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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Example Problems I

Slide Duration:

Table of Contents

I. Limits and Derivatives
Overview & Slopes of Curves

42m 8s

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

50m 53s

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

59m 12s

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

18m 43s

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

51m 53s

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

24m 43s

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

53m 48s

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

21m 22s

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

50m 1s

Intro
0:00
Limit as x Goes to Infinity
0:14
Limit as x Goes to Infinity
0:15
Let's Look at f(x) = 1 / (x-3)
1:04
Summary
9:34
Example I: Calculating Limits as x Goes to Infinity
12:16
Example II: Calculating Limits as x Goes to Infinity
21:22
Example III: Calculating Limits as x Goes to Infinity
24:10
Example IV: Calculating Limits as x Goes to Infinity
36:00
Example Problems for Limits at Infinity

36m 31s

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

53m

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

40m 2s

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

53m 45s

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

31m 38s

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

47m 35s

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

47m 25s

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

41m 8s

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

24m 56s

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
More Chain Rule Example Problems

25m 32s

Intro
0:00
Example I: Differentiate f(x) = sin(cos(tanx))
0:38
Example II: Find an Equation for the Line Tangent to the Given Curve at the Given Point
2:25
Example III: F(x) = f(g(x)), Find F' (6)
4:22
Example IV: Differentiate & Graph both the Function & the Derivative in the Same Window
5:35
Example V: Differentiate f(x) = ( (x-8)/(x+3) )⁴
10:18
Example VI: Differentiate f(x) = sec²(12x)
12:28
Example VII: Differentiate
14:41
Example VIII: Differentiate
19:25
Example IX: Find an Expression for the Rate of Change of the Volume of the Balloon with Respect to Time
21:13
Implicit Differentiation

52m 31s

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

47m 34s

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

45m 33s

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: Line-of-Sight
36:20
More Related Rates Examples

37m 17s

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

40m 44s

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

40m 44s

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

25m 54s

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

25m 54s

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

44m 58s

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

49m 19s

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

59m 1s

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

30m 9s

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

38m 14s

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

49m 59s

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

55m 10s

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

30m 22s

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
IV. Integrals
Antiderivatives

55m 26s

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

51m 3s

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

33m 7s

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

43m 19s

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

32m 14s

Intro
0:00
Example I: Approximate the Following Definite Integral Using Midpoints & Sub-intervals
0:11
Example II: Express the Following Limit as a Definite Integral
5:28
Example III: Evaluate the Following Definite Integral Using the Definition
6:28
Example IV: Evaluate the Following Integral Using the Definition
17:06
Example V: Evaluate the Following Definite Integral by Using Areas
25:41
Example VI: Definite Integral
30:36
The Fundamental Theorem of Calculus

24m 17s

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

25m 21s

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

34m 22s

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

27m 20s

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
V. Applications of Integration
Areas Between Curves

34m 56s

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

42m 55s

Intro
0:00
Instructions for the Example Problems
0:10
Example I: y = 7x - x² and y=x
0:37
Example II: x=y²-3, x=e^((1/2)y), y=-1, and y=2
6:25
Example III: y=(1/x), y=(1/x³), and x=4
12:25
Example IV: 15-2x² and y=x²-5
15:52
Example V: x=(1/8)y³ and x=6-y²
20:20
Example VI: y=cos x, y=sin(2x), [0,π/2]
24:34
Example VII: y=2x², y=10x², 7x+2y=10
29:51
Example VIII: Velocity vs. Time
33:23
Part A: At 2.187 Minutes, Which care is Further Ahead?
33:24
Part B: If We Shaded the Region between the Graphs from t=0 to t=2.187, What Would This Shaded Area Represent?
36:32
Part C: At 4 Minutes Which Car is Ahead?
37:11
Part D: At What Time Will the Cars be Side by Side?
37:50
Volumes I: Slices

