our answer is c.0584

They give you a particular graph, they tell you the bounds, and you just have to find a particular area.0589

Let us see what number 5 says.0600

Question number 5 on this particular version.0604

We are given a particular function, we want you to determine the change in y with respect to x.0606

They want us to find the derivative dy dx.0613

We have 3x³ - 4xy – 4y² is equal to 1.0619

We see that in this particular case, our function is given implicitly.0632

We are going to use implicit differentiation.0636

Let us go ahead and do that.0639

The derivative of this with respect to x.0641

We are looking for dy dx.0643

The question asks, determine the change in y with respect to x.0645

Dy dx is what we are looking for.0651

The derivative of this is going to be 9x² – 4.0653

I just tend to pull my constants.0661

It is going to be 4 ×, this is a function of x and this is a function of x.0664

It is going to be this × the derivative of that.0669

x × y’ which is dy dx, just a shortened version, + y × the derivative of that which is 1.0671

- the derivative of this is -8yy’.0681

The derivative of 1 is 0.0686

We have got 9x² - 4xy’ - 4y – 8yy’ is equal to 0.0690

I have got 9x² - 4y, I’m going to put the y’ on one side = 4xy’ + 8yy’.0703

Factor out the y’, I got 9x² – 4y = y’ × 4x + 8y.0720

I’m left with y’ is equal to 9x² – 4y/ 4x + 8y.0733

As far as our choices are concerned, that is equivalent to 9x² - 4y/ - and -4 x is -8y.0756

The answer that you get, depending on how you did it, which you move to the left or the right,0776

maybe slightly different on the choices that you have.0781

What is probably going to be the most tricky part of this is,0784

just because you got something that does not look like any of your choices, it does not mean you are wrong.0788

Just see if you can change what you got and if it is equivalent to one of your choices.0792

In this case, it is equal to one of the choices.0798

It is going to be d.0801

Do not freak out, if what you got is going to be different.0805

Clearly, you figured out by now, having gone through calculus that three of you can do the problem and all get it right.0808

And three of you have different answers, that is not a problem because there is a lot of symbolism going on.0815

The symbolism is going to take different shapes, depending on the particular approach that was used to solve the problem.0820

That was number 5, let us take a look at number 6.0828

We have got f(x) = 4 × sec(x) - 3 × csc(x).0837

We are asked to find the derivative of this.0848

Very straightforward, as long as you know your derivative formulas, = 4.0851

The derivative of sec x is sec tan.0858

4 sec x tan x -, let us do it this way,0861

Let me put -3.0881

The derivative of csc x is –csc cot.0883

-csc x cot x, you end up with 4 sec x tan(x) + 3 csc x cot x.0888

In this particular case, our answer is b.0905

Just a straight application, just have to watch the sign, that is about it.0908

That was number 6, let us go ahead and see what number 7 has to offer.0913

We are asked to compute an integral here.0922

We have the integral from 0 to 1/4 of 16/ 1 + 16t² dt.0926

You remember some of your integral formulas, there was an integral formula0941

where the integral of 1/ 1 + x² dx was equal to the inv tan(x) + c.0945

This thing, then I'm going to rewrite, I’m going to pull the 16 out.0961

16 × the integral from 0 to 1/4 of 1/,0967

I’m going to write this as 1 + 4t².0984

I just change the 16t², I wrote it as (4t)² dt.0994

I’m just going to do a little bit of u substitution here.0999

I’m going to call u for t, therefore du = 4 dt.1003

Therefore, dt is equal to du/ 4.1011

What we get is 16.1019

This integral with the u substitution, we get 16, 0 to ¼, 1/ 1 + u² × dt is du/ 4.1023

I go ahead and pull the 4 out.1041

I can write this as 16/ 4 × the integral from 0 to 1/ 4 of + 1/ 1 + u² du,1043

which is equal to 4 × the inv tan of u which is equal to 4 × the inv tan or u is 4t.1057

From 0 to 1/4 which is equal to, I will go to the next page, not a problem.1077

Which is equal to 4 × the inv tan of 1 – inv tan of 0.1088

We get = 4 × the inv tan of 1 is π/4 -,1110

The inv tan of 0 is 0 = π.1121

Our answer is d, that is it.1130

A little bit of a recognition of what the integral formula is for 1/ 1 + x².1132

In this case, 1 + 1/ u², slight manipulation.1138

And then, use the u substitution to go ahead and take care of that.1141

Let us move on to number 8.1147

What is question number 8 asking us to do.1156

We have to determine the derivative of this particular expression.1159

ddx, 2x⁴ – 2x/ 2x⁴ + 2x.1168

What is the best way to approach this?1184

What is the best way to do this?1197

Okay, that is not a problem.1204

Before we actually start with the quotient rule, more than likely we are going to be using the quotient rule here.1205

