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Raffi Hovasapian

Raffi Hovasapian

More Example Problems for The Derivative

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
Sat Sep 23, 2017 4:17 AM

Post by sorin dragon on September 16, 2017

Hello professor Raffi!

Can you please explain me again how i can graph the derivate f'(x)? It's kinda tricky for me and i couldn't understand very well from your first two examples.

You're doing a great job! Thank you very much!

0 answers

Post by Peter Fraser on April 11, 2017

Hi Raffi

At 8:35 I think the divisor for the complex fraction should not have changed to a negative sign for the x^-1/2 term; also at 8:58 I think the x^1/2 - (x + h)^1/2 expression should be x^1/2 + (x + h)^1/2.

Sorry in advance if I'm wrong in this.

Peter :)

1 answer

Last reply by: Professor Hovasapian
Fri Apr 7, 2017 6:44 PM

Post by Peter Fraser on April 6, 2017

3:33: That derivative shape's really interesting because I think it's suggesting that an antiderivative of a rational function such as 1/x ought to have that sort of red coloured graph.

More Example Problems for The Derivative

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: 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

Transcription: More Example Problems for The Derivative

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

Today, I thought we would do some more examples on the derivative.0004

Let us jump right on in.0009

Typical example, graphs are going to be very huge.0013

You are going to be given a function, find the derivative graph.0015

Or the derivative graph, find the function, work your way back.0018

The following is a graph of f(x), the original function.0022

On the same graph, sketch f’(x).0024

They just want a sketch of the derivative.0028

One of the things that we know -- when you are working with graphs and their derivatives, the most important thing is slope, where slope is 0.0033

We know that a slope of 0 means a horizontal tangent line.0042

On this graph, we have a horizontal tangent line here, here, and here.0046

At these x values, namely, let us say somewhere around there.0056

Here it looks like this is 0.0060

Here, those are the places where the derivative, because the derivative is the slope of the graph.0062

It is where the derivative graph hits 0.0068

Now to the left of this point, we notice that the slope is negative.0071

It is below the axis.0076

As x moves this way, as we hit this point, that is where we are going to hit 0.0080

From this point to this point, the slope is positive.0086

It changes and it increases a little bit, and it decreases towards 0.0090

Basically, it goes like this and then it passes 0 again.0097

Now you notice the slope is pass this point, the slope is negative.0102

It gets more negative and it starts to go positive again until it hits 0 again.0106

It gets more negative, it starts to go positive until it hits 0 again.0113

Then, it keeps going up.0117

Look for horizontal tangents, when you are working with graphs and their derivatives.0120

Whether it is first derivative or second derivative, or whatever.0125

There you go, that is the derivative.0128

This is black, this is your f’(x).0129

Same thing, the following is a graph of f(x), sketch f’(x).0136

Here, we notice that f(x) is not differentiable here.0142

Anytime you remember when you have a cusp,0147

that is one of the ways that a function cannot be differentiable at that point.0148

Everything else is fine.0152

We look at this and we see that the slope moving from of course left to right,0154

we are moving this way from negative values to positive values, the slope is negative.0159

It is actually becoming more negative.0165

At that point, it is not differentiable.0169

The derivative is not defined there.0174

It is a negative slope, it means it starts below the x axis, becomes more negative.0176

The derivative graph looks like that.0183

You are just graphing what the slope does.0185

That is what the f’ graph is, that is what the derivative graph is.0187

What is the slope doing?0191

Starting negative becoming more negative and not touching 0.0193

Here, the derivative just passes 0.0201

It starts really highly positive, it stays positive but it becomes less positive.0205

Highly positive becomes less positive, and that is it.0210

This hyperbola looking thing, this is your f’ graph.0217

That is all it is, what is the slope doing, that is all you have to ask yourself.0221

Find the derivative of the following function using the definition.0232

I think I’m going to work in blue here.0237

Again, it is always a great idea to write the definition.0239

Just get accustomed to writing it in a nice, just to be systematic in your approach.0242

We know that the definition f’(x), the derivative = the limit as h approaches 0 of the following quotient, f(x) + h.0247

It must be nice if I make my + signs legible.0257

x + h - f(x)/ h.0262

We form this quotient first, simplify as much as possible, and then we take the limit.0266

Let us do it, I'm going to go ahead and just work with the difference quotient first.0278

And then, I will do the limit at the end.0285

I do not want to keep writing limit as h goes to 0.0287

I will just work with the quotient first.0291

f(x) + h is going to be 1/ √x + h – f(x).0293

f(x) is 1/ √x/ h.0302

I take the common denominator for the top.0309

It becomes √x – √x + h/ √x × √x + h/ h.0312

I’m going to keep simplifying this here.0334

Actually you know what, it looks like this particular one going to cause a little bit of a problem0338

because we are still going to end up with, if we take h to 0,0349

it looks like here the denominator is 0 as h goes to 0, because when we put 0 in there for h.0353