34m 15s

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

51m 43s

Intro
0:00
Volumes II: Volumes by Washers
0:11
Rotating Region Bounded by y=x³ & y=x around the x-axis
0:12
Equation for Volumes by Washer
11:14
Process for Solving Volumes by Washer
13:40
Example I: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
15:58
Example II: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
25:07
Example III: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
34:20
Example IV: Find the Volume of the Solid Obtained by Rotating the Region Bounded by the Following Functions around the Given Axis
44:05
Volumes III: Solids That Are Not Solids-of-Revolution

49m 36s

Intro
0:00
Solids That Are Not Solids-of-Revolution
0:11
Cross-Section Area Review
0:12
Cross-Sections That Are Not Solids-of-Revolution
7:36
Example I: Find the Volume of a Pyramid Whose Base is a Square of Side-length S, and Whose Height is H
10:54
Example II: Find the Volume of a Solid Whose Cross-sectional Areas Perpendicular to the Base are Equilateral Triangles
20:39
Example III: Find the Volume of a Pyramid Whose Base is an Equilateral Triangle of Side-Length A, and Whose Height is H
29:27
Example IV: Find the Volume of a Solid Whose Base is Given by the Equation 16x² + 4y² = 64
36:47
Example V: Find the Volume of a Solid Whose Base is the Region Bounded by the Functions y=3-x² and the x-axis
46:13
Volumes IV: Volumes By Cylindrical Shells

50m 2s

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

32m 13s

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
VI. Techniques of Integration
Integration by Parts

50m 32s

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

24m 50s

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

22m 12s

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

17m 22s

Intro
0:00
Example I: ∫sin²(x)cos⁷(x)dx
0:14
Example II: ∫x sin²(x) dx
3:56
Example III: ∫csc⁴ (x/5)dx
8:39
Example IV: ∫( (1-tan²x)/(sec²x) ) dx
11:17
Example V: ∫ 1 / (sinx-1) dx
13:19
Integration by Partial Fractions I

55m 12s

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

42m 57s

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
VII. Differential Equations
Introduction to Differential Equations

46m 37s

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

28m 8s

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

51m 7s

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

24m 37s

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

45m 29s

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.

41m 55s

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

58m 47s

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

25m 40s

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

31m 20s

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
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Lecture Comments (5)

1 answer

Last reply by: Professor Hovasapian
Tue Jul 19, 2016 6:07 AM

Post by Peter Ke on July 15, 2016

For example V, how is the point of inflection is (0.7, -0.7)?

2 answers

Last reply by: Professor Hovasapian
Mon Jul 25, 2016 6:49 PM

Post by Acme Wang on April 7, 2016

Hi Professor,

I felt confused in finding the horizontal asymptote of 10x^2/(x^2+5). When x goes to infinity, would f(x) also approaches infinity since 10x^2 goes to infinity and (x^2+5) goes to infinity?

Thank you very much!

Sincerely,

Acme

Example Problems I

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • 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

Transcription: Example Problems I

Hello, and welcome back to www.educator.com, and welcome back to AP Calculus.0000

In the last couple of lessons, we discussed the first derivative and the second derivative.0005

And how they are used in order to decide what graph of a function looks like.0009

In today's lesson and in the next lesson, we are going to do example problems using these techniques,0015

getting progressively more complex as far as the functions that we are dealing with.0020

Let us jump right on in.0024

The following is the graph of the first derivative of some function,0027

on which the original function increasing or decreasing and are there any local maxes or mins?0032

With these particular problems, you have to be very careful to remember which function you are dealing with,0040

which graph they actually gave you.0046

In this case, they gave you the first derivative.0048

This is not the function itself, this is not the second derivative, it is the first derivative.0051

This is f’(x), we have to remember that.0057

In the process of deciding whether it is increasing/decreasing, concave up, concave down,0060

what is a local max and what is a local min.0067

It is going to start to get confusing because you are going to try to describe the behavior of the function,0069

but what you are given is the first derivative of the function.0075

We know that increasing is where the first derivative is greater than 0.0080

Decreasing is where the first derivative is less than 0.0085

This is the first derivative.0088

At this point, between this point and this point, the original function is increasing0090

because the first derivative is positive, it is above the x axis.0096

To the left of this point which is -1, the first derivative which is this graph is negative, it is below the x axis0100