Before we do that, let us see if there is something that we can actually do this function to make it a little easier on us.1210

Especially, when you take a look at the choices.1219

The choices have 2x³ + 2 in the denominator, except one of them which is 1 + x³.1222

Before we actually start taking the derivative, let us see if there is something we can do here.1230

This 2x⁴ - 2x/ 2x⁴ + 2x.1234

Let me factor out a 2x and I end up with x³ - 1, if I’m not mistaken.1247

I got a 2x here and I have got an x³ + 1 over here.1254

2x actually vanishes.1259

What I'm left with is x³ - 1/ x³ + 1.1261

Our f is the x³ - 1 and our g is our x³ + 1.1269

Now f’(x), the quotient rule is gf’ – fg’/ g².1274

We just have to work it out.1285

This is going to equal x³ + 1 × 3x² – fg’ - x³ - 1 × 3x²/ x³ + 1².1288

That is going to equal 3x⁵ + 3x² – 3x⁵ + 3x²/ x³ + 1².1310

3x⁵ and 3x⁵ goes away, that leaves us with a final answer of 6x²/ x³ + 1².1327

This happens to coercive with answer d.1339

That is about it, pretty straightforward.1342

If you just gone ahead and started doing that,1345

you are going to end up with a pretty complicated expression to have to simplify algebraically.1347

Would you have come up with the same answer, honestly,1354

I did not actually carry out that particular algebraic manipulation.1356

I took a look at the choices and I notice the x³.1362

I looked back at the function and thought can I factor it.1366

It turned out that I can.1369

Again, this is calculus we are dealing with.1372

There is a million ways to do something.1377

That is question number 8.1380

Let us go ahead and move on to question number 9 here.1382

How long is 9, that is not a problem.1389

Question number 9, in this particular case, we are given a function and1392

we are asked to find the equation of the normal line to the graph at a given point.1399

Our f(x) is equal to 3x × √x² + 6 – 3.1408

It is going to be at the point 0 – 3.1423

What we are going to do is we are going to find the slope of the tangent line which is just a derivative,1428

evaluated at a certain point.1439

The normal line, we know that the normal line is perpendicular to the tangent line.1441

All we are going to do is, the slope that we get for the tangent line, we are going to take the negative reciprocal of it.1455

That will give us the slope of the normal line, the slope of the normal line.1460

Because it is still passing through 0 and -3, we have the slope, we have a point.1468

We do y - y1 = m × x - x1.1472

We see the normal line is perpendicular the tangent line.1477

We take the negative reciprocal.1484

Let us go ahead and find f’(x) first.1490

F’(x), we got 3, we got a function of x × a function of x.1494

It is going to be a little long but not too big of a deal.1498

Again, I tend to take that out.1501

It is going to be this × the derivative of that + that × the derivative of this.1503

We are going to get x × 1/2 x² + 6⁻¹/2 × 2x + x² + 6 ^½ × 1.1509

That is going to equal 3 × x²/ √x² + 6 + √x² + 6.1528

We want to find f’ at 0, it is a function of x.1545

We are going to be using the 0 value.1549

That is going to equal 3 × 0/ √0 + 6 + √6.1551

We are going to end up with 3√6.1563

This is the slope of the tangent line, we want the negative reciprocal.1571

The normal line slope = -1/ 3√6.1576

Therefore, our line is equal to y - y1 - 3 = m which is -1/ 3√6 × x – 0.1587

Let us see if I can turn the page here.1605

I have a little difficulty getting to the next page.1607

We end up with y + 3 = -1/ 3√6 × x.1611

When you rearrange this, just multiply by 3√6.1625

Rearrange it to make it correspond with one of the choices.1628

Again, the answer that you got is not going to match one of the choices, but it is the same object.1631

You just have to rearrange it.1637

Rearranging to match one of the choices.1639

We get x – 3√6, y = 9√6 which is choice a.1641

It is going to happen quite a lot.1656

Your answer is going to be slightly different.1658

When you do the rearrangement, make sure you go very slowly.1660

In the choices that they give you, the differences are going to be very subtle.1663

