Raffi Hovasapian

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 Calculating Limits Mathematically

Section 1: Limits and Derivatives: Lecture 7 | 53:48 min

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2 answers

Last reply by: sorin dragon

Fri Sep 15, 2017 1:26 PM

Post by sorin dragon on September 13, 2017

hello professor Raffi. I have a quick question. How works the method you used for factoring the numerator in Example 1: Calculating Limits? Thank you.

1 answer

Last reply by: Professor Hovasapian

Thu Feb 11, 2016 11:05 PM

Post by Avijit Singh on February 10, 2016

Hi Professor Raffi. I am a student self studying for the AP Calculus AB exam in May 2016, and I just wanted to thank you for your great videos. They really explain the concepts well and give me confidence to study for this exam in the comfort of my home.

I was wondering if you have any advice as to how I could best study for AP Calc. I am hoping to get through the AB course really soon (I have graduated from high school and am just taking this course to prepare for university level calculus). I do have a lot of time on my hands so I am only focusing on AP Calc. Do you think it is a good idea to progress to AP Calculus BC if I manage to complete the AB course in the next month? Thanks very much.

1 answer

Last reply by: Professor Hovasapian

Thu Jan 7, 2016 11:06 PM

Post by Sohan Mugi on January 6, 2016

Hello Professor Hovasapian. Just had a quick question about this lecture. Will we have to possibly face a problem like the last one in this lecture during the AP Calculus AB Exam? Thank you.

1 answer

Last reply by: Professor Hovasapian

Wed Nov 25, 2015 12:35 AM

Post by peter alabi on November 23, 2015

Hi professor Raffi. I don't know how you derived 2x^2 from X^2 - X^2, i just want to know if it a simple mistake or am tripping.

second. last lecture you obliviously mentioned infinitesimal, am just wondering if you can be more elaborate for me, the definition of infinitesimals and hyper real numbers? My professor seems to care so much about it but i don't get it. Despite the minor request great lecture, i never knew math can be so explicitly explain thanks.