I’m going to try an alternate procedure here.0367

I’m going to start again.0369

I’m going to go 1/ √x + h - 1/ √x/ h.0372

This one, the reason I stop is because at this point, there is really not much you can do.0382

I mean, this is h/1.0389

When you flip it, you are going to end up with an h at the bottom.0391

When you take h to 0, you are going to end up with 0 in the denominator.0394

That is what is happening here.0398

When that happens, because you really cannot simplify this much more than that this.0399

Because we have 0 in the denominator, we cannot really do anything with that.0405

We have to try an alternate procedure, an alternate manipulation.0408

That is why we are going to try an alternate manipulation here.0412

I go back to the beginning, this thing.0415

I think what I’m going to do them is multiply it by the conjugate of the numerator.0417

It is going to be × 1/ √x + h + 1/ √x/ 1/ √x + h + 1/ √x.0422

This is going to end up equaling 1/ x + h - 1/ x/ h × 1/ √x + h + 1/ √x, when you simplify this.0441

Let us see what we have got here.0462

I might as well continue on.0465

This is going to equal x - x + h/ x × x + h.0468

I just found the common denominator for the numerator, and I leave the bottom as is.0478

1/ √x + h + 1/ √x.0484

This is going to end up equaling, x - x cancels, you are going to end up with -h/ x × x + h/ h × 1/ √x + h -1/ √x.0490

Now the h cancel and you are left with,0519

Now we have -1/ x × x + h/, do a common denominator here, √x - √x + h/ √ x × √x + h.0526

This = -1/ x × x + h × √x × √x + h.0548

Again, this is just algebraic manipulation, that is all it is, /√x + √x + h.0558

That is fine, I will go ahead and write out the whole thing, not a problem.0577

This is going to be -1/ x × x + h ×, this is √ × √, I can combine them.0579

It is just going to be x × x + h, under the radical together, / √x + √x + h = -1/,0589

This can cancel with that, one of these.0605

I’m left with √x × x + h.0612

On the bottom, this one, this cancels one of the radicals here.0618

This just leaves me with that.0622

This is going to be × √x + √x + h.0624

Now I take the limit of this.0633

I take the limit as h goes to 0.0637

This is just the same as f(x), I have just manipulated it.0640

The limit as h goes to 0.0644

Now I take h to 0 and I end up with -1/ √x² × √x + √x.0647

Let me go to the next page, it is not a problem.0669

It = -1/ √x × 2√x = -1/ 2, just √x, x².0671

x² × x is x³.0686

There you go, that is your derivative.0688

Or we can write it as 2x³/2, that is another way that we can write it.0693

Again, that is one possibility, or a third possibility is you can bring this up to the top.0703

You can write -, this x³/2 is going to be x⁻³/2 /2.0710

Any one of these is absolutely fine, it is not a problem.0717

Notice that f’(x), we have this thing right here.0721

If it is 0, if x is 0, we are going to end up with 0 in the denominator.0736

The original function f(x) is not differentiable at 0.0740

If you try to graph it, you can see that it is not going to be differentiable at 0.0749

Analytically, you can see that it is not going to be differentiable at 0.0752

Let us try something else here.0764

The following is a graph of f(x), f’, and f”.0765

Our first and second derivatives along with the original function.0770

Decide which is which and be able to explain why you made the choices that you made.0773

Again, when you are dealing with situations like this, you work with 0 slopes.0777

You look for a function, you look for the graph that has a horizontal tangent.0782

You see where it hits 0 on the x axis, that is going to be the derivative of that function.0788

Again, work with horizontal tangent which is 0 slopes.0793

Let me work in black actually because one of my graphs is blue.0814

I have a horizontal tangent here.0818

I find that the blue graph is where it hits 0.0822

The blue graph is the derivative of the red graph.0826

Now the blue graph has a horizontal tangent here.0829

As I go straight up from there, I notice the green graph has crosses 0 at the x value0834

where the blue graph has a horizontal tangent, which means that the green graph is the derivative of the blue graph.0843

What you get is f, the original function is the red graph.0851

The blue is f’, the first derivative.0856

The green is going to be the f”.0861

That is it, just work with horizontal tangents and see where it is.0866

The graph has a horizontal tangent, see where the x value crosses 0, that is the derivative of that function.0869

Find an equation for the tangent line to the graph of the following function at the given x value.0881

A tangent line, we need to find the derivative.0887

We need to find the y value, whatever that is going to be, for the original function.0891

And then, we are going to do y - y1 = the slope.0896

We are going to put the x value into the derivative function to get the slope, × x – x.0909

x1 is 4, y1 we are going to find.0915

Let us get started.0921

We know that f’(x) is equal to the limit as h approaches 0 of f(x) + h - f(x)/ h.0923