There are going to be + and -, all the symbols are going to be there.1667

9, √6, 3 √6, just be very careful with your choices.1673

Make sure you look at all of your choices to make sure that you have an excluded one.1678

Just because you think you found the right choice, there may be something else going on.1682

Make sure you look at all of your choices.1685

Question number 10, let us see what we got for question number 10.1690

Here we want to find the concavity of a particular graph.1696

In this particular case, our function f(x) is equal to 2 sin(x) + 3 × cos² x.1702

We want to find the concavity at a given point.1716

Let us find and we know that concavity has to do with the second derivative.1722

Let us find f"(x) and evaluate f” at the point π, to see what the concavity of π is.1728

F’x, the derivative of this is going to be 2 × cos(x) + cos(x) – 6 sin x cos x.1747

That is one of our f’.1780

We want to do our f”.1784

F”(x), I will just take the derivative of what it is that we just got.1787

We are going to end up with -2 × sin(x) - 6 × sin x × -sin x + cos x cos x,1791

which is equal to -2 × sin(x) + 6 sin² x - 6 cos² x.1812

Now we go ahead and evaluate f” at π.1826

When I put π in for this, -2 × sin(π) + 6 × sin² of π - 6 × cos² of π.1830

sin(π) is 0, sin(π)² is 0, cos(π) is -1.1851

-1² is 1, we get -6.1859

The answer is d because -6 is less than 0.1869

We are concave down.1876

That is it, nice, straight application.1878

Second derivative is positive, you are concave up.1880

Second derivative is negative, you are concave down.1883

Let us go to question number 11.1889

Here it looks like we are computing a derivative.1893

Number 11, our particular derivative, we are evaluating at an indefinite integral.1898

My apologies.1905

The integral of -3x² × √x³ + 3 dx.1906

I think it is just going to be the straight u substitution.1917

I noticed x³ and I noticed an x².1921

I’m going to try u is equal to x³ + 3, du = 3x² dx.1924

Our integral, -3x² × the integral of x³ + 3 dx, is actually going to equal -the integral,1939

3x² x is du, this is going to be e ^½.1955

We are accustomed to seeing du on the right.1965

It is not a big deal, the order does not matter because you are just multiplying two things.1968

The multiplication is commutative, it does not matter the order.1971

You are used to seeing the du or the dx, or the dy, on the right hand side in the integrand.1975

Let us do it that way.1983

- the integral of u ^½ du which is equal to -u³/2/ 3/2 + c which is equal to -2/3 u³/2 + c.1985

Of course, we plug the u back in, x³ + 3.2005

Our final answer is -2/3 × x³ + 3³/2 + c.2009

Our final answer, the choice is going to be e.2020

I hope that was reasonably straightforward.2024

Let us take a look at the problem number 12.2030

Here we have a particular function.2039

We want to give the value of x where the function has a local extrema.2042

In this particular case, a local maximum.2046

Local maximum means you take the derivative, you set the derivative equal to 0.2050

You draw yourself a number line and you check points to the left of the critical values,2058

to the right of the critical values, to see whether the derivative is increasing or decreasing.2064

You decide which is local min and local max.2068

f(x) is equal to x³ + 6x² + 9x + 4.2072

f’(x), very simple.2084

We have 3x² + 12x + 9.2087

We want to set the derivative equal to 0.2092

We can factor out the 3.2095

We are going to get the same roots.2096

We have got x² + 4x + 3 is equal to 0.2097

We can actually factor this one.2104

We get x + 3, we get x + 1.2107

Therefore, we have x is equal to -3 and we have x is equal to -1.2111

Go ahead and draw myself a little number line here.2119

I have got -3, I will put it over here.2122

I have got -1, I will put it over here.2125

I’m going to check a point here, check a point here, and check a point here, in those intervals.2127

I’m going to put them into the derivative.2132

To see if the derivative is less than 0, decreasing, or greater than 0, increasing.2134

That is all I'm doing.2140

Let me go ahead and rewrite f’(x), I’m going to use this version right here.2143

I should actually use the original version.2155

I should have actually written this as 3 × that.2157

This is 3 × x + 3 × x + 1.2162

To the left of -3, when I check something in this interval, over here, let us try -4.2171

When I plug in -4, I'm going to get, for x, I'm going to get 3 × -4 + 3 is a negative number.2177

