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# AP Calculus AB Online Prep CourseProf. Raffi Hovasapian Facebook Twitter More

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• 66 Lessons (44hr : 00min)
• Audio: English
• English

Professor Raffi Hovasapian will help you ace the AP Calculus AB exam in this video prep series. Raffi takes complex math concepts and distills them into easy-to-understand fundamentals that are illustrated with numerous applications and sample problems. He also walks through an entire real Advanced Placement test while dispensing useful test-taking strategies that will help you get that 5.

## Section 1: Limits and Derivatives

Overview & Slopes of Curves 42:08
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 50:53
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 59:12
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 18:43
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 51:53
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 24:43
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 53:48
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 21:22
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 50:01
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 36:31
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 53:00
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 40:02
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 53:45
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 31:38
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 47:35
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 47:25
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 41:08
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 IX: Evaluate 33:22
Example X: Evaluate 36:38
The Chain Rule 24:56
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
Example X: Differentiate f(x) = (x²-2)⁴ (5x² - 2x -2)⁸
More Chain Rule Example Problems 25:32
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 52:31
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 47:34
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 45:33
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 III: Water Tank 19:08
Example IV: Distance between Two Cars 29:27
Example V: Line-of-Sight 36:20
More Related Rates Examples 37:17
Intro 0:00
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 40:44
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 40:44
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 25:54
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 25:54
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 44:58
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 49:19
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 59:01
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 30:09
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 38:14
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 49:59
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 55:10
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 30:22
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 55:26
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 51:03
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 33:07
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 43:19
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 32:14
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 24:17
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 25:21
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 34:22
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 27:20
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 34:56
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 42:55
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 34:15
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 51:43
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 49:36
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 50:02
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 32:13
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 50:32
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 24:50
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 22:12
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 17:22
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 55:12
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
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 42:57
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 46:37
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 28:08
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 51:07
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 24:37
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 45:29
Intro 0:00
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. 41:55
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 58:47
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 25:40
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 31:20
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

Duration: 44 hours, 00 minutes

Number of Lessons: 66

This online course is crucial for students who want to score well on their AP Calculus AB exam. Although geared towards high school students taking the Calculus AB course and exam, this course is also indispensible for college students in the first semester of Calculus. Each topic begins with Raffi breaking down a concept into easily understood steps, followed by tons of examples that help cement learning. The course ends with a walkthrough of a real full-length AP test complete with useful strategies and tips.

• Free Sample Lessons

Topics Include:

• Slopes of Curves
• Continuity
• Derivatives
• Product, Power, & Quotient Rule
• Chain Rule
• Related Rates
• Maximum & Minimum Values
• Mean Value Theorem
• L'Hospital's Rule
• Antiderivatives

With his 15+ years teaching and tutoring experience coupled with triple majors in Chemistry, Mathematics, and Classics, Raffi's gift is explaining difficult concepts in simple-to-understand terms. Be sure to check out his other beloved math & science courses on Educator!

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#### Student Feedback

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By Roy JiangJanuary 3, 2019
Hello Professor Raffi,

I was just wondering if it was any coincidence that in example V, if I took Pi*30.4^2 - Pi*29.6^2 and divided the answer, I would get the possible error in area? And is this a valid way to approach a question like example V?

Thank you,

RJ
By Magic FuJuly 7, 2018
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).
By Patricia XiangApril 28, 2018
Thank you professor, I have got the refund!
By Patricia XiangApril 19, 2018
Hello professor,

Is Reduced Echelon form required in AP calculus? Or it is just helpful solving the problems? And if it helps a lot, should I acquire how to do it by hands or merely how to do it on the calculator?

Have a great day.
By Tewodros BelachewDecember 17, 2017
Ohh okay, I got it! Thank you

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