Pass the point 4, it is also negative.0110

The function itself is decreasing on this interval, increasing on this interval, decreasing on this interval.0117

Let me go ahead and work in blue here.0130

I’m just going to write right on top of the graph.0134

Our intervals of increase, it is when f’ is greater than 0.0137

Our interval of increase is from, this is -1.0148

I think I read the graph wrong.0161

It looks like this is -1 and this looks like this 4, from -1 to 4.0166

The interval of decrease is when the first derivative is actually less than 0.0177

In this case, decrease happens on the interval from -infinity to -1 union 4, all the way to infinity.0184

These points, where the first derivative actually equal 00211

because this is the first derivative of the graph, this one.0214

This -1 and this 4, those are the local extrema.0217

F’(x) = 0 at -1 and 4.0225

These are the local extrema, in other words the local mins and local max.0236

Let us see which one is which.0244

At -1, we have decreasing, increasing.0247

You have a local min at x = -1.0253

Over here, it is increasing/decreasing.0262

You have a local max at x = 4.0267

That is it, that is all that is happening here.0276

Just be very careful which graph they are giving you to read it.0279

In the process, you are going to get confused.0284

That is not a problem, that is the whole process.0287

We have all gone through the process, we have all gotten confused.0290

We have all make some minor mistakes and some gross mistakes, regarding this.0292

This is the process, just be extra careful.0297

You have to be very vigilant here.0300

Let us do another example, same type.0305

The following is a graph of the first derivative.0307

This is f’ of a function, on which intervals is the function, the original function increasing/decreasing.0312

Are there any local maxes or mins, you also want more information,0322

on which interval is the function concave up and concave down?0326

What are the x values of the points of inflection?0330

A lot of information that they require here.0333

Let us go through increasing/decreasing like we did before.0336

Once again, increasing is where f’ is greater than 0, decreasing is where f’ is less than 0.0339

This is the f’ graph, therefore here, between here and here, and from here onward, f’ is positive.0346

Therefore, the function is increasing.0361

Therefore, our increasing interval, I will write it over here, is going to be from, this looks roughly like -1.8 all the way to about here.0363

We can read off as, let us just call it 0.9.0380

This should be an open.0385

This is roughly 2.2, all the way to +infinity.0389

This is where the function, the original function is increasing.0399

It is increasing where the first derivative is positive, positive above the x axis.0404

Therefore, our interval of decrease, it is negative from -infinity all the way to this -1.8.0411

It is also below between this 0.9 and this 2.2.0422

These are our intervals of increase and decrease.0431

What we have here is decreasing on this interval, increasing on this interval, decreasing on this interval, increasing on that interval.0435

Therefore, that makes this point right here.0445

I will say the local maxes are going to be this one which is at x = 0.9.0456

Our local min, decreasing/increasing, decreasing/increasing.0470

We have -1.8 and our 2.2.0479

Remember, what we are talking about here is the actual original function.0486

What we had here is our first derivative graph.0492

This does not describe the graph.0497

We are using the graph to describe the original function and we are doing it analytically, things that we know.0500

Increasing/decreasing, positive, negative.0507

Let us move on to our next page here.0512

We have taken care of the local maxes and mins.0515

We have taken care of the increasing/decreasing.0517

Let us find some points of inflection and some intervals of concavity here.0519

Once again, let us remind ourselves actually that this is f’ not f”.0526

We want to talk about some inflection points.0535

Inflection points, we said that inflection points are points where f” is equal to 0.0538

F” is equal to 0, since this graph that we are looking at, since this graph is f’, f” is the slope of this graph.0553

F” is the slope of this graph.0571

I hope that make sense.0584

We want the slope of this graph, where is it the slope of this graph equal 0?0588

