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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Differentiation of Polynomials & Exponential Functions

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
Fri Apr 1, 2016 3:05 AM

Post by Kaushik Srinivasan on March 27, 2016

Hey!

In your example 9 (IX) , I got a=0.25, b=0, c=-3 and d=4.

This gives the equation 0.25x^3 - 3x + 4. This has tangents of gradient 0 (making horizontal tangents) at (-2,8) and (2,0)

I even graphed it on Desmos and it worked. Did I do something wrong and accidentally graph the derivative?

Differentiation of Polynomials & Exponential Functions

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
  • Differentiation of Polynomials & Exponential Functions 0:15
    • Derivative of a Function
    • Derivative of a Constant
    • Power Rule
    • If C is a Constant
    • Sum Rule
    • Exponential Functions
  • 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
    • Part B: Find the Acceleration after 3 Seconds
    • Part C: Find the Acceleration when the Velocity is 0
    • Part D: Graph the Position, Velocity, & Acceleration Graphs
  • Example IX: Find a Cubic Function Whose Graph has Horizontal Tangents 34:53
  • Example X: Find a Point on a Graph 42:31

Transcription: Differentiation of Polynomials & Exponential Functions

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

Today, we are going to start talking about ways of taking the derivative of a function, of different types of functions but doing it in a quick way.0005

Let us just jump right on in.0013

We have seen and we have been dealing with the definition of the derivatives.0016

I will go stick with blue today.0022

We have seen that the derivative of any function can be gotten from the definition of the derivative.0029

That is we take the limit as h approaches 0 of f(x) + h - f(x) divided by h.0048

We formed this difference quotient, we simplified it as much as possible.0061

The last thing that we do is we take h to 0 to see what happens.0064

We end up getting this new function.0069

We need a quicker way to find derivative.0072

We cannot just keep using the definition of the derivative.0075

You saw from previous lessons that it can get very tedious.0078

We need a quicker way, you need a quicker way to find derivatives.0082

In this particular lesson, we are going to lay out the basic rules for finding derivatives of polynomial functions and exponential functions.0101

For the majority of these lessons, I’m actually not going to be proving where we get these formulas0112

because our main concern is having the formulas at our disposal, being able to use them.0116

We want to develop technique.0121

For those of you that go onto higher mathematics, you actually revisit this stuff again0123

and you will spend time actually proving all of these things.0127

If you want to take a look at the proofs, they are in your textbook.0131

They are very straightforward, it is actually no different than anything that you have done so far.0135

You are just using the definition of a derivative in its most general way.0139

And then, you run through it, you prove it, they are very easy to follow.0144

But again, it just tends to be a little bit tedious. We would not worry about proofs.0147

The first thing we are going to talk about is the derivative of a constant, let us start there.0153

The derivative of a constant.0162

If you guys do not know already from previous work, the derivative with respect to the variable of a constant C is just equal to 0.0166

The derivative of 5 is 0, the derivative of E is 0, the derivative of 9.7 is 0.0179

Number 2, we will just call it the power rule.0188

This says that if n is any real number, it can be an integer, it could be a fraction, it could be radical to, whatever, any real number.0208

Then, ddx of x ⁺n is equal to n × x ⁺n-1.0225

In other words, we take the n, we bring it out front, turn it into a coefficient.0235

And then, we change the exponent, we drop it by 1.0240

If the exponent is 5, the exponent becomes 4.0244

If the exponent is 1/3, the exponent, the n-1 becomes 1/3 -3/3 becomes -2/3, that is it.0246

This is the basic rule for all polynomials.0255

Let us see, next rule, if C is a constant then the derivative of C × some function f(x), whatever it happens to be.0261

Basically, it just says that the derivative of a constant × a function is the constant × the derivative of a function.0284

We can pull out the constant and then just save it and multiply it by everything, after we take the derivative of a function, equal C × ddx of whatever f happens to be.0290

I’m going to leave out the x just to make it a little bit more notationally tractable.0303

Let us do the sum rule.0321

If f and g are differentiable, then the derivative of f + g is just equal to,0329

