Section 1: Limits and Derivatives 

Overview & Slopes of Curves 
42:08 
 
Intro 
0:00  
 
Overview & Slopes of Curves 
0:21  
 
 Differential and Integral 
0:22  
 
 Fundamental Theorem of Calculus 
6:36  
 
 Differentiation or Taking the Derivative 
14:24  
 
 What Does the Derivative Mean and How do We Find it? 
15:18  
 
 Example: f'(x) 
19:24  
 
 Example: f(x) = sin (x) 
29:16  
 
 General Procedure for Finding the Derivative of f(x) 
37:33  

More on Slopes of Curves 
50:53 
 
Intro 
0:00  
 
Slope of the Secant Line along a Curve 
0:12  
 
 Slope of the Tangent Line to f(x) at a Particlar Point 
0:13  
 
 Slope of the Secant Line along a Curve 
2:59  
 
Instantaneous Slope 
6:51  
 
 Instantaneous Slope 
6:52  
 
 Example: Distance, Time, Velocity 
13:32  
 
 Instantaneous Slope and Average Slope 
25:42  
 
Slope & Rate of Change 
29:55  
 
 Slope & Rate of Change 
29:56  
 
 Example: Slope = 2 
33:16  
 
 Example: Slope = 4/3 
34:32  
 
 Example: Slope = 4 (m/s) 
39:12  
 
 Example: Density = Mass / Volume 
40:33  
 
 Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change 
47:46  

Example Problems for Slopes of Curves 
59:12 
 
Intro 
0:00  
 
Example I: Water Tank 
0:13  
 
 Part A: Which is the Independent Variable and Which is the Dependent? 
2:00  
 
 Part B: Average Slope 
3:18  
 
 Part C: Express These Slopes as RatesofChange 
9:28  
 
 Part D: Instantaneous Slope 
14:54  
 
Example II: y = √(x3) 
28:26  
 
 Part A: Calculate the Slope of the Secant Line 
30:39  
 
 Part B: Instantaneous Slope 
41:26  
 
 Part C: Equation for the Tangent Line 
43:59  
 
Example III: Object in the Air 
49:37  
 
 Part A: Average Velocity 
50:37  
 
 Part B: Instantaneous Velocity 
55:30  

Desmos Tutorial 
18:43 
 
Intro 
0:00  
 
Desmos Tutorial 
1:42  
 
 Desmos Tutorial 
1:43  
 
Things You Must Learn To Do on Your Particular Calculator 
2:39  
 
 Things You Must Learn To Do on Your Particular Calculator 
2:40  
 
Example I: y=sin x 
4:54  
 
Example II: y=x³ and y = d/(dx) (x³) 
9:22  
 
Example III: y = x² {5 <= x <= 0} and y = cos x {0 < x < 6} 
13:15  

The Limit of a Function 
51:53 
 
Intro 
0:00  
 
The Limit of a Function 
0:14  
 
 The Limit of a Function 
0:15  
 
 Graph: Limit of a Function 
12:24  
 
 Table of Values 
16:02  
 
 lim x→a f(x) Does not Say What Happens When x = a 
20:05  
 
Example I: f(x) = x² 
24:34  
 
Example II: f(x) = 7 
27:05  
 
Example III: f(x) = 4.5 
30:33  
 
Example IV: f(x) = 1/x 
34:03  
 
Example V: f(x) = 1/x² 
36:43  
 
The Limit of a Function, Cont. 
38:16  
 
 Infinity and Negative Infinity 
38:17  
 
 Does Not Exist 
42:45  
 
Summary 
46:48  

Example Problems for the Limit of a Function 
24:43 
 
Intro 
0:00  
 
Example I: Explain in Words What the Following Symbols Mean 
0:10  
 
Example II: Find the Following Limit 
5:21  
 
Example III: Use the Graph to Find the Following Limits 
7:35  
 
Example IV: Use the Graph to Find the Following Limits 
11:48  
 
Example V: Sketch the Graph of a Function that Satisfies the Following Properties 
15:25  
 
Example VI: Find the Following Limit 
18:44  
 
Example VII: Find the Following Limit 
20:06  

Calculating Limits Mathematically 
53:48 
 
Intro 
0:00  
 
Plugin Procedure 
0:09  
 
 Plugin Procedure 
0:10  
 
Limit Laws 
9:14  
 
 Limit Law 1 
10:05  
 
 Limit Law 2 
10:54  
 
 Limit Law 3 
11:28  
 
 Limit Law 4 
11:54  
 
 Limit Law 5 
12:24  
 
 Limit Law 6 
13:14  
 
 Limit Law 7 
14:38  
 
Plugin Procedure, Cont. 
16:35  
 
 Plugin Procedure, Cont. 
16:36  
 
Example I: Calculating Limits Mathematically 
20:50  
 
Example II: Calculating Limits Mathematically 
27:37  
 
Example III: Calculating Limits Mathematically 
31:42  
 
Example IV: Calculating Limits Mathematically 
35:36  
 
Example V: Calculating Limits Mathematically 
40:58  
 
Limits Theorem 
44:45  
 
 Limits Theorem 1 
44:46  
 
 Limits Theorem 2: Squeeze Theorem 
46:34  
 
Example VI: Calculating Limits Mathematically 
49:26  

Example Problems for Calculating Limits Mathematically 
21:22 
 
Intro 
0:00  
 
Example I: Evaluate the Following Limit by Showing Each Application of a Limit Law 
0:16  
 
