Raffi Hovasapian

Example Problems for Calculating Limits Mathematically

Slide Duration:Table of Contents

Section 1: Limits and Derivatives

Overview & Slopes of Curves

42m 8s

- Intro0:00
- Overview & Slopes of Curves0:21
- Differential and Integral0:22
- Fundamental Theorem of Calculus6:36
- Differentiation or Taking the Derivative14:24
- 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

- Intro0:00
- Slope of the Secant Line along a Curve0:12
- Slope of the Tangent Line to f(x) at a Particlar Point0:13
- Slope of the Secant Line along a Curve2:59
- Instantaneous Slope6:51
- Instantaneous Slope6:52
- Example: Distance, Time, Velocity13:32
- Instantaneous Slope and Average Slope25:42
- Slope & Rate of Change29:55
- Slope & Rate of Change29:56
- Example: Slope = 233:16
- Example: Slope = 4/334:32
- Example: Slope = 4 (m/s)39:12
- Example: Density = Mass / Volume40:33
- Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change47:46

Example Problems for Slopes of Curves

59m 12s

- Intro0:00
- Example I: Water Tank0:13
- Part A: Which is the Independent Variable and Which is the Dependent?2:00
- Part B: Average Slope3:18
- Part C: Express These Slopes as Rates-of-Change9:28
- Part D: Instantaneous Slope14:54
- Example II: y = √(x-3)28:26
- Part A: Calculate the Slope of the Secant Line30:39
- Part B: Instantaneous Slope41:26
- Part C: Equation for the Tangent Line43:59
- Example III: Object in the Air49:37
- Part A: Average Velocity50:37
- Part B: Instantaneous Velocity55:30

Desmos Tutorial

18m 43s

- Intro0:00
- Desmos Tutorial1:42
- Desmos Tutorial1:43
- Things You Must Learn To Do on Your Particular Calculator2:39
- Things You Must Learn To Do on Your Particular Calculator2:40
- Example I: y=sin x4:54
- 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

- Intro0:00
- The Limit of a Function0:14
- The Limit of a Function0:15
- Graph: Limit of a Function12:24
- Table of Values16:02
- lim x→a f(x) Does not Say What Happens When x = a20:05
- Example I: f(x) = x²24:34
- Example II: f(x) = 727:05
- Example III: f(x) = 4.530:33
- Example IV: f(x) = 1/x34:03
- Example V: f(x) = 1/x²36:43
- The Limit of a Function, Cont.38:16
- Infinity and Negative Infinity38:17
- Does Not Exist42:45
- Summary46:48

Example Problems for the Limit of a Function

24m 43s

- Intro0:00
- Example I: Explain in Words What the Following Symbols Mean0:10
- Example II: Find the Following Limit5:21
- Example III: Use the Graph to Find the Following Limits7:35
- Example IV: Use the Graph to Find the Following Limits11:48
- Example V: Sketch the Graph of a Function that Satisfies the Following Properties15:25
- Example VI: Find the Following Limit18:44
- Example VII: Find the Following Limit20:06

Calculating Limits Mathematically

53m 48s

- Intro0:00
- Plug-in Procedure0:09
- Plug-in Procedure0:10
- Limit Laws9:14
- Limit Law 110:05
- Limit Law 210:54
- Limit Law 311:28
- Limit Law 411:54
- Limit Law 512:24
- Limit Law 613:14
- Limit Law 714:38
- Plug-in Procedure, Cont.16:35
- Plug-in Procedure, Cont.16:36
- Example I: Calculating Limits Mathematically20:50
- Example II: Calculating Limits Mathematically27:37
- Example III: Calculating Limits Mathematically31:42
- Example IV: Calculating Limits Mathematically35:36
- Example V: Calculating Limits Mathematically40:58
- Limits Theorem44:45
- Limits Theorem 144:46
- Limits Theorem 2: Squeeze Theorem46:34
- Example VI: Calculating Limits Mathematically49:26

Example Problems for Calculating Limits Mathematically

21m 22s

- Intro0:00
- Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law0:16
- Example II: Evaluate the Following Limit1:51
- Example III: Evaluate the Following Limit3:36
- Example IV: Evaluate the Following Limit8:56
- Example V: Evaluate the Following Limit11:19
- Example VI: Calculating Limits Mathematically13:19
- Example VII: Calculating Limits Mathematically14:59