Again, I'm going to work with just the difference quotient.0940

I'm going to get the f(x) + h is 5 × x + h/ x + h + 3² - f(x) which is 5x/ x + 3², and all of that is going to be /h.0946

I'm going to end up with, I’m going to take a common denominator here.0975

I’m going to get 5 × x + h × x + 3² - 5x × x + h + 3²/ x + h + 3 × x + 3²/ h.0978

I’m not going to do the expansion.1026

I’m going to leave the algebra to you.1027

When you expand, when you multiply everything out, when you expand, multiply, and simplify, you get the following.1029

Once you simplify it, you take the limit as h goes to 0.1052

In other words, you put 0 in wherever h shows up in the expression.1055

You are going to get your derivative.1058

You get f’(x) is equal to -5x + 15/ x + 3³, that is the derivative of the function.1061

Now the original f(x), that was equal to 5x/ x + 3².1073

We need to find the y value for where the original function = 4.1088

This is going to be the hardest part, when dealing with derivatives and functions1094

is making sure you keep the original function separate from the derivative function.1098

They are not the same.1105

Here we want to find f(4), the original function.1107

You are going to end up with 20/49, when you put it in.1110

The tangent line touches the graph of the original function at the point 4, 20/49.1114

Now the slope, that is the derivative.1136

We want f’ at 4.1138

That is going to equal -5 × 4, we are using this one, + 15/ 4 + 3³.1142

It is going to end up equaling -5/ 343, if I done my arithmetic correctly which I often do not.1156

Please check that.1161

My slope is -5.1163

The equation of the tangent line to the original function at x = 4 is, we said it is y - y1 = the slope × x - x1.1169

It is going to be y - 20/ 49, that is our y value = our slope -5/ 343 × x – 4.1188

That is it, nice and simple.1202

You find the derivative, you find the slope.1203

You find the x and y values of the point that it passes through.1205

Now you got a slope and you got a point, you are done.1208

The following graph shows a distance vs. time graph for two runners.1219

One runner is the blue.1223

One runner is our blue graph and runner two is our red.1225

Answer the following questions.1229

Describe the race from beginning to end.1231

Distance vs. time.1232

Distant on the y axis, time on the x axis.1234

The slope of the graph is the y axis/ the x axis.1240

y/x, Δ y/ Δ x.1245

Distance/ time, we already know what distance/ time is.1248

Distance/ time = velocity.1256

The slope at any point along the graph tells me how fast the runner is going.1259

It is always going to be that way.1264

The derivative is always going to be the slope, you know that already.1266

In this case, it is a physical application.1269

Here the y value is distance, the x value is time.1271

Distance/ time, velocity.1274

In this case, the slope gives me the velocity of the person at any given moment.1276

We notice that the red runner, it is constant velocity.1281

It is one speed.1284

The blue runner starts slowly, his slope is almost horizontal but he speeds up.1285

At some point, his slope is actually steeper than the slope of the red.1293

It is actually going faster than the other person.1299

What is happening here is very simple.1304

They start running, the person in red is running faster initially.1306

At some point, where the slopes are the same, they are the same speed but the red is further ahead.1311

However, as the blue starts to speed up his velocity, eventually, at this value,1323

it looks like maybe 14 or 15 seconds, he catches up with the first one.1329

They are actually at the same distance, that is what is happening.1334

Red, blue catches up eventually, that is what is happening.1338

That takes care of part A, when do the runners have the same velocity?1344

Velocity is a slope, they have the same velocity when the slope of the blue graph is the same as,1347

We look for something like that when they are parallel.1353

Maybe somewhere around 8.2 seconds or something like that.1356

When is the distance between the runners greatest, the part c.1363

The distance is greatest when the gap between them, when that distance is the greatest.1367

It looks like it is somewhere around 8 seconds.1376

You can argue, maybe it is 7 or 9, something like that.1379

But basically you are looking for the gap between the two.1384

The distance, that is the y axis, when is that distance the greatest.1388

That is it, nice and simple.1395

Again, slope is the y axis/ the x axis.1397

In any given real world situation, it might be temperature time, it might be distance time, whatever it is.1401

In this case, it was distance/ time.1408

Distance/ time, we know it is velocity.1411

The displacement of a particle from the origin.1420

As it moves along the x axis it is given by x(t) = t³ – 7t² + 12t – 3.1422

What this means is that as time goes from 0 forward, 1, 2, 3, 4, 5, 6, 7 seconds,1430

when I put both t values in here, the number that I get, the x tells me where on the x axis I am.1437

It gives the x coordinate, that is what this means.1444

Find the displacement, the velocity, and the acceleration of the particle at t = 5 seconds.1449

It is asking me, at 5 seconds, where is it, how fast is it going, in what direction, and is it speeding up or slowing down.1453