Example II: Evaluate the Following Limit 
1:51  
 
Example III: Evaluate the Following Limit 
3:36  
 
Example IV: Evaluate the Following Limit 
8:56  
 
Example V: Evaluate the Following Limit 
11:19  
 
Example VI: Calculating Limits Mathematically 
13:19  
 
Example VII: Calculating Limits Mathematically 
14:59  

Calculating Limits as x Goes to Infinity 
50:01 
 
Intro 
0:00  
 
Limit as x Goes to Infinity 
0:14  
 
 Limit as x Goes to Infinity 
0:15  
 
 Let's Look at f(x) = 1 / (x3) 
1:04  
 
 Summary 
9:34  
 
Example I: Calculating Limits as x Goes to Infinity 
12:16  
 
Example II: Calculating Limits as x Goes to Infinity 
21:22  
 
Example III: Calculating Limits as x Goes to Infinity 
24:10  
 
Example IV: Calculating Limits as x Goes to Infinity 
36:00  

Example Problems for Limits at Infinity 
36:31 
 
Intro 
0:00  
 
Example I: Calculating Limits as x Goes to Infinity 
0:14  
 
Example II: Calculating Limits as x Goes to Infinity 
3:27  
 
Example III: Calculating Limits as x Goes to Infinity 
8:11  
 
Example IV: Calculating Limits as x Goes to Infinity 
14:20  
 
Example V: Calculating Limits as x Goes to Infinity 
20:07  
 
Example VI: Calculating Limits as x Goes to Infinity 
23:36  

Continuity 
53:00 
 
Intro 
0:00  
 
Definition of Continuity 
0:08  
 
 Definition of Continuity 
0:09  
 
 Example: Not Continuous 
3:52  
 
 Example: Continuous 
4:58  
 
 Example: Not Continuous 
5:52  
 
 Procedure for Finding Continuity 
9:45  
 
Law of Continuity 
13:44  
 
 Law of Continuity 
13:45  
 
Example I: Determining Continuity on a Graph 
15:55  
 
Example II: Show Continuity & Determine the Interval Over Which the Function is Continuous 
17:57  
 
Example III: Is the Following Function Continuous at the Given Point? 
22:42  
 
Theorem for Composite Functions 
25:28  
 
 Theorem for Composite Functions 
25:29  
 
Example IV: Is cos(x³ + ln x) Continuous at x=π/2? 
27:00  
 
Example V: What Value of A Will make the Following Function Continuous at Every Point of Its Domain? 
34:04  
 
Types of Discontinuity 
39:18  
 
 Removable Discontinuity 
39:33  
 
 Jump Discontinuity 
40:06  
 
 Infinite Discontinuity 
40:32  
 
Intermediate Value Theorem 
40:58  
 
 Intermediate Value Theorem: Hypothesis & Conclusion 
40:59  
 
 Intermediate Value Theorem: Graphically 
43:40  
 
Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6] 
47:46  

Derivative I 
40:02 
 
Intro 
0:00  
 
Derivative 
0:09  
 
 Derivative 
0:10  
 
Example I: Find the Derivative of f(x)=x³ 
2:20  
 
Notations for the Derivative 
7:32  
 
 Notations for the Derivative 
7:33  
 
Derivative & Rate of Change 
11:14  
 
 Recall the Rate of Change 
11:15  
 
 Instantaneous Rate of Change 
17:04  
 
 Graphing f(x) and f'(x) 
19:10  
 
Example II: Find the Derivative of x⁴  x² 
24:00  
 
Example III: Find the Derivative of f(x)=√x 
30:51  

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

More Example Problems for The Derivative 
31:38 
 
Intro 
0:00  
 
Example I: Sketch f'(x) 
0:10  
 
Example II: Sketch f'(x) 
2:14  
 
Example III: Find the Derivative of the Following Function sing the Definition 
3:49  
 
Example IV: Determine f, f', and f'' on a Graph 
12:43  
 
Example V: Find an Equation for the Tangent Line to the Graph of the Following Function at the Given xvalue 
13:40  
 
Example VI: Distance vs. Time 
20:15  
 
Example VII: Displacement, Velocity, and Acceleration 
23:56  
 
Example VIII: Graph the Displacement Function 
28:20  
Section 2: Differentiation 

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

The Product, Power & Quotient Rules 
47:25 
 
Intro 
0:00  
 
The Product, Power and Quotient Rules 
0:19  
 
 Differentiate Functions 
0:20  
 
 Product Rule 
5:30  
 
 Quotient Rule 
9:15  
 
 Power Rule 
10:00  
 
Example I: Product Rule 
13:48  
 
Example II: Quotient Rule 
16:13  
 
Example III: Power Rule 
18:28  
 
Example IV: Find dy/dx 
19:57  
 
Example V: Find dy/dx 
24:53  
 
Example VI: Find dy/dx 
28:38  
 
Example VII: Find an Equation for the Tangent to the Curve 
34:54  
 
Example VIII: Find d²y/dx² 
38:08  

Derivatives of the Trigonometric Functions 
41:08 
 
Intro 
0:00  
 
Derivatives of the Trigonometric Functions 
0:09  
 
 Let's Find the Derivative of f(x) = sin x 
0:10  
 
 Important Limits to Know 
4:59  
 
 d/dx (sin x) 
6:06  
 
 d/dx (cos x) 
6:38  
 
 d/dx (tan x) 
6:50  
 
 d/dx (csc x) 
7:02  
 
 d/dx (sec x) 
7:15  
 
 d/dx (cot x) 
7:27  
 
Example I: Differentiate f(x) = x²  4 cos x 
7:56  
 
Example II: Differentiate f(x) = x⁵ tan x 
9:04  
 
Example III: Differentiate f(x) = (cos x) / (3 + sin x) 
10:56  
 
Example IV: Differentiate f(x) = e^x / (tan x  sec x) 
14:06  
 
Example V: Differentiate f(x) = (csc x  4) / (cot x) 
15:37  
 
Example VI: Find an Equation of the Tangent Line 
21:48  
 
Example VII: For What Values of x Does the Graph of the Function x + 3 cos x Have a Horizontal Tangent? 
25:17  
 