Calculating Limits as x Goes to Infinity

50m 1s

- Intro0:00
- Limit as x Goes to Infinity0:14
- Limit as x Goes to Infinity0:15
- Let's Look at f(x) = 1 / (x-3)1:04
- Summary9:34
- Example I: Calculating Limits as x Goes to Infinity12:16
- Example II: Calculating Limits as x Goes to Infinity21:22
- Example III: Calculating Limits as x Goes to Infinity24:10
- Example IV: Calculating Limits as x Goes to Infinity36:00

Example Problems for Limits at Infinity

36m 31s

- Intro0:00
- Example I: Calculating Limits as x Goes to Infinity0:14
- Example II: Calculating Limits as x Goes to Infinity3:27
- Example III: Calculating Limits as x Goes to Infinity8:11
- Example IV: Calculating Limits as x Goes to Infinity14:20
- Example V: Calculating Limits as x Goes to Infinity20:07
- Example VI: Calculating Limits as x Goes to Infinity23:36

Continuity

53m

- Intro0:00
- Definition of Continuity0:08
- Definition of Continuity0:09
- Example: Not Continuous3:52
- Example: Continuous4:58
- Example: Not Continuous5:52
- Procedure for Finding Continuity9:45
- Law of Continuity13:44
- Law of Continuity13:45
- Example I: Determining Continuity on a Graph15:55
- Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous17:57
- Example III: Is the Following Function Continuous at the Given Point?22:42
- Theorem for Composite Functions25:28
- Theorem for Composite Functions25:29
- 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 Discontinuity39:18
- Removable Discontinuity39:33
- Jump Discontinuity40:06
- Infinite Discontinuity40:32
- Intermediate Value Theorem40:58
- Intermediate Value Theorem: Hypothesis & Conclusion40:59
- Intermediate Value Theorem: Graphically43:40
- Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6]47:46

Derivative I

40m 2s

- Intro0:00
- Derivative0:09
- Derivative0:10
- Example I: Find the Derivative of f(x)=x³2:20
- Notations for the Derivative7:32
- Notations for the Derivative7:33
- Derivative & Rate of Change11:14
- Recall the Rate of Change11:15
- Instantaneous Rate of Change17:04
- Graphing f(x) and f'(x)19:10
- Example II: Find the Derivative of x⁴ - x²24:00
- Example III: Find the Derivative of f(x)=√x30:51

Derivatives II

53m 45s

- Intro0:00
- Example I: Find the Derivative of (2+x)/(3-x)0:18
- Derivatives II9:02
- f(x) is Differentiable if f'(x) Exists9:03
- Recall: For a Limit to Exist, Both Left Hand and Right Hand Limits Must Equal to Each Other17:19
- Geometrically: Differentiability Means the Graph is Smooth18:44
- Example II: Show Analytically that f(x) = |x| is Nor Differentiable at x=020:53
- Example II: For x > 023:53
- Example II: For x < 025:36
- Example II: What is f(0) and What is the lim |x| as x→0?30:46
- Differentiability & Continuity34:22
- Differentiability & Continuity34:23
- 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 Derivatives41:58
- Higher Derivatives41:59
- Derivative Operator45:12
- Example III: Find (dy)/(dx) & (d²y)/(dx²) for y = x³49:29

More Example Problems for The Derivative

31m 38s

- Intro0:00
- 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 Definition3:49
- Example IV: Determine f, f', and f'' on a Graph12:43
- Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given x-value13:40
- Example VI: Distance vs. Time20:15
- Example VII: Displacement, Velocity, and Acceleration23:56
- Example VIII: Graph the Displacement Function28:20

