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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Using Derivatives to Graph Functions, Part 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 (2)

1 answer

Last reply by: Professor Hovasapian
Thu Apr 7, 2016 1:37 AM

Post by Acme Wang on April 5, 2016

Hi Professor,

In example II, I am a bit confused about your solving way. I know -2.33 and other 3 values are critical values. Why can we just simply choose a random number such as -4 or -2.48 to discuss the increasing/decreasing property at the left side of -2.33? Would there be the situation that f'(-4) is positive and f'(-2.48) is negative? Then what's the increasing/decreasing property at the left side of this critical point, -2.33? I don't know whether I have made clear about my problem. Hope you can understand and clear my confusion. Thank you very much!

Sincerely,

Acme

Using Derivatives to Graph Functions, Part 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
  • Using Derivatives to Graph Functions, Part I 0:12
    • Increasing/ Decreasing Test
  • 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

Transcription: Using Derivatives to Graph Functions, Part I

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

Today, we are going to talk about using the derivative to actually help us graph a function.0005

Let us jump right on in.0011

I will go ahead and stick with black here.0017

No, I think I will go back to blue.0020

We can use first and second derivatives along with any other information that we have0024

for a function to complete the graph of a function.0062

You are going to be spending a fair amount of time doing this.0080

We ourselves are going to be spending a fair amount of time doing this.0082

The first thing we are going to talk about is something called the increasing/decreasing test.0087

We have the increasing/decreasing test.0092

If f’(x) is greater than 0 on an interval… and remember, when we speak about interval, we are talking about the x axis, the domain.0105

If f’(x) happens to be greater than 0 on an interval, then f is increasing on that interval.0123

The original function, the function itself is on its way up.0135

Again, we are moving from left to right, from negative values towards the positive direction.0138

It is increasing on that interval.0144

If f’(x) is less than 0 on an interval and f is decreasing on that interval, it is on its way down decreasing on that interval.0153

You already know this geometrically.0180

If you have some function, slope is positive 0, decreasing.0181

Now the slope is negative, it is changing, 0, the slope is positive again.0188

It is increasing, function is increasing, function is decreasing, function is increasing.0194

You know this intuitively, in terms of the derivative, what is happening.0199

Let us do an example, find the intervals over which the function f(x) = 2x⁴ – 4x³ – 10 x² + 5 is increasing and decreasing.0206

Find the intervals of increase and find the intervals of which the function is decreasing.0217

When a function goes from increasing to decreasing, in other words,0241

when it goes from f’ greater than 0 to f’ less than 0, it has to pass through f’ = 0.0262

Or it goes from decreasing to increasing, meaning f’ less than 0 going to f’ greater than 0.0275

Then, f’ passes through 0, in other words, there is some point such that f’ is equal to 0.0292

Let us set f’(x) equal to 0.0310

Let us find those points such that f’ is 0.0316

We are going to check points to the left of that and to the right of that, to see whether the derivative is positive or negative.0322

That is going to tell us whether the function is increasing or decreasing.0329

F’(x) is equal to 8x³ - 12x² - 20x.0337

We are going to set that equal to 0.0355

I’m going to factor out a 4x.0357

We have 4x × 2x² - 3x – 5, equal to 0.0360

Therefore, I have 4x ×, I can factor this one, 2x - 5 × x + 1 is equal to 0.0373

Therefore, I have x = 0, x = 5/2, and I have x = -1.0383

This happened to be the critical points of this function.0395

Remember, back when we talked about critical points, maxima and minima,0397

the critical points are the points where the derivative is equal to 0.0400

We know the derivative = 0, the slope is 0.0404

I’m going to check to see a point to the left of these points.0411

I’m going to check a point to the right of these points, and to see if the derivative actually changes sign.0414

If it goes from negative to positive, or it goes from positive to negative.0419

If that is the case, then it will me whether it is increasing or decreasing to the left or right of these points.0423

Let us go ahead and do that.0430

I’m going to go ahead and draw the number line.0435

I have my points, -1, 0, and 5/2.0439

Great, I’m going to check a point here, I’m going to check a point here, check a point here,0449

and I'm going to check a point over there.0454

Let us rewrite f’(x), we said that f’.0458

Let me do it over here actually.0465

We said that f’(x) is equal to 4x × 2x - 5 × x + 1.0468

I’m going to check a point to the left of -1.0480

It is going to pick -2.0483

When I do f’ at -2, in other words I’m just putting -2 in for the x values here.0485