Example VIII: Ladder Problem 
28:23  
 
Example IX: Evaluate 
33:22  
 
Example X: Evaluate 
36:38  

The Chain Rule 
24:56 
 
Intro 
0:00  
 
The Chain Rule 
0:13  
 
 Recall the Composite Functions 
0:14  
 
 Derivatives of Composite Functions 
1:34  
 
Example I: Identify f(x) and g(x) and Differentiate 
6:41  
 
Example II: Identify f(x) and g(x) and Differentiate 
9:47  
 
Example III: Differentiate 
11:03  
 
Example IV: Differentiate f(x) = 5 / (x² + 3)³ 
12:15  
 
Example V: Differentiate f(x) = cos(x² + c²) 
14:35  
 
Example VI: Differentiate f(x) = cos⁴x +c² 
15:41  
 
Example VII: Differentiate 
17:03  
 
Example VIII: Differentiate f(x) = sin(tan x²) 
19:01  
 
Example IX: Differentiate f(x) = sin(tan² x) 
21:02  
 
Example X: Differentiate f(x) = (x²2)⁴ (5x²  2x 2)⁸ 
 

More Chain Rule Example Problems 
25:32 
 
Intro 
0:00  
 
Example I: Differentiate f(x) = sin(cos(tanx)) 
0:38  
 
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point 
2:25  
 
Example III: F(x) = f(g(x)), Find F' (6) 
4:22  
 
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window 
5:35  
 
Example V: Differentiate f(x) = ( (x8)/(x+3) )⁴ 
10:18  
 
Example VI: Differentiate f(x) = sec²(12x) 
12:28  
 
Example VII: Differentiate 
14:41  
 
Example VIII: Differentiate 
19:25  
 
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time 
21:13  

Implicit Differentiation 
52:31 
 
Intro 
0:00  
 
Implicit Differentiation 
0:09  
 
 Implicit Differentiation 
0:10  
 
Example I: Find (dy)/(dx) by both Implicit Differentiation and Solving Explicitly for y 
12:15  
 
Example II: Find (dy)/(dx) of x³ + x²y + 7y² = 14 
19:18  
 
Example III: Find (dy)/(dx) of x³y² + y³x² = 4x 
21:43  
 
Example IV: Find (dy)/(dx) of the Following Equation 
24:13  
 
Example V: Find (dy)/(dx) of 6sin x cos y = 1 
29:00  
 
Example VI: Find (dy)/(dx) of x² cos² y + y sin x = 2sin x cos y 
31:02  
 
Example VII: Find (dy)/(dx) of √(xy) = 7 + y²e^x 
37:36  
 
Example VIII: Find (dy)/(dx) of 4(x²+y²)² = 35(x²y²) 
41:03  
 
Example IX: Find (d²y)/(dx²) of x² + y² = 25 
44:05  
 
Example X: Find (d²y)/(dx²) of sin x + cos y = sin(2x) 
47:48  
Section 3: Applications of the Derivative 

Linear Approximations & Differentials 
47:34 
 
Intro 
0:00  
 
Linear Approximations & Differentials 
0:09  
 
 Linear Approximations & Differentials 
0:10  
 
Example I: Linear Approximations & Differentials 
11:27  
 
Example II: Linear Approximations & Differentials 
20:19  
 
Differentials 
30:32  
 
 Differentials 
30:33  
 
Example III: Linear Approximations & Differentials 
34:09  
 
Example IV: Linear Approximations & Differentials 
35:57  
 
Example V: Relative Error 
38:46  

Related Rates 
45:33 
 
Intro 
0:00  
 
Related Rates 
0:08  
 
 Strategy for Solving Related Rates Problems #1 
0:09  
 
 Strategy for Solving Related Rates Problems #2 
1:46  
 
 Strategy for Solving Related Rates Problems #3 
2:06  
 
 Strategy for Solving Related Rates Problems #4 
2:50  
 
 Strategy for Solving Related Rates Problems #5 
3:38  
 
Example I: Radius of a Balloon 
5:15  
 
Example II: Ladder 
12:52  
 
Example III: Water Tank 
19:08  
 
Example IV: Distance between Two Cars 
29:27  
 
Example V: LineofSight 
36:20  

More Related Rates Examples 
37:17 
 
Intro 
0:00  
 
Example I: Shadow 
0:14  
 
Example II: Particle 
4:45  
 
Example III: Water Level 
10:28  
 
Example IV: Clock 
20:47  
 
Example V: Distance between a House and a Plane 
29:11  

Maximum & Minimum Values of a Function 
40:44 
 
Intro 
0:00  
 
Maximum & Minimum Values of a Function, Part 1 
0:23  
 
 Absolute Maximum 
2:20  
 
 Absolute Minimum 
2:52  
 
 Local Maximum 
3:38  
 
 Local Minimum 
4:26  
 
Maximum & Minimum Values of a Function, Part 2 
6:11  
 
 Function with Absolute Minimum but No Absolute Max, Local Max, and Local Min 
7:18  
 