Section 2: Differentiation

Differentiation of Polynomials & Exponential Functions

47m 35s

- Intro0:00
- Differentiation of Polynomials & Exponential Functions0:15
- Derivative of a Function0:16
- Derivative of a Constant2:35
- Power Rule3:08
- If C is a Constant4:19
- Sum Rule5:22
- Exponential Functions6:26
- Example I: Differentiate7:45
- Example II: Differentiate12:38
- Example III: Differentiate15:13
- Example IV: Differentiate16:20
- Example V: Differentiate19:19
- Example VI: Find the Equation of the Tangent Line to a Function at a Given Point12:18
- Example VII: Find the First & Second Derivatives25:59
- Example VIII27:47
- Part A: Find the Velocity & Acceleration Functions as Functions of t27:48
- Part B: Find the Acceleration after 3 Seconds30:12
- Part C: Find the Acceleration when the Velocity is 030:53
- Part D: Graph the Position, Velocity, & Acceleration Graphs32:50
- Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents34:53
- Example X: Find a Point on a Graph42:31

The Product, Power & Quotient Rules

47m 25s

- Intro0:00
- The Product, Power and Quotient Rules0:19
- Differentiate Functions0:20
- Product Rule5:30
- Quotient Rule9:15
- Power Rule10:00
- Example I: Product Rule13:48
- Example II: Quotient Rule16:13
- Example III: Power Rule18:28
- Example IV: Find dy/dx19:57
- Example V: Find dy/dx24:53
- Example VI: Find dy/dx28:38
- Example VII: Find an Equation for the Tangent to the Curve34:54
- Example VIII: Find d²y/dx²38:08

Derivatives of the Trigonometric Functions

41m 8s

- Intro0:00
- Derivatives of the Trigonometric Functions0:09
- Let's Find the Derivative of f(x) = sin x0:10
- Important Limits to Know4:59
- 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 x7:56
- Example II: Differentiate f(x) = x⁵ tan x9:04
- 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 Line21:48
- 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 Problem28:23
- Example IX: Evaluate33:22
- Example X: Evaluate36:38

The Chain Rule

24m 56s

- Intro0:00
- The Chain Rule0:13
- Recall the Composite Functions0:14
- Derivatives of Composite Functions1:34
- Example I: Identify f(x) and g(x) and Differentiate6:41
- Example II: Identify f(x) and g(x) and Differentiate9:47
- Example III: Differentiate11:03
- Example IV: Differentiate f(x) = -5 / (x² + 3)³12:15
- Example V: Differentiate f(x) = cos(x² + c²)14:35
- Example VI: Differentiate f(x) = cos⁴x +c²15:41
- Example VII: Differentiate17:03
- 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

- Intro0:00
- 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 Point2:25
- Example III: F(x) = f(g(x)), Find F' (6)4:22
- Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window5:35
- Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴10:18
- Example VI: Differentiate f(x) = sec²(12x)12:28
- Example VII: Differentiate14:41
- Example VIII: Differentiate19:25
- Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time21:13

Implicit Differentiation

52m 31s

- Intro0:00
- Implicit Differentiation0:09
- Implicit Differentiation0:10
- Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y12:15
- Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 1419:18
- Example III: Find (dy)/(dx) of x³y² + y³x² = 4x21:43
- Example IV: Find (dy)/(dx) of the Following Equation24:13
- Example V: Find (dy)/(dx) of 6sin x cos y = 129:00
- Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y31:02
- Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x37:36
- Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²-y²)41:03
- Example IX: Find (d²y)/(dx²) of x² + y² = 2544:05
- Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x)47:48

Section 3: Applications of the Derivative

Linear Approximations & Differentials

47m 34s

- Intro0:00
- Linear Approximations & Differentials0:09
- Linear Approximations & Differentials0:10
- Example I: Linear Approximations & Differentials11:27
- Example II: Linear Approximations & Differentials20:19
- Differentials30:32
- Differentials30:33
- Example III: Linear Approximations & Differentials34:09
- Example IV: Linear Approximations & Differentials35:57
- Example V: Relative Error38:46

Related Rates

45m 33s

- Intro0:00
- Related Rates0:08
- Strategy for Solving Related Rates Problems #10:09
- Strategy for Solving Related Rates Problems #21:46
- Strategy for Solving Related Rates Problems #32:06
- Strategy for Solving Related Rates Problems #42:50
- Strategy for Solving Related Rates Problems #53:38
- Example I: Radius of a Balloon5:15
- Example II: Ladder12:52
- Example III: Water Tank19:08
- Example IV: Distance between Two Cars29:27
- Example V: Line-of-Sight36:20