I do not have to worry about what the values are.0492

I just need to know whether it is positive or negative.0494

4 × -2 is negative, 2 × -2 - 5 is also negative, -2 + 1 is negative.0498

Negative × a negative × a negative, it is going to be negative, which means that f’ is less than 0.0508

Anything to the left of -1, the function is decreasing.0522

The function is going down, it is decreasing.0526

I’m going to pick a point, I’m going to check this region right here between -1 and 0.0530

I’m going to pick -0.5.0535

F’(-0.5), this is going to be a negative number, this is going to be a negative number, this is going to be a positive number.0539

Negative × negative × positive is a positive number.0549

In other words, f’ is greater than 0, the function is increasing.0554

On this interval, from -1 to 0, the function is increasing.0559

From 0 to 5/2, I’m going to go ahead and do f’(1).0565

I’m going to just choose the point 1.0569

This is positive, this is negative, this is positive, this is negative.0574

Positive × negative × positive is a negative number.0581

Therefore, f’ is less than 0, decreasing.0585

On this interval, between 0, the function is decreasing.0590

I have to pick a point that is in this region right here, greater than 5/2.0595

I’m just going to go ahead and pick 3, f’ at 3.0601

The first one is positive, positive, positive.0606

I have positive, f’ is greater than 0, it is increasing.0611

Therefore, the intervals of increase are, we have -1 to 0 union 5/2 to +infinity.0620

And the intervals over which the function is decreasing is -infinity to -1 union 0 to 5/2.0639

It is at these points -1, 0, 5/2, f’ is equal to 0.0656

The slope is horizontal, decreasing, increasing.0663

Slope is 0 here, slope is 0 here, the slope is 0 here.0669

Just by looking at this, I can tell that I have a minimum, maximum, minimum.0673

Local min, local max, local min.0678

That is our next discussion.0682

Let us go ahead and take a look real quickly what this function actually looks like.0685

This is the function that we had.0689

Again, at -1, it hits a local min, 0 hit a max, and 5/2 it hits a local min.0691

The function is decreasing, it is increasing, it is decreasing, and it is increasing.0698

-infinity to -1, the function is decreasing.0706

-1 to 0, the function is increasing.0709

From 0 to 5/2, the function is decreasing.0711

From 5/2 to infinity, the function is increasing.0716

That is all that is going on here.0721

We are moving in this direction.0721

The graph is going like that because we always move from lower to higher numbers.0724

Let us see here, in this example, we had something that look like this.0736

This procedure that we just followed, finding the critical values,0762

checking points to the left or right of it to see whether increasing or decreasing, plugging values of the first derivative.0768

That is what we are checking.0773

This procedure also tells us, based on what we saw from the example.0775

Also tells us which critical values these are local maxes and local mins.0779

If you go from decreasing to increasing, you are going to hit a local min.0804

If you go from increasing to decreasing, you are going to hit a local max.0808

Increasing to increasing, local min.0811

This is the basis of the first derivative test.0816

It seems like it is a little pointless to actually, like formalize a state, the first derivative test, increasing/decreasing test.0821

These are things that we just sort of layoff for the sake of that being there.0830

The ideas to understand what is happening.0833

Understanding what is happening, you do not to call it a first derivative test, an increasing/decreasing test.0835

This theorem, that theorem, the names are irrelevant.0841

What is important is the concept.0846

Historically, these things evolve and given names, first derivative test.0850

It is like this, it is exactly what we just said.0855

If f’ goes from positive to negative at c, in other words, as you pass through c to the left of c, to the right of c.0859

If you are going from positive to negative, from your perspective positive to negative, then c is a local max.0875

That is here, it is going from positive to negative, the derivative, the slope.0890

The slope is positive, slope is negative, and it passes through c which is that point where the derivative equal 0.0896