It equal 0 there, there, there, and there.0593

What are these x values, it is going to be some place like right there, right there, right there, and right there.0602

Therefore, at x equals -1.25, -0.6, 0, it looks like about 1.6.0613

At these points, the second derivative equal 0.0633

The second derivative is the slope of this graph which is the graph of the first derivative.0637

These are points of inflection.0642

We take those points of inflection, we put them on a number line.0644

We evaluate whether the second derivative is positive or negative.0652

To the left or right of those points.0660

I have got -1.25, -0.6, 0, and 1.6.0662

I need to check where the second derivative.0674

We are dealing with the second derivative here.0677

I need to check this region, this region, this interval, this interval, and this interval,0679

to see whether it is concave up or concave down.0687

I look to the left of -1.25, the slope here is positive, it is concave up.0692

From -1.25 to -0.6, slope is negative, it is concave down.0704

Not the graph, what we are talking about here is the original function.0711

This is where the confusion lies in.0715

This is not concave up but because this is the f’ graph, f” of the original function is the slope of this graph.0717

The slope is positive, therefore, the original function is concave up.0727

Let us write concave up, concave down.0733

From 0.6 to 0, the slope is positive, this is concave up.0736

From 0 to about 1.6, the slope is negative, we are looking at concave down.0741

From here onward, the slope of this is positive.0749

F’ is positive, this means it is concave up.0753

Therefore, our intervals of concavity, concave up are, -infinity to -1.25 union -0.6 to 0 union 1.6 to +infinity.0757

We are going to be concave down from -1.25 all the way to -0.6.0777

And union where it is negative, it is going to be from 0 all the way to +1.6.0786

I hope it makes sense what it is that we have down here.0795

There you go, you got your points of inflection, you have your intervals of concavity0803

We found our local maxes and mins, and we found out the intervals of increase and decrease of the actual function.0809

It looks like we have everything.0815

Let us try another one.0820

The following is a graph of the first derivative.0823

Let us remind ourselves, we are looking at f’ of a function.0828

On which intervals of the function increasing/decreasing?0832

Are there local maxes and mins, on which intervals of the function concave up and concave down?0835

What are the x values of the points of inflection?0840

The exact same thing as what we just did, we are going to do it again.0842

Increasing/decreasing is where the first derivative is positive/negative respectively.0850

Right about there, it looks like about 2.8, 2.7, 2.8, something like that.0858

To the right of that, that is where the first root is positive.0864

To the left of that, it is all below the x axis.0868

F’ is negative, it is decreasing.0874

Do not let this fool you, just read right off the graph.0878

Trust the math, do not trust your instinct.0883

Your instinct is going to want you to see this as the function.0887

This is not the function, this is the derivative of the function.0891

Our intervals of increase is 2.8 all the way to +infinity, 2.8 onward, that way.0895

Our intervals of decrease, we have -infinity to 2.8.0908

-infinity to 2.8.0914

This is going to be, it is negative so it is decreasing.0921

Here it is increasing, we are going to have a local min at 2.8.0925

That is it, local min at x = 2.8.0931

There are no local maxes because there is no place where it goes from increasing to decreasing.0942

In this case, there is no local max.0947

Let us go ahead and talk about some inflection points.0954

I think I can go ahead and do this one in red.0957

Inflection points, we said inflection points are points where the second derivative is equal to 0.0961

Inflection points, there are places where f” is actually equal to 0.0974

F” is f’ of f’, it is going to be the derivative of this graph.0985

It is going to be the slope of that graph.0997

F” which is the slope of this graph is equal to 0 at 2 points.1004

That right there and about right there.1015

Points of inflection are going to be x = -0.9 and x = roughly 1.5.1020

Once again, we have our -0.9, we have our 1.5.1045

Let us go ahead and do our f’ check.1050

We have -0.9 and we have our 1.5.1058

I need to check this region, this region, and this region.1063

To the left of 0.9, the slope of this graph is positive that means the f” is positive.1069