More Related Rates Examples

37m 17s

- Intro0:00
- Example I: Shadow0:14
- Example II: Particle4:45
- Example III: Water Level10:28
- Example IV: Clock20:47
- Example V: Distance between a House and a Plane29:11

Maximum & Minimum Values of a Function

40m 44s

- Intro0:00
- Maximum & Minimum Values of a Function, Part 10:23
- Absolute Maximum2:20
- Absolute Minimum2:52
- Local Maximum3:38
- Local Minimum4:26
- Maximum & Minimum Values of a Function, Part 26:11
- Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min7:18
- Function with Local Max & Min but No Absolute Max & Min8:48
- Formal Definitions10:43
- Absolute Maximum11:18
- Absolute Minimum12:57
- Local Maximum14:37
- Local Minimum16:25
- Extreme Value Theorem18:08
- Theorem: f'(c) = 024:40
- 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 + sinx29:51
- 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

- Intro0:00
- Example I: Identify Absolute and Local Max & Min on the Following Graph0:11
- Example II: Sketch the Graph of a Continuous Function3:11
- Example III: Sketch the Following Graphs4:40
- Example IV: Find the Critical Values of f (x) = 3x⁴ - 7x³ + 4x²6:13
- Example V: Find the Critical Values of f(x) = |2x - 5|8:42
- Example VI: Find the Critical Values11:42
- 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

- Intro0:00
- Rolle's Theorem0:08
- Rolle's Theorem: If & Then0:09
- Rolle's Theorem: Geometrically2:06
- There May Be More than 1 c Such That f'( c ) = 03:30
- Example I: Rolle's Theorem4:58
- The Mean Value Theorem9:12
- The Mean Value Theorem: If & Then9:13
- The Mean Value Theorem: Geometrically11:07
- Example II: Mean Value Theorem13:43
- Example III: Mean Value Theorem21:19

Using Derivatives to Graph Functions, Part I

25m 54s

- Intro0:00
- Using Derivatives to Graph Functions, Part I0:12
- Increasing/ Decreasing Test0:13
- Example I: Find the Intervals Over Which the Function is Increasing & Decreasing3:26
- Example II: Find the Local Maxima & Minima of the Function19:18
- Example III: Find the Local Maxima & Minima of the Function31:39

Using Derivatives to Graph Functions, Part II

44m 58s

- Intro0:00
- Using Derivatives to Graph Functions, Part II0:13
- Concave Up & Concave Down0:14
- What Does This Mean in Terms of the Derivative?6:14
- Point of Inflection8:52
- Example I: Graph the Function13:18
- Example II: Function x⁴ - 5x²19:03
- Intervals of Increase & Decrease19:04
- Local Maxes and Mins25:01
- Intervals of Concavity & X-Values for the Points of Inflection29:18
- Intervals of Concavity & Y-Values for the Points of Inflection34:18
- Graphing the Function40:52

Example Problems I

49m 19s

- Intro0:00
- Example I: Intervals, Local Maxes & Mins0:26
- Example II: Intervals, Local Maxes & Mins5:05
- Example III: Intervals, Local Maxes & Mins, and Inflection Points13:40
- Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity23:02
- Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity34:36

Example Problems III

59m 1s

- Intro0:00
- Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes0:11
- Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes21:24
- Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D37:56
- Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes46:19

L'Hospital's Rule

30m 9s

- Intro0:00
- L'Hospital's Rule0:19
- Indeterminate Forms0:20
- L'Hospital's Rule3:38
- Example I: Evaluate the Following Limit Using L'Hospital's Rule8:50
- Example II: Evaluate the Following Limit Using L'Hospital's Rule10:30
- Indeterminate Products11:54
- Indeterminate Products11:55
- Example III: L'Hospital's Rule & Indeterminate Products13:57
- Indeterminate Differences17:00
- Indeterminate Differences17:01
- Example IV: L'Hospital's Rule & Indeterminate Differences18:57
- Indeterminate Powers22:20
- Indeterminate Powers22:21
- Example V: L'Hospital's Rule & Indeterminate Powers25:13