 Function with Local Max & Min but No Absolute Max & Min 
8:48  
 
Formal Definitions 
10:43  
 
 Absolute Maximum 
11:18  
 
 Absolute Minimum 
12:57  
 
 Local Maximum 
14:37  
 
 Local Minimum 
16:25  
 
 Extreme Value Theorem 
18:08  
 
 Theorem: f'(c) = 0 
24:40  
 
 Critical Number (Critical Value) 
26:14  
 
 Procedure for Finding the Critical Values of f(x) 
28:32  
 
Example I: Find the Critical Values of f(x) x + sinx 
29:51  
 
Example II: What are the Absolute Max & Absolute Minimum of f(x) = x + 4 sinx on [0,2π] 
35:31  

Example Problems for Max & Min 
40:44 
 
Intro 
0:00  
 
Example I: Identify Absolute and Local Max & Min on the Following Graph 
0:11  
 
Example II: Sketch the Graph of a Continuous Function 
3:11  
 
Example III: Sketch the Following Graphs 
4:40  
 
Example IV: Find the Critical Values of f (x) = 3x⁴  7x³ + 4x² 
6:13  
 
Example V: Find the Critical Values of f(x) = 2x  5 
8:42  
 
Example VI: Find the Critical Values 
11:42  
 
Example VII: Find the Critical Values f(x) = cos²(2x) on [0,2π] 
16:57  
 
Example VIII: Find the Absolute Max & Min f(x) = 2sinx + 2cos x on [0,(π/3)] 
20:08  
 
Example IX: Find the Absolute Max & Min f(x) = (ln(2x)) / x on [1,3] 
24:39  

The Mean Value Theorem 
25:54 
 
Intro 
0:00  
 
Rolle's Theorem 
0:08  
 
 Rolle's Theorem: If & Then 
0:09  
 
 Rolle's Theorem: Geometrically 
2:06  
 
 There May Be More than 1 c Such That f'( c ) = 0 
3:30  
 
Example I: Rolle's Theorem 
4:58  
 
The Mean Value Theorem 
9:12  
 
 The Mean Value Theorem: If & Then 
9:13  
 
 The Mean Value Theorem: Geometrically 
11:07  
 
Example II: Mean Value Theorem 
13:43  
 
Example III: Mean Value Theorem 
21:19  

Using Derivatives to Graph Functions, Part I 
25:54 
 
Intro 
0:00  
 
Using Derivatives to Graph Functions, Part I 
0:12  
 
 Increasing/ Decreasing Test 
0:13  
 
Example I: Find the Intervals Over Which the Function is Increasing & Decreasing 
3:26  
 
Example II: Find the Local Maxima & Minima of the Function 
19:18  
 
Example III: Find the Local Maxima & Minima of the Function 
31:39  

Using Derivatives to Graph Functions, Part II 
44:58 
 
Intro 
0:00  
 
Using Derivatives to Graph Functions, Part II 
0:13  
 
 Concave Up & Concave Down 
0:14  
 
 What Does This Mean in Terms of the Derivative? 
6:14  
 
 Point of Inflection 
8:52  
 
Example I: Graph the Function 
13:18  
 
Example II: Function x⁴  5x² 
19:03  
 
 Intervals of Increase & Decrease 
19:04  
 
 Local Maxes and Mins 
25:01  
 
 Intervals of Concavity & XValues for the Points of Inflection 
29:18  
 
 Intervals of Concavity & YValues for the Points of Inflection 
34:18  
 
 Graphing the Function 
40:52  

Example Problems I 
49:19 
 
Intro 
0:00  
 
Example I: Intervals, Local Maxes & Mins 
0:26  
 
Example II: Intervals, Local Maxes & Mins 
5:05  
 
Example III: Intervals, Local Maxes & Mins, and Inflection Points 
13:40  
 
Example IV: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity 
23:02  
 
Example V: Intervals, Local Maxes & Mins, Inflection Points, and Intervals of Concavity 
34:36  

Example Problems III 
59:01 
 
Intro 
0:00  
 
Example I: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes 
0:11  
 
Example II: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes 
21:24  
 
Example III: Cubic Equation f(x) = Ax³ + Bx² + Cx + D 
37:56  
 
Example IV: Intervals, Local Maxes & Mins, Inflection Points, Intervals of Concavity, and Asymptotes 
46:19  

L'Hospital's Rule 
30:09 
 
Intro 
0:00  
 
L'Hospital's Rule 
0:19  
 
 Indeterminate Forms 
0:20  
 
 L'Hospital's Rule 
3:38  
 
Example I: Evaluate the Following Limit Using L'Hospital's Rule 
8:50  
 
Example II: Evaluate the Following Limit Using L'Hospital's Rule 
10:30  
 
Indeterminate Products 
11:54  
 
 Indeterminate Products 
11:55  
 
Example III: L'Hospital's Rule & Indeterminate Products 
13:57  
 
Indeterminate Differences 
17:00  
 
 Indeterminate Differences 
17:01  
 
Example IV: L'Hospital's Rule & Indeterminate Differences 
18:57  
 
Indeterminate Powers 
22:20  
 
 Indeterminate Powers 
22:21  
 
Example V: L'Hospital's Rule & Indeterminate Powers 
25:13  

Example Problems for L'Hospital's Rule 
38:14 
 
Intro 
0:00  
 
Example I: Evaluate the Following Limit 
0:17  
 
Example II: Evaluate the Following Limit 
2:45  
 
Example III: Evaluate the Following Limit 
6:54  
 
Example IV: Evaluate the Following Limit 
8:43  
 
Example V: Evaluate the Following Limit 
11:01  
 
Example VI: Evaluate the Following Limit 
14:48  
 
Example VII: Evaluate the Following Limit 
17:49  
 
Example VIII: Evaluate the Following Limit 
20:37  
 
Example IX: Evaluate the Following Limit 
25:16  
 
Example X: Evaluate the Following Limit 
32:44  

Optimization Problems I 
49:59 
 
Intro 
0:00  
 
Example I: Find the Dimensions of the Box that Gives the Greatest Volume 
1:23  
 
Fundamentals of Optimization Problems 
18:08  
 
 Fundamental #1 
18:33  
 
 Fundamental #2 
19:09  
 
 Fundamental #3 
19:19  
 
 Fundamental #4 
20:59  
 
 Fundamental #5 
21:55  
 
 Fundamental #6 
23:44  
 
Example II: Demonstrate that of All Rectangles with a Given Perimeter, the One with the Largest Area is a Square 
24:36  
 