Therefore, the slope is concave up.1077

From -0.9 to 1.5, from here to here, the slope is negative, this means it is concave down.1082

The slope, remember this is f’.1093

The slope is f” of the original function, concave down.1098

From this point onward, we have a positive slope, positive slope, it is concave up.1103

Therefore, our intervals of concavity are concave up from -infinity to -0.9 union at 1.5, all the way to +infinity.1111

We have concave down from -0.9 all the way to 1.5.1126

Again, be very careful, when you know you are dealing with f’.1138

Now I’m going to show you the image of all three graphs right on top of each other, to see what is going on.1143

This was the graph that we were given, this is f’.1155

This is the graph of f”, in other words, it is the slope of this purple.1162

This is actually f(x), this is the one whose behavior we listed and elucidated.1170

Let us double check.1178

Let us go back to blue here.1180

We said the following, we said our interval of increase is 2.8 to +infinity.1182

We look at our original function, it is this one the blue.1211

Yes, decrease, decrease.1215

Yes, from 2.8 onward.1217

That checks out, very nice.1221

We said that it decreased from -infinity all the way to this 2.8.1225

Sure enough, the function from -infinity as we move from left to right,1230

the function is decreasing, decreasing, decreasing, decreasing, until it hits 2.8.1235

Yes, that checks out.1239

We said that we had a local min at 2.8.1242

Yes, there is our local min at 2.8, right there, that checks.1248

We said we had inflection points, let us see what we have got.1255

Inflection points, we said that we have an inflection point at x = -0.9.1261

We also said we had one at x = 1.5.1267

Let us go to -0.9, roughly right about there.1275

Yes, there it is, our blue.1279

There is our inflection point, it changes from concave up to concave down.1280

And then, roughly around 1.5, right about there.1285

There we go at 1.5, at this point.1289

It goes from concave down to concave up.1291

Yes, these two check out.1295

And then, we had our intervals of concavity.1298

We said that we have an interval of concavity from -infinity to -0.9 union at 1.5, all the way to +infinity.1301

Let us double check.1311

Concave up, concave up from -infinity to -0.9.1312

From 1.5 all the way to +infinity, 1.5 to +infinity.1317

Yes, that checks out.1321

The last thing we want to double check is our concave down, from negative 0.9 all the way to 1.5.1323

Yes, from -0.9 all the way to about 1.5.1332

Yes, the graph is concave down, there we go.1337

Here, we see them all together.1342

The graph they gave us was this one.1344

This is the graph they gave us.1350

From that, I was able to elucidate all of this information that corroborated the actual function, which is this.1352

Be very careful with this, that is the take home lesson, just be vigilant.1361

As in all things with, when it comes down to higher math and higher science.1369

You just have to be extra vigilant, there is a lot happening.1373

You have to keep track of every little thing.1376

Let us go ahead and do some analytical work here.1380

For the function 10x²/ x² + 5, find the intervals of increase and decrease,1385

the local maxes and mins, the points of inflection, and the intervals of concavity.1395

Use this information to actually draw the graph.1400

Let us go ahead and do it.1402

I’m going to go back to blue here.1406

I have got f’(x) is equal to, I have this × the derivative of that - that × the derivative of this/ that².1409

I got x² + 5 × 20x - 10x².1420

I really hope to God that I did my arithmetic correctly.1427

All over x² + 5, I’m going to rely on you to double check that for me.1431

All of that is equal to, when I multiply it out, I get 20x³ + 100x - 20x³/ x² + 5².1439

Those cancel, I'm left with 100x/ x² + 5².1456

This is my first derivative, I’m going to set that first derivative equal to 0.1470

When I set it equal to 0, the denominator was irrelevant.1474

It is only 0 when the numerator is 0.1478

100x = 0, which means that x = 0, that is my critical value.1481

I need to check values to the left of 0, values to the right of 0, to see whether I have a local max or min, and increasing/decreasing.1492