It is a local max.0903

B, if f’ goes from negative to positive at c, then, c is a local min.0906

Negative slope, positive slope.0924

If it passes from negative to positive, it is a local min.0928

And part c, if f’ does not change sign as it passes from the left of c to the right of c,0934

if it goes from negative to negative, -0 negative, positive 0 positive.0946

If it does not change sign, if f’ does not change sign at c, in other words as it passes through c,0950

then c is neither a local max nor a local min.0968

Part c is a reminder that just because c is a critical point,0992

in other words a place where the derivative = 0 or the derivative fails to exist, in this particular case, if the derivative equal 0.1014

Just because c is a critical number, a critical value of the function, it does not mean that it is a local max or min.1022

Just because something is a critical value, it does not mean that it is a local max or min.1048

If it is a local max or min, it is a critical value but not the other way around.1053

You can have a critical value which is not a local max or min.1059

For example, let us take the function f(x) is equal to x³.1064

F’(x) is equal to 3x², when we set that derivative equal to 0, we see that x = 0 is a critical value.1076

If x is a critical value, let us go ahead and check.1088

To the left of it, to the right of it, to see what happens.1092

0, let us just take the point -1.1096

When I put -1 to the derivative, -1² is 1, 3 × 1 is 3.1099

This is positive, the function is increasing.1104

At 0, the slope is horizontal.1109

Let us check 1, after 1 in there, we get 3, it is positive.1114

The function is increasing again.1119

I know I already know what this graph looks like, it looks like this.1122

It looks like that so can have an increasing positive slope.1128

Horizontal slope and then positive.1132

From positive to positive, there is no local max or min here.1134

It is neither local max nor local min, it does not change sign, that is what part c says.1138

Just because you have a critical value, it does not mean that it is a local max or min.1143

Let us go ahead and do an example here.1153

Find the local maxima and minima of the function f(x) = x + 9 sin 2x/ the interval –π to π.1160

Now, we want to find the local maxes and mins.1170

We are going to find the critical values, we are going to check points to the left and right of them1175

to see what they are and to see whether things change sign, as you pass through those critical points.1180

Let us go ahead and do that.1188

First thing we want to do is find the critical values.1190

Let us go ahead and find f’(x).1194

F’(x) is equal to 1 + 18 × cos(2x), we are going to set that equal to 0.1198

We have 18 × cos(2x) = -1.1214

We have cos(2x) = -1/18.1220

I’m going to set 2x equal to a.1225

What I actually have is the cos(a) is equal to -1/18.1230

A is equal to the inv cos(-1/18).1245

The inv cos(1/18), just the numerical part is going to equal 1.515 rad, it is an angle.1256

However, it is negative.1273

Because it is negative, we are dealing with an angle in the 3rd quadrant and the 4th quadrant.1277

Therefore, what I’m actually going to get is a, the angle a that I’m looking for which is this angle and that angle, is 1.515 rad.1287

That is actually this reference angle right here.1299

A itself, this angle, the one from the positive x axis, it is actually equal to π - 1.515.1302

That is equal to 1.627.1312

A is also equal to this angle which is π + 1.515 which is equal to 4.657.1316

Those are my two values of a.1328

In other words, when I take the inv cos of that, I’m getting this and this.1330

I have just gone through a more basic process and I found the actual reference angle.1337

And then, because I know that it has to be negative, I know that is going to be the 3rd and 4th quadrant.1345

I have just taken the 180 - the reference angle 180 + the reference angle to get my values of a.1349

Because we are talking about a trigonometric function,1359

a is actually equal to 1.627 + multiples of 2 π and a is equal to 4.657 + multiples of 2 π.1362

But a is equal to 2x, and it is x that we are interested in.1378

Therefore, I have got 2x is equal to 1.627 + n 2 π.1385

I have 2x is equal to 4.657 + n 2 π.1401

This is going through these, I mean all of these should be familiar to you from trigonometry, from pre-calculus.1407

I find that it is nice to go through the process itself.1414

This was the calculus part, this is just the trigonometry part.1418

Therefore, x is equal to 0. 814 + n π and x is equal to 2.329 + n π.1423

The possible values, let us list the first couple of possible values.1439

If n is 0, n is 1, n is 2, do it that way.1444

This is going to be + or -.1451

We have 0.814, we have 3.96, we have -2.33.1454

Here we have x = 2.33, - 0.814, 5.47.1465

This is etc, etc.1482

Let me rewrite these values for the x that I got.1491

X is equal to 0. 814, 2.33.1494

Let us do +, it is going to be 3.96, -2.33, etc.1505

X = 2.33, - 0.814, 5.47, etc.1514

The only values that are in –π to π are that one, that one, and that one.1526

X = -2.33, -0.814, + 0.814, and +2.33.1551

These are my critical values, this is what I have found here.1574

I have got -2.33, -0.814, + 0.814, 2.33.1578

-2.33, -0.814, 0.814, and 2.33.1587

I have to check this interval, this interval, this interval, and this interval.1597