Example III: Find the Points on the Ellipse 9x² + y² = 9 Farthest Away from the Point (1,0) 
35:13  
 
Example IV: Find the Dimensions of the Rectangle of Largest Area that can be Inscribed in a Circle of Given Radius R 
43:10  

Optimization Problems II 
55:10 
 
Intro 
0:00  
 
Example I: Optimization Problem 
0:13  
 
Example II: Optimization Problem 
17:34  
 
Example III: Optimization Problem 
35:06  
 
Example IV: Revenue, Cost, and Profit 
43:22  

Newton's Method 
30:22 
 
Intro 
0:00  
 
Newton's Method 
0:45  
 
 Newton's Method 
0:46  
 
Example I: Find x2 and x3 
13:18  
 
Example II: Use Newton's Method to Approximate 
15:48  
 
Example III: Find the Root of the Following Equation to 6 Decimal Places 
19:57  
 
Example IV: Use Newton's Method to Find the Coordinates of the Inflection Point 
23:11  
Section 4: Integrals 

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

The Area Under a Curve 
51:03 
 
Intro 
0:00  
 
The Area Under a Curve 
0:13  
 
 Approximate Using Rectangles 
0:14  
 
 Let's Do This Again, Using 4 Different Rectangles 
9:40  
 
Approximate with Rectangles 
16:10  
 
 Left Endpoint 
18:08  
 
 Right Endpoint 
25:34  
 
 Left Endpoint vs. Right Endpoint 
30:58  
 
 Number of Rectangles 
34:08  
 
True Area 
37:36  
 
 True Area 
37:37  
 
 Sigma Notation & Limits 
43:32  
 
 When You Have to Explicitly Solve Something 
47:56  

Example Problems for Area Under a Curve 
33:07 
 
Intro 
0:00  
 
Example I: Using Left Endpoint & Right Endpoint to Approximate Area Under a Curve 
0:10  
 
Example II: Using 5 Rectangles, Approximate the Area Under the Curve 
11:32  
 
Example III: Find the True Area by Evaluating the Limit Expression 
16:07  
 
Example IV: Find the True Area by Evaluating the Limit Expression 
24:52  

The Definite Integral 
43:19 
 
Intro 
0:00  
 
The Definite Integral 
0:08  
 
 Definition to Find the Area of a Curve 
0:09  
 
 Definition of the Definite Integral 
4:08  
 
 Symbol for Definite Integral 
8:45  
 
 Regions Below the xaxis 
15:18  
 
 Associating Definite Integral to a Function 
19:38  
 
 Integrable Function 
27:20  
 
Evaluating the Definite Integral 
29:26  
 
 Evaluating the Definite Integral 
29:27  
 
Properties of the Definite Integral 
35:24  
 
 Properties of the Definite Integral 
35:25  

Example Problems for The Definite Integral 
32:14 
 
Intro 
0:00  
 
Example I: Approximate the Following Definite Integral Using Midpoints & Subintervals 
0:11  
 