Let us see here, I'm going to go ahead and check.1506

Let us just go ahead and check -1.1515

When we put -1 into the first derivative, you are going to end up with a negative number on top.1516

It is going to be positive, this is going to be negative.1532

This is increasing.1537

If I check the number 1, which is to the right of 0, you are going to end up with a positive number on top.1539

Positive, positive, this is going to be increasing.1544

Therefore, we have our increasing interval from 0 to +infinity.1548

We have our decreasing interval from -infinity to 0.1559

That takes care of our increasing/decreasing of the actual function.1565

We also know that this is decreasing, this is increasing.1572

We know that there is a local min at x = 0.1576

This critical value is a local min.1584

Let us go ahead and find f”(x).1588

Let us go back to blue.1590

F”(x), we are looking for inflection points, in order to check intervals of concavity.1594

We are left with this thing.1603

It is going to be this × the derivative of that - that × the derivative of this/ this².1604

We get x² + 5² × 100 - 100x × 2 × x² + 5 × 2x/ x² + 5⁴.1609

I’m going to factor out an x² + 5 here, which leaves me with x² + 5 × 100 - 400x²/,1633

I multiply this, this, this, to get 400²/,1649

I’m sorry, I multiply this, this, and this.1656

I have factored this out as here.1659

I get x² + 5⁴, this cancels with one of these, leaving 3.1661

I need that equal to 0.1670

We have f”(x) is equal to 100x² + 500 – 400x²/ x² + 5³.1677

Set that equal to 0 and I’m going to get -300x² + 500 = 0 because it is only the numerator that matters.1699

I have got 300x² is equal to 500.1712

I get x² is equal to 5/3 which gives me the x is equal to + or - 5/3, which is approximately equal to + or -1.3.1717

I’m going to set up my -1.3, +1.3.1731

I’m doing f”, I need to check points there, points there, and points there.1739

Let us go ahead and actually do this one.1747

F”(x)is equal to -300x² + 500/ x² + 5³.1750

I’m going to check the point -2.1766

When I check the point -2, I'm going to get a negative number/ a positive number1769

which is a negative number, which is concave down.1779

When I check 0, I’m going to end up with a positive number on top of a positive number.1783

When I put 0 into the second derivative, which means positive, which means concave up.1789

When I check 2, I'm going to get a negative/ a positive number that is going to be negative, this is going to be concave down.1796

Let us move on to the next page here.1811

I have got concave down from -infinity to -1.3 union at 1.3 to +infinity.1814

I have got concave up from -1.3 all the way to 1.3.1826

Let us see what happens.1835

We are dealing with a rational function.1839

We have to check the asymptotic behavior.1841

We have to check to see what happens when x gets really big, in a positive or negative direction.1843

Let us see what happens when x goes to + or -infinity.1850

F(x) is equal to 10x²/ x² + 5.1866

As x goes to infinity, this drops out.1874

The x² cancel and you are left with f(x) ends up approaching 10.1878

10 is the horizontal asymptote of this function.1884

What do we have, let us go back and put it all together.1890

We have a local min at 0.1898

When we check f (0), it equal 0.1905

We are looking at the point 0,0.1908

We know that the function is increasing from 0 to +infinity.1912

We know the function is actually decreasing from -infinity to 0.1918

We know that we have points of inflection at x = -1.3.1926

When I check the y value, I get 2.53.1941

Therefore, at -1.3 and 2.53 is my actual point of inflection, both xy coordinate.1944

My other one is at +1.3 and the y value is 2.53.1955

1.3 and 2.53 is my other point of inflection.1963

I am concave down from -infinity to -1.3 union 1.3 to +infinity.1969

I am concave up from -1.3 all the way to +1.3.1981

I have a horizontal asymptote at y is equal to 10.1990

When I put all of this together, what I end up getting is the following graph.2007