I have to find values that I put into the derivative f’ to check to see if they are positive or negative.1604

To see if they are increasing or decreasing.1611

Let us go ahead and do that.1615

Let us go to the next page here and write.1620

We have f’(x), we set it is equal to 1 + 18 × cos(2x).1627

We have this number line that we were going to check.1638

The one point, one point, one point, one point.1641

This was our -2.33, -0.814, 0.814, and 2.33.1645

I need to check a point over to the left of this.1657

I’m going to check f’(-3).1660

When I do that and I put it in, I end up with 18.28.1665

Just put -3 into here, make sure that your calculator is in radian mode because these are all radians, real numbers, radians.1668

This is greater than 0, it is increasing.1677

It is increasing on that interval.1681

I’m going to check something here.1685

I’m going to use -1.1688

F’(-1), it ends up being -6.49, it is less than 0.1690

It is decreasing, decreasing on that interval.1696

That makes -2.33 a local max.1702

I’m going to check a point in between the -0.814 and 0.814.1708

I guess we can actually use 0, it is probably the easiest.1727

I, actually, turns out I did not use 0 but we can go ahead and do it here.1731

F’(0), the cos(2) × 0 is 0, the cos(0) is 1.1735

18 × 1 is 18, 18 + 1 is 19.1743

It is 19, it is greater than 0, it is increasing on this interval.1746

That makes -0.814 a local min.1751

I go ahead and check another point.1759

Now between here and here, I’m going to use the point 1.1760

F’(1) is going to equal -6.49, less than 0, decreasing.1765

It is decreasing on that interval which makes 0.814 a local max.1773

If I check f(3), it is going to end up being 18.28 which is greater than 0, which means that it is increasing.1778

That makes 2.33 a local min.1788

There you go, -2.33 is a local max, -0.814 local min, using this increasing/decreasing, instead of positive negative.1793

I think it is better because you can actually see pictorially going up, coming down.1813

You can see the max and min.1818

That is a local min, and then 0.814 is another local max.1819

2.33 happens to be another local min.1828

Take the function, find the critical values, examine points to the left and right of those critical values1836

to see if your derivative is positive or negative.1843

If it is positive, the function is increasing.1846

If it is negative, the function is decreasing.1848

If the function passes from positive to negative, from your perspective, it is a local max,1851

If it passes from negative to positive, it is a local min.1858

Let us go ahead and take a look at what it looks like.1865

π is somewhere around like there, π is somewhere around here.1871

Here we have our -2.33 local max, -0.814 local min, 0.814 local max, 2.33 local min.1877

There we go, that is it, nice and straightforward, nothing strange about it.1890

Let us go ahead and do another example.1899

Find the local maxima and minima of the function f(x) = x³ + 4x²/ x² + 2, rational function.1901

Let us go ahead and find the derivative of this.1910

F’(x), it is going to be this × the derivative of that - that × the derivative this/ this².1913

We are going to get x² + 1.1922

I think I used slightly different set of numbers here.1930

Let me just double check to make that I’m not going to end up with some values that are different than normal.1944

This 3x², 3x².1953

You know what, I actually ended up doing my arithmetic wrong on this one.1955

That is not a problem, we will just go ahead and do it.1959

We will finish it off based on what is written here.1962

It is going to be x² + 2 × the derivative of this which is going to be 3x² + 8x – the numerator × the derivative of the denominator.1965