-4 + 1 is a negative number.2185

3 × a negative × a negative is definitely a positive number.2189

Here the slope is increasing.2194

Or if you want I can put a positive sign.2197

Some of you use positive, some of you use increasing/decreasing with an arrow.2199

It is that way.2203

Let us try a point in between here.2205

Let us try -2.2207

When I plug -2 in to the derivative, I get 3, -2 + 3 is a positive number.2210

-2 + 1 is a negative number.2218

3 × a positive × a negative gives me a number that is a negative.2220

It is less than 0, it is decreasing there or negative slope.2226

There you go, that pretty much takes care of it.2229

Because it is increasing, the graph is increasing.2232

Then, the graph is decreasing, that is a local max.2235

To the left it is increasing, to the right it is decreasing.2242

That means it is hitting a maximum point.2245

There is a local max at x = -3.2251

I think the particular choice was choice e.2260

Nothing too crazy so far.2266

Here we have number 13 and let us see what number 13 is asking us.2269

The slope of the tangent line of the graph that will give the value of c.2277

We are give a particular function.2283

We have 4x² + cx + 2, e ⁺y is equal to 2.2284

They are telling us that the slope of the tangent line of this graph, at x = 0 is 4.2298

They are asking us to find c.2320

The slope of the tangent line, for this graph, at x = 0 is equal to 4.2326

They want us to find the value of c.2331

Let us see what we can do.2338

Let us go ahead and take the derivative.2342

In this particular case, let us go ahead and take the derivative with respect to,2344

8x + c +, this is the function of y, we are going to do implicit differentiation here.2348

This is going to be 2e ⁺yy’ is equal to 0.2356

When I rearrange this, I'm going to get y’ is equal to -8x - c/ 2 × e ⁺y.2363

I will get 2 × e ⁺y.2377

2 × e ⁺y, if I move these two over to the right, 2 × e ⁺y from the original function is equal to 2 - 4x² – cx.2386

Therefore, if I plug in this into here, I get y’ is equal to -8x - c/ 2 – 4x² – c ⁺x.2401

They are telling me that y’ at 0 is equal to 4.2421

y' at 0 is equal to -8.2426

y’ at 0 is equal to 0 - c/ 2 - 0 – 0.2431

They are telling me that this is actually equal to 4.2440

I get - c/ 2 is equal to 4 which implies that c is equal to 8.2445

Our choice is e.2457

Just differentiate, in this particular case, you are going to get something which is 2e ⁺y.2460

You notice the 2e ⁺y is actually separate here.2464

You can move these over, plug in for x, and then solve for c.2467

I hope that made sense.2472

Let us try number 14, see how we are doing here.2477

Number 14, what is number 14 asking us.2483

It looks like it is asking us to evaluate this particular integral.2486

Evaluate the integral of 7 ⁺x - 4e⁷ ln x dx.2492

Let us see what we can do with this.2509

I’m going to separate this out.2511

The integral is linear, the linear of the integral of the sum is the sum of the integrals.2512

I’m going to do this as, this is the integral of 7 ⁺x dx - 4 × the integral of e⁷ ln x dx.2518

Let us go ahead and deal with the first integral.2534

This one right here.2536

The first integral, the integral of 7 ⁺x dx.2538

I’m going to do a u substitution here.2548

I’m going to let u equal to 7 ⁺x.2550

Du is going to be 7 ⁺x ln 7 dx, du/ ln 7 = 7 ⁺x dx.2554

Therefore, this integral actually = the integral of du/ ln 7 = 1/ ln 7 × the integral of du = 1/ ln 7 u + c.2572

I plug u back in which is 7 ⁺x.2601

I get 7 ⁺x/ ln 7 + c, that is our first integral.2604

Let me see here, where am I?2614

The second integral is -4, it is that one, × the integral of e ⁺ln x dx = -4 × the integral of e ⁺ln(x)⁷ dx2619

= 4 × the integral of e ⁺ln is just x⁷ dx = -4x⁸/ 8 + some constant c.2648

The c and c are not the same.2664

If you want you can just c1 and c2, it does not really matter.2666

This is this one, which is –x⁸/ 2 + c.2672

Now we just put them back together.2691

We have the integral that we wanted, it is equal to 7 ⁺x/ ln 7 – x⁸/ 2 + c.2694

It looks like that is option a.2709

That takes care of the first of 14 of the problems.2714

For the next lesson, we are just going to continue on with this particular section and go on with the practice exam.2718

Thank you so much for joining us here at www.educator.com.2725

We will see you next time, bye.2728

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