Here is my local min, here is one of my points of inflection.2017

Here is my other point of inflection.2024

It is decreasing all the way to 0, increasing past 0.2028

It is concave down all the way to -1.3, concave down up to this point.2035

It is concave up from this point to this point.2044

It is concave down again, notice that it approaches 10.2047

That is it, first derivative, second derivative, local maxes and mins, points of inflection, intervals of concavity.2055

I need horizontal asymptotes, vertical asymptotes, everything that I need in order to graph this function.2064

Let us go ahead and try another example here.2073

For the function x² ln 1/3 x, find the intervals of increase/decrease,2077

local maxes and mins, points of inflection, intervals of concavity.2081

Use this information to draw the graph.2085

Let us go ahead and do it.2088

F’(x) is equal to x, this is a product function.2091

This × the derivative of that + that × the derivative of this.2097

X² × 1/ 1/3 x × 1/3 + ln of 1/3 x × 2x.2101

1/3, 1/3, x, x, what I should end up with is x + 2x ln 1/3 (x).2116

We want to set that equal to 0.2129

I'm going to go ahead and factor out an x.2131

I get x × 1 + 2 ln of 1/3 x that is equal to 0, that gives me x = 0.2134

It gives me 1 + 2 ln of 1/3 x is equal to 0.2153

0 is one of my critical points.2160

When I solve this one, I will go ahead and solve it up here.2165

I get 2 × ln of 1/3 x is equal to -1.2169

Ln of 1/3 x is equal to -1/2.2177

I exponentiate both sides, I get 1/3 x = e ^-½.2181

I get x is equal to 3 × e⁻¹/2.2190

X is approximately equal to 1.8.2194

Let us go ahead and do it.2202

These give me my critical points.2204

I have 0 and I have got 1.8.2206

I need to check a point here, check a point here, check a point here.2210

Put them into my first derivative to see whether I get a positive or negative value.2213

Let me go ahead and write it out.2223

F’(x), I’m going to write out the multiplied form, = x × 1 + 2 ln of 1/3 x, that is my f’(x).2225

I’m going to pick a point again here, here, here.2243

Put it into this to see what I get.2246

When I check the point, this here is not the domain.2247

I do not have to check a point there.2257

The reason is I cannot take the log of a negative number.2258

That is not a problem, I do not have to check a point here and here.2262

I'm going to go ahead and check 1.2265

For 1, when I put 1 into here, this is going to be positive, this is going to be negative.2268

Therefore, it is going to be decreasing.2274

We are going to be decreasing on that interval.2277

When I check the number 2, I get positive and positive which means it is increasing on that interval.2279

I’m decreasing from 0 to 1.8, I’m increasing from 1.8 to +infinity.2294

I have a local from decreasing to increasing.2300

I have a local min at 1.8.2302

That is what this information tells me.2305

Let us write that down.2307

I’m decreasing from 0 to 1.8, I am increasing from 1.8 to +infinity.2309

I have a local min at x = 1.8.2324

The y value at x = 1.8, that is just the original function.2335

F(1.8), it gives me -1.7.2345

My local min is going to be the point 1.8, -1.7.2353

Notice that we have x = 0, what is the other critical point.2364

But I cannot say that there is actually a local max there.2368

The reason I cannot say that is because there is nothing to the left of 0, the domain.2374

In order to have a local max or a local min, I have to have the point2378

defined to the left on to the right of that particular critical point.2381

Here it is only defined to the right.2385

The positive and negatives do not count.2389

Let me write that out.2395

We cannot say that there is a local max.2400

In some sense there is, but not by definition, that there is a local max at x = 0 because f is not defined for x less than 0.2410

In other words, our function, we know there is a local min at 1.8 and -1.7.2435

We know it is here.2442

We know it is going to be something like this.2444

We cannot necessarily say that this is a local max because it is not defined over to the left of it.2446