Example II: Express the Following Limit as a Definite Integral 
5:28  
 
Example III: Evaluate the Following Definite Integral Using the Definition 
6:28  
 
Example IV: Evaluate the Following Integral Using the Definition 
17:06  
 
Example V: Evaluate the Following Definite Integral by Using Areas 
25:41  
 
Example VI: Definite Integral 
30:36  

The Fundamental Theorem of Calculus 
24:17 
 
Intro 
0:00  
 
The Fundamental Theorem of Calculus 
0:17  
 
 Evaluating an Integral 
0:18  
 
 Lim as x → ∞ 
12:19  
 
 Taking the Derivative 
14:06  
 
 Differentiation & Integration are Inverse Processes 
15:04  
 
1st Fundamental Theorem of Calculus 
20:08  
 
 1st Fundamental Theorem of Calculus 
20:09  
 
2nd Fundamental Theorem of Calculus 
22:30  
 
 2nd Fundamental Theorem of Calculus 
22:31  

Example Problems for the Fundamental Theorem 
25:21 
 
Intro 
0:00  
 
Example I: Find the Derivative of the Following Function 
0:17  
 
Example II: Find the Derivative of the Following Function 
1:40  
 
Example III: Find the Derivative of the Following Function 
2:32  
 
Example IV: Find the Derivative of the Following Function 
5:55  
 
Example V: Evaluate the Following Integral 
7:13  
 
Example VI: Evaluate the Following Integral 
9:46  
 
Example VII: Evaluate the Following Integral 
12:49  
 
Example VIII: Evaluate the Following Integral 
13:53  
 
Example IX: Evaluate the Following Graph 
15:24  
 
 Local Maxs and Mins for g(x) 
15:25  
 
 Where Does g(x) Achieve Its Absolute Max on [0,8] 
20:54  
 
 On What Intervals is g(x) Concave Up/Down? 
22:20  
 
 Sketch a Graph of g(x) 
24:34  

More Example Problems, Including Net Change Applications 
34:22 
 
Intro 
0:00  
 
Example I: Evaluate the Following Indefinite Integral 
0:10  
 
Example II: Evaluate the Following Definite Integral 
0:59  
 
Example III: Evaluate the Following Integral 
2:59  
 
Example IV: Velocity Function 
7:46  
 
 Part A: Net Displacement 
7:47  
 
 Part B: Total Distance Travelled 
13:15  
 
Example V: Linear Density Function 
20:56  
 
Example VI: Acceleration Function 
25:10  
 
 Part A: Velocity Function at Time t 
25:11  
 
 Part B: Total Distance Travelled During the Time Interval 
28:38  

Solving Integrals by Substitution 
27:20 
 
Intro 
0:00  
 
Table of Integrals 
0:35  
 
Example I: Evaluate the Following Indefinite Integral 
2:02  
 
Example II: Evaluate the Following Indefinite Integral 
7:27  
 
Example IIII: Evaluate the Following Indefinite Integral 
10:57  
 
Example IV: Evaluate the Following Indefinite Integral 
12:33  
 
Example V: Evaluate the Following 
14:28  
 
Example VI: Evaluate the Following 
16:00  
 
Example VII: Evaluate the Following 
19:01  
 
Example VIII: Evaluate the Following 
21:49  
 
Example IX: Evaluate the Following 
24:34  
Section 5: Applications of Integration 

Areas Between Curves 
34:56 
 
Intro 
0:00  
 
Areas Between Two Curves: Function of x 
0:08  
 
 Graph 1: Area Between f(x) & g(x) 
0:09  
 
 Graph 2: Area Between f(x) & g(x) 
4:07  
 
 Is It Possible to Write as a Single Integral? 
8:20  
 
 Area Between the Curves on [a,b] 
9:24  
 
 Absolute Value 
10:32  
 
 Formula for Areas Between Two Curves: Top Function  Bottom Function 
17:03  
 
Areas Between Curves: Function of y 
17:49  
 
 What if We are Given Functions of y? 
17:50  
 
 Formula for Areas Between Two Curves: Right Function  Left Function 
21:48  
 
 Finding a & b 
22:32  

Example Problems for Areas Between Curves 
42:55 
 
Intro 
0:00  
 
Instructions for the Example Problems 
0:10  
 
Example I: y = 7x  x² and y=x 
0:37  
 
Example II: x=y²3, x=e^((1/2)y), y=1, and y=2 
6:25  
 
Example III: y=(1/x), y=(1/x³), and x=4 
12:25  
 
Example IV: 152x² and y=x²5 
15:52  
 
Example V: x=(1/8)y³ and x=6y² 
20:20  
 
Example VI: y=cos x, y=sin(2x), [0,π/2] 
24:34  
 
Example VII: y=2x², y=10x², 7x+2y=10 
29:51  
 
Example VIII: Velocity vs. Time 
33:23  
 
 Part A: At 2.187 Minutes, Which care is Further Ahead? 
33:24  
 
 Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent? 
36:32  
 
 Part C: At 4 Minutes Which Car is Ahead? 
37:11  
 
 Part D: At What Time Will the Cars be Side by Side? 
37:50  

Volumes I: Slices 
34:15 
 
Intro 
0:00  
 
Volumes I: Slices 
0:18  
 
 Rotate the Graph of y=√x about the xaxis 
0:19  
 
 How can I use Integration to Find the Volume? 
3:16  
 
 Slice the Solid Like a Loaf of Bread 
5:06  
 
 Volumes Definition 
8:56  
 
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation 
12:18  
 
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation 
19:05  
 
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Given Functions about the Given Line of Rotation 
25:28  

Volumes II: Volumes by Washers 
51:43 
 
Intro 
0:00  
 
Volumes II: Volumes by Washers 
0:11  
 
 Rotating Region Bounded by y=x³ & y=x around the xaxis 
0:12  
 
 Equation for Volumes by Washer 
11:14  
 
 Process for Solving Volumes by Washer 
13:40  
 
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis 
15:58  
 
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis 
25:07  
 
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis 
34:20  
 
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis 
44:05  

Volumes III: Solids That Are Not SolidsofRevolution 
49:36 
 
Intro 
0:00  
 
Solids That Are Not SolidsofRevolution 
0:11  
 
 CrossSection Area Review 
0:12  
 
 CrossSections That Are Not SolidsofRevolution 
7:36  
 
Example I: Find the Volume of a Pyramid Whose Base is a Square of Sidelength S, and Whose Height is H 
10:54  
 
Example II: Find the Volume of a Solid Whose Crosssectional Areas Perpendicular to the Base are Equilateral Triangles 
20:39  
 
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of SideLength A, and Whose Height is H 
29:27  
 
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64 
36:47  
 
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3x² and the xaxis 
46:13  

Volumes IV: Volumes By Cylindrical Shells 
50:02 
 
Intro 
0:00  
 
Volumes by Cylindrical Shells 
0:11  
 
 Find the Volume of the Following Region 
0:12  
 
 Volumes by Cylindrical Shells: Integrating Along x 
14:12  
 
 Volumes by Cylindrical Shells: Integrating Along y 
14:40  
 
 Volumes by Cylindrical Shells Formulas 
16:22  
 
Example I: Using the Method of Cylindrical Shells, Find the Volume of the Solid 
18:33  
 
Example II: Using the Method of Cylindrical Shells, Find the Volume of the Solid 
25:57  
 
Example III: Using the Method of Cylindrical Shells, Find the Volume of the Solid 
31:38  
 
Example IV: Using the Method of Cylindrical Shells, Find the Volume of the Solid 
38:44  
 