Example Problems for L'Hospital's Rule

38m 14s

- Intro0:00
- Example I: Evaluate the Following Limit0:17
- Example II: Evaluate the Following Limit2:45
- Example III: Evaluate the Following Limit6:54
- Example IV: Evaluate the Following Limit8:43
- Example V: Evaluate the Following Limit11:01
- Example VI: Evaluate the Following Limit14:48
- Example VII: Evaluate the Following Limit17:49
- Example VIII: Evaluate the Following Limit20:37
- Example IX: Evaluate the Following Limit25:16
- Example X: Evaluate the Following Limit32:44

Optimization Problems I

49m 59s

- Intro0:00
- Example I: Find the Dimensions of the Box that Gives the Greatest Volume1:23
- Fundamentals of Optimization Problems18:08
- Fundamental #118:33
- Fundamental #219:09
- Fundamental #319:19
- Fundamental #420:59
- Fundamental #521:55
- Fundamental #623:44
- Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square24:36
- 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 R43:10

Optimization Problems II

55m 10s

- Intro0:00
- Example I: Optimization Problem0:13
- Example II: Optimization Problem17:34
- Example III: Optimization Problem35:06
- Example IV: Revenue, Cost, and Profit43:22

Newton's Method

30m 22s

- Intro0:00
- Newton's Method0:45
- Newton's Method0:46
- Example I: Find x2 and x313:18
- Example II: Use Newton's Method to Approximate15:48
- Example III: Find the Root of the Following Equation to 6 Decimal Places19:57
- Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point23:11

Section 4: Integrals

Antiderivatives

55m 26s

- Intro0:00
- Antiderivatives0:23
- Definition of an Antiderivative0:24
- Antiderivative Theorem7:58
- Function & Antiderivative12:10
- x^n12:30
- 1/x13:00
- e^x13:08
- cos x13:18
- sin x14:01
- sec² x14:11
- secxtanx14:18
- 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 Functions15:07
- Function 1: f(x) = x³ -6x² + 11x - 915:42
- Function 2: f(x) = 14√(x) - 27 4√x19:12
- Function 3: (fx) = cos x - 14 sinx20:53
- 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) = 4027:55
- Function 2: f'(x) 3 sinx + sec²x, f(π/6) = 530:34
- Function 3: f''(x) = 8x - cos x, f(1.5) = 12.7, f'(1.5) = 4.232:54
- Function 4: f''(x) = 5/(√x), f(2) 15, f'(2) = 737:54
- Example III: Falling Object41:58
- Problem 1: Find an Equation for the Height of the Ball after t Seconds42:48
- 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

- Intro0:00
- The Area Under a Curve0:13
- Approximate Using Rectangles0:14
- Let's Do This Again, Using 4 Different Rectangles9:40
- Approximate with Rectangles16:10
- Left Endpoint18:08
- Right Endpoint25:34
- Left Endpoint vs. Right Endpoint30:58
- Number of Rectangles34:08
- True Area37:36
- True Area37:37
- Sigma Notation & Limits43:32
- When You Have to Explicitly Solve Something47:56

Example Problems for Area Under a Curve

33m 7s

- Intro0:00
- Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve0:10
- Example II: Using 5 Rectangles, Approximate the Area Under the Curve11:32
- Example III: Find the True Area by Evaluating the Limit Expression16:07
- Example IV: Find the True Area by Evaluating the Limit Expression24:52

The Definite Integral

43m 19s

- Intro0:00
- The Definite Integral0:08
- Definition to Find the Area of a Curve0:09
- Definition of the Definite Integral4:08
- Symbol for Definite Integral8:45
- Regions Below the x-axis15:18
- Associating Definite Integral to a Function19:38
- Integrable Function27:20
- Evaluating the Definite Integral29:26
- Evaluating the Definite Integral29:27
- Properties of the Definite Integral35:24
- Properties of the Definite Integral35:25

Example Problems for The Definite Integral

32m 14s

- Intro0:00
- Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals0:11
- Example II: Express the Following Limit as a Definite Integral5:28
- Example III: Evaluate the Following Definite Integral Using the Definition6:28
- Example IV: Evaluate the Following Integral Using the Definition17:06
- Example V: Evaluate the Following Definite Integral by Using Areas25:41
- Example VI: Definite Integral30:36

