Raffi Hovasapian
Optimization Problems II
Slide Duration:Table of Contents
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
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
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 RatesofChange9:28
 Part D: Instantaneous Slope14:54
 Example II: y = √(x3)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
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
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
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
53m 48s
 Intro0:00
 Plugin Procedure0:09
 Plugin 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
 Plugin Procedure, Cont.16:35
 Plugin 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
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
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 / (x3)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
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
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
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
53m 45s
 Intro0:00
 Example I: Find the Derivative of (2+x)/(3x)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
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 xvalue13:40
 Example VI: Distance vs. Time20:15
 Example VII: Displacement, Velocity, and Acceleration23:56
 Example VIII: Graph the Displacement Function28:20
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
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
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
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
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) = ( (x8)/(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
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(x²y²)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
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
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: LineofSight36:20
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
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
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  58:42
 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
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
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
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 & XValues for the Points of Inflection29:18
 Intervals of Concavity & YValues for the Points of Inflection34:18
 Graphing the Function40:52
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
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
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
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
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
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
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
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/√(1x²)14:26
 1/(1+x²)14:36
 1/√(1x²)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
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
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
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 xaxis15:18
 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
32m 14s
 Intro0:00
 Example I: Approximate the Following Definite Integral Using Midpoints & Subintervals0:11
 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
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
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
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
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
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
42m 55s
 Intro0:00
 Instructions for the Example Problems0:10
 Example I: y = 7x  x² and y=x0:37
 Example II: x=y²3, 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: 152x² and y=x²515:52
 Example V: x=(1/8)y³ and x=6y²20:20
 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
34m 15s
 Intro0:00
 Volumes I: Slices0:18
 Rotate the Graph of y=√x about the xaxis0:19
 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
51m 43s
 Intro0:00
 Volumes II: Volumes by Washers0:11
 Rotating Region Bounded by y=x³ & y=x around the xaxis0:12
 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
49m 36s
 Intro0:00
 Solids That Are Not SolidsofRevolution0:11
 CrossSection Area Review0:12
 CrossSections That Are Not SolidsofRevolution7:36
 Example I: Find the Volume of a Pyramid Whose Base is a Square of Sidelength S, and Whose Height is H10:54
 Example II: Find the Volume of a Solid Whose Crosssectional Areas Perpendicular to the Base are Equilateral Triangles20:39
 Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of SideLength 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=3x² and the xaxis46:13
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
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
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
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
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
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: ∫( (1tan²x)/(sec²x) ) dx11:17
 Example V: ∫ 1 / (sinx1) dx13:19
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
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
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
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
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
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
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
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
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
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
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
1 answer
Last reply by: Sarmad Khokhar
Sun Apr 30, 2017 7:35 AM
Post by Sarmad Khokhar on April 30, 2017
In Example we could have also taken the second derivative to check our x value.
1 answer
Last reply by: Professor Hovasapian
Wed Jan 18, 2017 7:55 PM
Post by Rohit Kumar on January 8, 2017
What would the domain be for the last example and how would you find it?
1 answer
Last reply by: Professor Hovasapian
Tue Apr 12, 2016 3:33 PM
Post by Acme Wang on April 12, 2016
In example IV, does the price function denote the price per unit?
0 answers
Post by Gautham Padmakumar on December 5, 2015
Also, you made an error in using the quadratic formula. Its supposed to be
x = 8.01 and x = 6.09
1 answer
Last reply by: Professor Hovasapian
Thu Dec 17, 2015 12:26 AM
Post by Gautham Padmakumar on December 5, 2015
Isn't it a problem to work with different units like that in Example 2? We have both 1.5 km and 7 miles?
Thank you