Example V: Using the Method of Cylindrical Shells, Find the Volume of the Solid 
44:03  

The Average Value of a Function 
32:13 
 
Intro 
0:00  
 
The Average Value of a Function 
0:07  
 
 Average Value of f(x) 
0:08  
 
 What if The Domain of f(x) is Not Finite? 
2:23  
 
 Let's Calculate Average Value for f(x) = x² [2,5] 
4:46  
 
 Mean Value Theorem for Integrate 
9:25  
 
Example I: Find the Average Value of the Given Function Over the Given Interval 
14:06  
 
Example II: Find the Average Value of the Given Function Over the Given Interval 
18:25  
 
Example III: Find the Number A Such that the Average Value of the Function f(x) = 4x² + 8x + 4 Equals 2 Over the Interval [1,A] 
24:04  
 
Example IV: Find the Average Density of a Rod 
27:47  
Section 6: Techniques of Integration 

Integration by Parts 
50:32 
 
Intro 
0:00  
 
Integration by Parts 
0:08  
 
 The Product Rule for Differentiation 
0:09  
 
 Integrating Both Sides Retains the Equality 
0:52  
 
 Differential Notation 
2:24  
 
Example I: ∫ x cos x dx 
5:41  
 
Example II: ∫ x² sin(2x)dx 
12:01  
 
Example III: ∫ (e^x) cos x dx 
18:19  
 
Example IV: ∫ (sin^1) (x) dx 
23:42  
 
Example V: ∫₁⁵ (lnx)² dx 
28:25  
 
Summary 
32:31  
 
Tabular Integration 
35:08  
 
 Case 1 
35:52  
 
 Example: ∫x³sinx dx 
36:39  
 
 Case 2 
40:28  
 
 Example: ∫e^(2x) sin 3x 
41:14  

Trigonometric Integrals I 
24:50 
 
Intro 
0:00  
 
Example I: ∫ sin³ (x) dx 
1:36  
 
Example II: ∫ cos⁵(x)sin²(x)dx 
4:36  
 
Example III: ∫ sin⁴(x)dx 
9:23  
 
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sin^m) (x) (cos^p) (x) dx 
15:59  
 
 #1: Power of sin is Odd 
16:00  
 
 #2: Power of cos is Odd 
16:41  
 
 #3: Powers of Both sin and cos are Odd 
16:55  
 
 #4: Powers of Both sin and cos are Even 
17:10  
 
Example IV: ∫ tan⁴ (x) sec⁴ (x) dx 
17:34  
 
Example V: ∫ sec⁹(x) tan³(x) dx 
20:55  
 
Summary for Evaluating Trigonometric Integrals of the Following Type: ∫ (sec^m) (x) (tan^p) (x) dx 
23:31  
 
 #1: Power of sec is Odd 
23:32  
 
 #2: Power of tan is Odd 
24:04  
 
 #3: Powers of sec is Odd and/or Power of tan is Even 
24:18  

Trigonometric Integrals II 
22:12 
 
Intro 
0:00  
 
Trigonometric Integrals II 
0:09  
 
 Recall: ∫tanx dx 
0:10  
 
 Let's Find ∫secx dx 
3:23  
 
Example I: ∫ tan⁵ (x) dx 
6:23  
 
Example II: ∫ sec⁵ (x) dx 
11:41  
 
Summary: How to Deal with Integrals of Different Types 
19:04  
 
 Identities to Deal with Integrals of Different Types 
19:05  
 
Example III: ∫cos(5x)sin(9x)dx 
19:57  

More Example Problems for Trigonometric Integrals 
17:22 
 
Intro 
0:00  
 
Example I: ∫sin²(x)cos⁷(x)dx 
0:14  
 
Example II: ∫x sin²(x) dx 
3:56  
 
Example III: ∫csc⁴ (x/5)dx 
8:39  
 
Example IV: ∫( (1tan²x)/(sec²x) ) dx 
11:17  
 
Example V: ∫ 1 / (sinx1) dx 
13:19  

Integration by Partial Fractions I 
55:12 
 
Intro 
0:00  
 
Integration by Partial Fractions I 
0:11  
 
 Recall the Idea of Finding a Common Denominator 
0:12  
 
 Decomposing a Rational Function to Its Partial Fractions 
4:10  
 
 2 Types of Rational Function: Improper & Proper 
5:16  
 
Improper Rational Function 
7:26  
 
 Improper Rational Function 
7:27  
 
Proper Rational Function 
11:16  
 
 Proper Rational Function & Partial Fractions 
11:17  
 
 Linear Factors 
14:04  
 
 Irreducible Quadratic Factors 
15:02  
 
Case 1: G(x) is a Product of Distinct Linear Factors 
17:10  
 
Example I: Integration by Partial Fractions 
20:33  
 
Case 2: D(x) is a Product of Linear Factors 
40:58  
 
Example II: Integration by Partial Fractions 
44:41  

Integration by Partial Fractions II 
42:57 
 
Intro 
0:00  
 
Case 3: D(x) Contains Irreducible Factors 
0:09  
 
Example I: Integration by Partial Fractions 
5:19  
 
Example II: Integration by Partial Fractions 
16:22  
 
Case 4: D(x) has Repeated Irreducible Quadratic Factors 
27:30  
 
Example III: Integration by Partial Fractions 
30:19  
Section 7: Differential Equations 