That is all that is going on here.2452

Let us do point of inflection.2454

Let me go back to red.2457

Points of inflection, we have f’(x) is equal to x + 2 ln 1/3 x.2459

Therefore, f”(x), I’m going to take the derivative of this.2471

It is going to be 1 + 2 ×, this is 2x, 1 + 2 × x × the derivative.2476

X × 1/1/3 x × 1/3 + the nat-log of 1/3 x × 1.2488

I just pulled out the 2, just for the hell of it.2505

1/3, 1/3, x, x, this becomes 1.2508

You end up with 1 + 2 × 1 is going to be 2 + 2 × the ln of 1/3 x.2512

We are going to get 3 + 2 × the ln of 1/3 x.2527

That is our second derivative.2535

We need to set that equal to 0.2540

F”(x)is equal to 3 + 2 × the ln of 1/3 x.2545

We need to set that equal to 0.2554

We get 2 ln 1/3 x is equal to -3, ln of 1/3 x = -3/2.2556

We exponentiate both sides, we get 1/3 x is equal to e⁻³/22566

which gives us x is equal to 3 × e⁻³/2, which is approximately equal to 0.7.2581

We have to check, this is a point of inflection.2593

We need to check a point to the left of 0.7 to the right of 0.7, to see whether the second derivative is positive or negative.2596

In other words, concave up or concave down, respectively.2603

We have got 0.7 here, we are checking f”.2609

F” is equal to 3 + 2 × the nat-log of 1/3 x.2615

When I check the value, I will just check 0.2626

I check the value 0, it is going to be concave down.2633

I’m not going to check 0 actually.2649

I will check my 0.5.2652

0.5, you are going to end up getting a negative number which is going to be concave down.2654

And then, I'm going to go ahead and check some other number 1, 2, 3, does not really matter.2666

Let us just check 3, positive, that is going to be concave up.2672

In other words, I’m putting these values into the second derivative to tell me whether something is positive or negative.2677

This is going to be positive which is going to be concave up.2686

That takes care of that.2692

Now I have my intervals of concavity.2706

It is going to be concave down from 0 all the way to this 0.7.2708

Let me double check, yes.2724

And then, concave up from 0.7 to +infinity.2728

Let us find where f(x) actually equal 0.2742

Let us see where it actually crosses the x axis, if in fact it actually does so.2746

Let me go ahead and do that on the next page here.2753

F(x) is equal to x² ln 1/3 x, we want to set that to equal to 0.2758

That gives us x² is equal to 0 and it also gives us ln of 1/3 x is equal to 0.2767

This is 0, it attaches at 0,0.2776

This one you get 1/3 x e⁰ is 1, that means x = 3.2779

It touches the x axis at 0 and 3.2788

Let us list what we have got.2794

Our root x = 0, x = 3.2800

We have a local min at the point .1.8, -1.7.2809

We have a point of inflection at 0.7, -0.7.2819

We are concave down from 0 to 1.4.2828

I’m sorry not 1.4, it is going to be 0.7.2841

Concave down from 0 to 0.7.2844

We are concave up from 0.7 to +infinity.2847

The function is decreasing from 0 to 1.8.2856

The function is increasing from 1.8 to +infinity.2862

When we put all of that together, local min at 1.8, -1.7.2875

1.8, -1.7 probably puts us right there.2884

Point of inflection at 0.7, -0.7, 0.7, -0.7, somewhere around there.2887

Our graph goes something like this.2894

It is decreasing from 0 to 1.8, decreasing, concave down from 0 to 0.7.2896

It is concave down here, concave up from 0.7 to infinity.2907

That is our graph, let us see a better version of it.2915

Here is our root, here is our root.2921

Points of inflection is somewhere around there.2925

Local min somewhere around there.2931

As you can see, we have concave down from here to here.2933

Concave up from here to here, and continuously concave up.2937

The graph goes, passes through 3.2941

There you go, that is it, wonderful.2946

Thank you so much for joining us here at www.educator.com2949

We will see you next time for a continuation of example problems on using the derivative to graph functions.2952

Take care, bye.2959

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