Displacement, where is it.1461

Velocity, how fast is it moving and in what direction.1463

Positive velocity to the right.1468

Negative velocity to the left.1469

We are on the x axis, we are moving this way and this way.1471

Positive velocity is that way, negative velocity is that way.1475

Acceleration, if it is positive, it is speeding up at that point.1479

If it is negative, it is slowing down.1483

You can have a particle with a positive velocity and a negative acceleration.1487

It is moving to the right but it is slowing down.1490

It is getting ready to stop.1494

f(t), displacement.1500

It is t³ – 7t² + 12t – 3.1504

We want to know what x of 5 is.1512

When we put it in, we get 7.1514

At 5 seconds, on the x axis, that point, my point is there.1517

Now velocity, the velocity is a function of time, it is equal to the first derivative of the displacement function.1527

The derivative of this function, you can either do it by doing the definition of derivative1544

or you can do that thing that we told you about.1551

3t² - 14t + 12, remember, I think we discussed that real quickly at the end of one lesson1554

or at the end of the problem lessons.1564

You take this number, you bring it down here.1566

You multiply it by that and you reduce the degree by 1.1568

This is the velocity function.1572

At any time t, this function tells me how fast I’m moving.1573

The velocity at 5 = 17.1579

This is positive which means that I'm here and I’m actually moving to the right at 17 m/s, ft/s, mph, whatever my unit is.1583

Now the acceleration function.1596

The acceleration at any given time t is equal to the derivative of the velocity,1598

which is equal to the second derivative of the displacement.1604

Now I take the derivative of that.1610

2 × 3 is 6, 6t, drop the degree by 1.1612

1 × 14 is 14, -14, drop it by 1, it is just t⁰ which is 1.1617

It just stays 14, that is my acceleration function.1626

My acceleration at 5 is equal to 16.1631

This is positive.1638

I’m at 7, I'm traveling at 7, let us say m/s.1642

I’m accelerating, I’m speeding up going to the right.1646

Not only am I going to the right, but I’m actually getting faster going to the right, which means 16 m/s² acceleration.1650

At t = 5, I’m at 17.1659

At 6 seconds, 1 second later, I’m at 17 + 16.1662

I’m 33 m/s.1666

One second after that, I’m at 33 + 16, I’m 49 m/s.1669

I'm here, I’m traveling this fast, and I’m speeding up that fast per second.1675

Let us say this is meters, this is going to be meters per second, this is going to be meters per square second.1683

Displacement, when you take the first derivative, you get the velocity.1691

When you take the derivative of velocity, you get the acceleration.1694

Now graph the displacement function from the previous problem.1701

Then use the graph to describe the motion of the particle during the first 5 seconds.1705

The function was x(t) = t³ – 7t² + 12t – 3.1711

Here is the graph of the function.1716

We want you to describe what is happening to this particle.1718

Describe its motion during the first 5 seconds.1722

5 seconds is here.1727

Between 0 seconds and 5 seconds, what is the particle doing?1729

At t = 0, I'm at, it looks like -3.1735

Let me draw it up here.1742

Here is my 0, let us say this is -3.1746

I start at -3, that is what this graph is telling time.1753

At time = 0, it is this way.1756

Now my slope is positive which means that the particle is actually going to be moving to the right.1758

It is positive but it is slowing down.1769

At 1 second, the velocity hits 0.1773

At 1 second, the particle actually stops momentarily.1776

The velocity which is the slope becomes negative and becomes more negative.1782

Now the particle is moving this way.1787

Again, but it starts to slow down and at just about 3.5 seconds, it hits 0 again.1792

The slope is 0, the velocity is 0 again.1800

Now the velocity becomes positive.1803

It turns around again and starts moving to the right.1805

This time the slope starts accelerating to the right.1809

That is what is happening.1812

At 5 seconds, it actually hits 7.1814

It starts at -3, it starts moving to the right.1820

Stops momentarily then starts moving to the left.1829

Stops again and starts moving to the right.1832

At 5 seconds, I’m at 7, that is what is happening.1835

Once again, if this is my 0, let us say this is my 3.1840

Let us say this is my 7.1845

My particle starts here, moves to the right, moves to the left, moves to the right.1847

At 5 seconds, I'm there, that is what is happening.1854

That is what the graph is telling me.1857

Again, distance vs. time.1858

Time is on this axis, x is the displacement.1861

It is the distance, it is how far I am along the x axis.1864

The graph is not describing the path that the particle is following.1868

It is not moving in two dimensions.1875

It is moving in one dimension.1876

The path is telling you what is happening in that one dimension.1878

The slope is the velocity.1882

Positive velocity moving to the right.1884

0 velocity, it stops momentarily.1887

Negative velocity, it is heading the other direction, that is what is happening.1889

Thank you so much for joining us here at www.educator.com.1895

We will see you next time, bye.1897

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