Introduction to Differential Equations 
46:37 
 
Intro 
0:00  
 
Introduction to Differential Equations 
0:09  
 
 Overview 
0:10  
 
 Differential Equations Involving Derivatives of y(x) 
2:08  
 
 Differential Equations Involving Derivatives of y(x) and Function of y(x) 
3:23  
 
 Equations for an Unknown Number 
6:28  
 
 What are These Differential Equations Saying? 
10:30  
 
Verifying that a Function is a Solution of the Differential Equation 
13:00  
 
 Verifying that a Function is a Solution of the Differential Equation 
13:01  
 
 Verify that y(x) = 4e^x + 3x² + 6x + e^π is a Solution of this Differential Equation 
17:20  
 
 General Solution 
22:00  
 
 Particular Solution 
24:36  
 
 Initial Value Problem 
27:42  
 
Example I: Verify that a Family of Functions is a Solution of the Differential Equation 
32:24  
 
Example II: For What Values of K Does the Function Satisfy the Differential Equation 
36:07  
 
Example III: Verify the Solution and Solve the Initial Value Problem 
39:47  

Separation of Variables 
28:08 
 
Intro 
0:00  
 
Separation of Variables 
0:28  
 
 Separation of Variables 
0:29  
 
Example I: Solve the Following g Initial Value Problem 
8:29  
 
Example II: Solve the Following g Initial Value Problem 
13:46  
 
Example III: Find an Equation of the Curve 
18:48  

Population Growth: The Standard & Logistic Equations 
51:07 
 
Intro 
0:00  
 
Standard Growth Model 
0:30  
 
 Definition of the Standard/Natural Growth Model 
0:31  
 
 Initial Conditions 
8:00  
 
 The General Solution 
9:16  
 
Example I: Standard Growth Model 
10:45  
 
Logistic Growth Model 
18:33  
 
 Logistic Growth Model 
18:34  
 
 Solving the Initial Value Problem 
25:21  
 
 What Happens When t → ∞ 
36:42  
 
Example II: Solve the Following g Initial Value Problem 
41:50  
 
Relative Growth Rate 
46:56  
 
 Relative Growth Rate 
46:57  
 
 Relative Growth Rate Version for the Standard model 
49:04  

Slope Fields 
24:37 
 
Intro 
0:00  
 
Slope Fields 
0:35  
 
 Slope Fields 
0:36  
 
 Graphing the Slope Fields, Part 1 
11:12  
 
 Graphing the Slope Fields, Part 2 
15:37  
 
 Graphing the Slope Fields, Part 3 
17:25  
 
Steps to Solving Slope Field Problems 
20:24  
 
Example I: Draw or Generate the Slope Field of the Differential Equation y'=x cos y 
22:38  
Section 8: AP Practic Exam 

AP Practice Exam: Section 1, Part A No Calculator 
45:29 
 
Intro 
0:00  
 
Exam Link 
0:10  
 
Problem #1 
1:26  
 
Problem #2 
2:52  
 
Problem #3 
4:42  
 
Problem #4 
7:03  
 
Problem #5 
10:01  
 
Problem #6 
13:49  
 
Problem #7 
15:16  
 
Problem #8 
19:06  
 
Problem #9 
23:10  
 
Problem #10 
28:10  
 
Problem #11 
31:30  
 
Problem #12 
33:53  
 
Problem #13 
37:45  
 
Problem #14 
41:17  

AP Practice Exam: Section 1, Part A No Calculator, cont. 
41:55 
 
Intro 
0:00  
 
Problem #15 
0:22  
 
Problem #16 
3:10  
 
Problem #17 
5:30  
 
Problem #18 
8:03  
 
Problem #19 
9:53  
 
Problem #20 
14:51  
 
Problem #21 
17:30  
 
Problem #22 
22:12  
 
Problem #23 
25:48  
 
Problem #24 
29:57  
 
Problem #25 
33:35  
 
Problem #26 
35:57  
 
Problem #27 
37:57  
 
Problem #28 
40:04  

AP Practice Exam: Section I, Part B Calculator Allowed 
58:47 
 
Intro 
0:00  
 
Problem #1 
1:22  
 
Problem #2 
4:55  
 
Problem #3 
10:49  
 
Problem #4 
13:05  
 
Problem #5 
14:54  
 
Problem #6 
17:25  
 
Problem #7 
18:39  
 
Problem #8 
20:27  
 
Problem #9 
26:48  
 
Problem #10 
28:23  
 
Problem #11 
34:03  
 
Problem #12 
36:25  
 
Problem #13 
39:52  
 
Problem #14 
43:12  
 
Problem #15 
47:18  
 
Problem #16 
50:41  
 
Problem #17 
56:38  

AP Practice Exam: Section II, Part A Calculator Allowed 
25:40 
 
Intro 
0:00  
 
Problem #1: Part A 
1:14  
 
Problem #1: Part B 
4:46  
 
Problem #1: Part C 
8:00  
 
Problem #2: Part A 
12:24  
 
Problem #2: Part B 
16:51  
 
Problem #2: Part C 
17:17  
 
Problem #3: Part A 
18:16  
 
Problem #3: Part B 
19:54  
 
Problem #3: Part C 
21:44  
 
Problem #3: Part D 
22:57  

AP Practice Exam: Section II, Part B No Calculator 
31:20 
 
Intro 
0:00  
 
Problem #4: Part A 
1:35  
 
Problem #4: Part B 
5:54  
 
Problem #4: Part C 
8:50  
 
Problem #4: Part D 
9:40  
 
Problem #5: Part A 
11:26  
 
Problem #5: Part B 
13:11  
 
Problem #5: Part C 
15:07  
 
Problem #5: Part D 
19:57  
 
Problem #6: Part A 
22:01  
 
Problem #6: Part B 
25:34  
 
Problem #6: Part C 
28:54  