The Fundamental Theorem of Calculus

24m 17s

- Intro0:00
- The Fundamental Theorem of Calculus0:17
- Evaluating an Integral0:18
- Lim as x → ∞12:19
- Taking the Derivative14:06
- Differentiation & Integration are Inverse Processes15:04
- 1st Fundamental Theorem of Calculus20:08
- 1st Fundamental Theorem of Calculus20:09
- 2nd Fundamental Theorem of Calculus22:30
- 2nd Fundamental Theorem of Calculus22:31

Example Problems for the Fundamental Theorem

25m 21s

- Intro0:00
- Example I: Find the Derivative of the Following Function0:17
- Example II: Find the Derivative of the Following Function1:40
- Example III: Find the Derivative of the Following Function2:32
- Example IV: Find the Derivative of the Following Function5:55
- Example V: Evaluate the Following Integral7:13
- Example VI: Evaluate the Following Integral9:46
- Example VII: Evaluate the Following Integral12:49
- Example VIII: Evaluate the Following Integral13:53
- Example IX: Evaluate the Following Graph15:24
- 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

- Intro0:00
- Example I: Evaluate the Following Indefinite Integral0:10
- Example II: Evaluate the Following Definite Integral0:59
- Example III: Evaluate the Following Integral2:59
- Example IV: Velocity Function7:46
- Part A: Net Displacement7:47
- Part B: Total Distance Travelled13:15
- Example V: Linear Density Function20:56
- Example VI: Acceleration Function25:10
- Part A: Velocity Function at Time t25:11
- Part B: Total Distance Travelled During the Time Interval28:38

Solving Integrals by Substitution

27m 20s

- Intro0:00
- Table of Integrals0:35
- Example I: Evaluate the Following Indefinite Integral2:02
- Example II: Evaluate the Following Indefinite Integral7:27
- Example IIII: Evaluate the Following Indefinite Integral10:57
- Example IV: Evaluate the Following Indefinite Integral12:33
- Example V: Evaluate the Following14:28
- Example VI: Evaluate the Following16:00
- Example VII: Evaluate the Following19:01
- Example VIII: Evaluate the Following21:49
- Example IX: Evaluate the Following24:34

Section 5: Applications of Integration

Areas Between Curves

34m 56s

- Intro0:00
- Areas Between Two Curves: Function of x0:08
- 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 Value10:32
- Formula for Areas Between Two Curves: Top Function - Bottom Function17:03
- Areas Between Curves: Function of y17:49
- What if We are Given Functions of y?17:50
- Formula for Areas Between Two Curves: Right Function - Left Function21:48
- Finding a & b22:32

Example Problems for Areas Between Curves

42m 55s

- Intro0:00
- Instructions for the Example Problems0:10
- Example I: y = 7x - x² and y=x0:37
- Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=26:25
- Example III: y=(1/x), y=(1/x³), and x=412:25
- Example IV: 15-2x² and y=x²-515:52
- Example V: x=(1/8)y³ and x=6-y²20:20
- Example VI: y=cos x, y=sin(2x), [0,π/2]24:34
- Example VII: y=2x², y=10x², 7x+2y=1029:51
- Example VIII: Velocity vs. Time33:23
- 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

- Intro0:00
- Volumes I: Slices0:18
- Rotate the Graph of y=√x about the x-axis0:19
- How can I use Integration to Find the Volume?3:16
- Slice the Solid Like a Loaf of Bread5:06
- Volumes Definition8:56
- Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation12:18
- Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation19:05
- Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation25:28

Volumes II: Volumes by Washers

51m 43s

- Intro0:00
- Volumes II: Volumes by Washers0:11
- Rotating Region Bounded by y=x³ & y=x around the x-axis0:12
- Equation for Volumes by Washer11:14
- Process for Solving Volumes by Washer13:40
- Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis15:58
- Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis25:07
- Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis34:20
- Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis44:05

Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

- Intro0:00
- Solids That Are Not Solids-of-Revolution0:11
- Cross-Section Area Review0:12
- Cross-Sections That Are Not Solids-of-Revolution7:36
- Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H10:54
- Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles20:39
- Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H29:27
- Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 6436:47
- Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis46:13