It is –x³ + 4x² × 2x, all divided by x + 2², the denominator².1982

We are going to set that equal to 0.2001

I'm going to go ahead and multiply it out.2005

X² and 3x², this × that is going to be 3x⁴.2010

X² × 8x is going to be + 8x³.2020

2 × 3x² is going to be 6x².2028

2 × 8x is going to be 16x – x³ × 2x.2036

It is going to be -2x⁴.2045

4x² × 2x is going to be -8x³/ x + 2² equal to 0.2050

The 8x³ and 8x³ go away.2068

I have 3x⁴ – 2x⁴, that takes care of the x⁴.2071

I have a 6x², that takes care of that.2078

I have + 16x/ x + 2² = 0.2090

The denominator goes away, it is just x⁴ + 6x² + 16x is equal to 0.2105

When I factor out the x, I’m going to get x × x³ + 6x + 16 is equal to 0.2116

I know that one of the values is equal to 0.2129

Again, because I did my arithmetic on my piece of paper here incorrectly,2131

I used an x² + 1 instead of x² + 2, basically, we just have to find the other root of whatever it happens to be.2139

Let us say it happens to be, my guess is only one root here.2151

Let us just call it x = a.2156

Let us just call it a.2161

When we have that, we are going to end up doing this.2164

We have x = 0 and x = a are our two critical points.2170

We are going to end up with 0 and we are going to end up with a.2174

We are going to have to check a value in this interval, a value in this interval,2178

and a value in this interval and plug in the derivative to see what they are.2183

We can go ahead and do this interval right here, it is not a problem.2194

Let us check, I’m not even sure whether a is positive or negative.2197

You know what, let us take that back and let us do it this way.2204

We said that, we have x = 0 and we have x is equal to some number a.2212

A itself can be a positive or negative root.2221

I’m not exactly sure which it is.2223

If a is a positive root, what I have is 0 and I have a over here to the right of 0.2227

I have to check a value there, check a value there, and check value there,2232

to see whether the function is increasing, decreasing, or otherwise.2236

If it is negative number, and of course 0 is here, and is going to be to the left of 0.2243

I have to check a point here, check a point here, and check a point here.2249

Since I do not know what a is because of my arithmetic, I cannot go ahead and at the very least,2255

check what is to the right and left of 0.2262

Let me go ahead and do that at least, that should not be a problem.2265

Let me go ahead and take a negative one and a positive one.2270

If I do f’(-1), what did we say what f’ was, let us make sure we have that right.2274

Let us go ahead and write what f’ is.2288

We have an f’(x) is equal to x⁴, we said we have x⁴ + 3x² + 8x/ x + 2².2290

This was our f’.2313

Again, I do not know what a is because of the arithmetic issue but I can at least check to the left of 0 and to the right of 0.2317

Let us take f’(-1).2323

When I put -1 into the f’, I’m going to end up with, this is going to be 1,2329

this is going to be +3, -8, this is always going to be + because it is square.2336

1 + 3 is 4, 4 – a, I’m going to end up with which is negative number/ a positive number which is a negative number.2352

F’ at -1 is going to be less than 0, this is going to be decreasing.2360

To the left, it is going to be decreasing.2367

Let us go ahead and check f’ at 1.2370

It is going to be 1, this is positive, this is positive, this is positive.2374

This is going to be 0, it is going to be increasing.2377

At the very least, I can tell you that my 0 is going to be a local minimum.2381

Because I do not what a is, I'm not exactly sure.2393

But again, if you go back and find the root of that particular function that we had,2395

you will find what a is an it will tell you whether it is either positive or negative.2400

And then, you can check points to the left or right of that.2405

My guess is that no matter what a is positive or negative, it is going to end up being actually a local max.2408

It will probably be increasing here and decreasing here.2415

Or it is going to be increasing to the left and decreasing.2419

It is going to make a local max.2423

My guess is that it actually going to be a local max.2425

The only other option is that it ends up not changing sign at all.2428

It ends up going from negative to negative or positive to positive.2431

In which case, it is neither a local max nor a local min.2435

But I can say with certainty to 0 is a local min.2438

There you have it and I apologize for the arithmetic error.2442

Let us go ahead and take a look at the graph as is, to confirm whether it is local max.2449

Yes, it is going to be a local max.2454

It is increasing, it looks like the root was somewhere around -2.2456

Therefore, we know 0 was the local min and sure enough this is the actual function, the correct original function.2465

Yes, it looks like our root is going to be here, the derivative is going to be 0 at, it looks like somewhere like -1., something like that.2476

There you have it, local max local min.2487

Thank you so much for joining us here at www.educator.com.2491

We will see you next time, bye.2493

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