Volumes IV: Volumes By Cylindrical Shells

50m 2s

- Intro0:00
- Volumes by Cylindrical Shells0:11
- Find the Volume of the Following Region0:12
- Volumes by Cylindrical Shells: Integrating Along x14:12
- Volumes by Cylindrical Shells: Integrating Along y14:40
- Volumes by Cylindrical Shells Formulas16:22
- Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid18:33
- Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid25:57
- Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid31:38
- Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid38:44
- Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid44:03

The Average Value of a Function

32m 13s

- Intro0:00
- The Average Value of a Function0:07
- 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 Integrate9:25
- Example I: Find the Average Value of the Given Function Over the Given Interval14:06
- Example II: Find the Average Value of the Given Function Over the Given Interval18:25
- 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 Rod27:47

Section 6: Techniques of Integration

Integration by Parts

50m 32s

- Intro0:00
- Integration by Parts0:08
- The Product Rule for Differentiation0:09
- Integrating Both Sides Retains the Equality0:52
- Differential Notation2:24
- Example I: ∫ x cos x dx5:41
- Example II: ∫ x² sin(2x)dx12:01
- Example III: ∫ (e^x) cos x dx18:19
- Example IV: ∫ (sin^-1) (x) dx23:42
- Example V: ∫₁⁵ (lnx)² dx28:25
- Summary32:31
- Tabular Integration35:08
- Case 135:52
- Example: ∫x³sinx dx36:39
- Case 240:28
- Example: ∫e^(2x) sin 3x41:14

Trigonometric Integrals I

24m 50s

- Intro0:00
- Example I: ∫ sin³ (x) dx1:36
- Example II: ∫ cos⁵(x)sin²(x)dx4:36
- Example III: ∫ sin⁴(x)dx9:23
- Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx15:59
- #1: Power of sin is Odd16:00
- #2: Power of cos is Odd16:41
- #3: Powers of Both sin and cos are Odd16:55
- #4: Powers of Both sin and cos are Even17:10
- Example IV: ∫ tan⁴ (x) sec⁴ (x) dx17:34
- Example V: ∫ sec⁹(x) tan³(x) dx20:55
- Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx23:31
- #1: Power of sec is Odd23:32
- #2: Power of tan is Odd24:04
- #3: Powers of sec is Odd and/or Power of tan is Even24:18

Trigonometric Integrals II

22m 12s

- Intro0:00
- Trigonometric Integrals II0:09
- Recall: ∫tanx dx0:10
- Let's Find ∫secx dx3:23
- Example I: ∫ tan⁵ (x) dx6:23
- Example II: ∫ sec⁵ (x) dx11:41
- Summary: How to Deal with Integrals of Different Types19:04
- Identities to Deal with Integrals of Different Types19:05
- Example III: ∫cos(5x)sin(9x)dx19:57

More Example Problems for Trigonometric Integrals

17m 22s

- Intro0:00
- Example I: ∫sin²(x)cos⁷(x)dx0:14
- Example II: ∫x sin²(x) dx3:56
- Example III: ∫csc⁴ (x/5)dx8:39
- Example IV: ∫( (1-tan²x)/(sec²x) ) dx11:17
- Example V: ∫ 1 / (sinx-1) dx13:19

Integration by Partial Fractions I

55m 12s

- Intro0:00
- Integration by Partial Fractions I0:11
- Recall the Idea of Finding a Common Denominator0:12
- Decomposing a Rational Function to Its Partial Fractions4:10
- 2 Types of Rational Function: Improper & Proper5:16
- Improper Rational Function7:26
- Improper Rational Function7:27
- Proper Rational Function11:16
- Proper Rational Function & Partial Fractions11:17
- Linear Factors14:04
- Irreducible Quadratic Factors15:02
- Case 1: G(x) is a Product of Distinct Linear Factors17:10
- Example I: Integration by Partial Fractions20:33
- Case 2: D(x) is a Product of Linear Factors40:58
- Example II: Integration by Partial Fractions44:41

Integration by Partial Fractions II

42m 57s

- Intro0:00
- Case 3: D(x) Contains Irreducible Factors0:09
- Example I: Integration by Partial Fractions5:19
- Example II: Integration by Partial Fractions16:22
- Case 4: D(x) has Repeated Irreducible Quadratic Factors27:30
- Example III: Integration by Partial Fractions30:19

Section 7: Differential Equations

Introduction to Differential Equations

46m 37s

- Intro0:00
- Introduction to Differential Equations0:09
- Overview0:10
- 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 Number6:28
- What are These Differential Equations Saying?10:30
- Verifying that a Function is a Solution of the Differential Equation13:00
- Verifying that a Function is a Solution of the Differential Equation13:01
- Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation17:20
- General Solution22:00
- Particular Solution24:36
- Initial Value Problem27:42
- Example I: Verify that a Family of Functions is a Solution of the Differential Equation32:24
- Example II: For What Values of K Does the Function Satisfy the Differential Equation36:07
- Example III: Verify the Solution and Solve the Initial Value Problem39:47

Separation of Variables

28m 8s

- Intro0:00
- Separation of Variables0:28
- Separation of Variables0:29
- Example I: Solve the Following g Initial Value Problem8:29
- Example II: Solve the Following g Initial Value Problem13:46
- Example III: Find an Equation of the Curve18:48

Population Growth: The Standard & Logistic Equations

51m 7s

- Intro0:00
- Standard Growth Model0:30
- Definition of the Standard/Natural Growth Model0:31
- Initial Conditions8:00
- The General Solution9:16
- Example I: Standard Growth Model10:45
- Logistic Growth Model18:33
- Logistic Growth Model18:34
- Solving the Initial Value Problem25:21
- What Happens When t → ∞36:42
- Example II: Solve the Following g Initial Value Problem41:50
- Relative Growth Rate46:56
- Relative Growth Rate46:57
- Relative Growth Rate Version for the Standard model49:04

Slope Fields

24m 37s

- Intro0:00
- Slope Fields0:35
- Slope Fields0:36
- Graphing the Slope Fields, Part 111:12
- Graphing the Slope Fields, Part 215:37
- Graphing the Slope Fields, Part 317:25
- Steps to Solving Slope Field Problems20:24
- Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y22:38

Section 8: AP Practic Exam

AP Practice Exam: Section 1, Part A No Calculator

45m 29s

- Intro0:00
- Exam Link0:10
- Problem #11:26
- Problem #22:52
- Problem #34:42
- Problem #47:03
- Problem #510:01
- Problem #613:49
- Problem #715:16
- Problem #819:06
- Problem #923:10
- Problem #1028:10
- Problem #1131:30
- Problem #1233:53
- Problem #1337:45
- Problem #1441:17

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

41m 55s

- Intro0:00
- Problem #150:22
- Problem #163:10
- Problem #175:30
- Problem #188:03
- Problem #199:53
- Problem #2014:51
- Problem #2117:30
- Problem #2222:12
- Problem #2325:48
- Problem #2429:57
- Problem #2533:35
- Problem #2635:57
- Problem #2737:57
- Problem #2840:04

AP Practice Exam: Section I, Part B Calculator Allowed

58m 47s

- Intro0:00
- Problem #11:22
- Problem #24:55
- Problem #310:49
- Problem #413:05
- Problem #514:54
- Problem #617:25
- Problem #718:39
- Problem #820:27
- Problem #926:48
- Problem #1028:23
- Problem #1134:03
- Problem #1236:25
- Problem #1339:52
- Problem #1443:12
- Problem #1547:18
- Problem #1650:41
- Problem #1756:38

AP Practice Exam: Section II, Part A Calculator Allowed

25m 40s

- Intro0:00
- Problem #1: Part A1:14
- Problem #1: Part B4:46
- Problem #1: Part C8:00
- Problem #2: Part A12:24
- Problem #2: Part B16:51
- Problem #2: Part C17:17
- Problem #3: Part A18:16
- Problem #3: Part B19:54
- Problem #3: Part C21:44
- Problem #3: Part D22:57

AP Practice Exam: Section II, Part B No Calculator

31m 20s

- Intro0:00
- Problem #4: Part A1:35
- Problem #4: Part B5:54
- Problem #4: Part C8:50
- Problem #4: Part D9:40
- Problem #5: Part A11:26
- Problem #5: Part B13:11
- Problem #5: Part C15:07
- Problem #5: Part D19:57
- Problem #6: Part A22:01
- Problem #6: Part B25:34
- Problem #6: Part C28:54

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For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

# AP Calculus AB Example Problems for Calculating Limits Mathematically

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