I’m going to leave it as it is.0356

The derivative of f + the derivative of g.0362

In other words, if I have some sum of a function, I can just take the derivative individually and add them up, that is it.0368

The derivative of the sum is the sum of the derivatives, that is all.0374

You can think of it as distributing the derivative operator over what the sum is.0378

We can differentiate term by term and add.0386

The last is the exponential functions.0389

The exponential functions, we have the derivative of e ⁺x = e ⁺x.0400

The exponential function is very special.0409

When you take the derivative of it, you get the function back.0414

That is very special, it is going to play a very huge role in all the mathematics that we do.0417

This is going to be ddx of a ⁺x.0423

In this case, we have just e, 2.718, it is this.0429

Whenever there is a different constant, whatever it is, some a ⁺x, the derivative of that is,0434

the a ⁺x stays but we have to multiply it by the nat-log of the base a.0442

These are our basic rules for finding the derivatives of polynomial and the exponential functions.0452

With that, let us just launch right into our examples because that is when everything starts to make sense.0460

The first example, 9x⁵ – 4x³ + 14x² – x – 12.0467

Pretty standard polynomial, 5th degree in this case.0476

We have a sum, this is one polynomial but it is made up of 12345 terms.0480

We are going to use the sum rule.0487

We are just going to differentiate one term at a time.0488

A combination of a power rule and the sum rule.0492

The power rule says, take this coefficient, bring it down here, and then subtract by 1.0496

Because of this 9 in front of it, it is a constant, we can pull out and save it for later.0503

This is actually going to be 9 × 5x⁴ -4 ×, the 3 comes down, 4 × 3x².0507

Subtract the exponent by 1, + 14, I bring the 2 down, I put here and this becomes just x¹.0522

Same thing here, I bring the 1 over here and this is going to be x¹ -1 is 0, x⁰ is just 1.0534

Basically, any time you are taking the derivative of 1x, 2x, 3x, 4x, it is just the number itself, the coefficient.0544

The derivative of a constant is 0.0551

Of course, we simplify it.0554

We have 45x⁴ -12x², 14 × 2x + 28x – 1.0556

There you go, that is the derivative of that polynomial, nice, straightforward, very easy.0572

An application of the power rule, an application of the constant multiplied by a function, and application of the sum rule.0579

When we do these, we are sort of doing them.0588

We can say yes, we are applying this rule or that rule.0592

After you do four of these, you are expert at it already.0596

Let us look at a portion of the graph just for this particular one.0605

I just wanted to take a look at it.0609

What I’m going to do is we are going to go ahead and take a look at what the graph of that looks like,0611

and then, the graph of the derivative on the same graph.0615

Let us see, there we go.0622

We have this one right here which is our f(x) that was the original function.0626

And then, this right here, this is actually the derivative.0634

This is f’(x).0637

We have seen this before, we have dealt with it before,0643

when we dealt with graphs and taking a look at the function and the derivative on the same graph.0645

Notice, here the slope of this original function is positive, one is positive.0652

It is positive but as it comes up here, the slope is actually decreasing.0660

The line from your perspective, the slope is actually going like this, it is becoming 0.0665

The slope is positive, this is the derivative.0671

That is what the derivative is, it is a slope, it is positive.0674

When it hits the top of the graph, it hits 0.0677

Of course now the slope of this curve is negative.0680

It is negative, it goes down.0683

At some point, it starts to turn and starts to turn and become 0 again.0686

As this point rises and it hits 0, and then from that point on, the function is rising, the slope is rising.0692

The line is here, the slope is increasing and becoming positive.0704

That is why you have this.0711

The red is the original function, the blue is the derivative.0714

It is kind of interesting, even after all these years, whenever I look at a function and its derivative on the same graph,0721

it always takes me a minute to look at it and make sure I'm concentrating on the right thing.0726

When you are looking at the original function, you are looking at the derivative.0734

What the derivative is describing is the behavior of the slope on the original function.0739

This line, that is the tangent one that is becoming that way and going down that way, and this way, and this way.0743

That is what you are doing, that is what the derivative represents.0753

Let us go ahead and differentiate this function, √x - 4√x.0761

This radical symbol is a leftover from years and years ago, hundreds of years ago.0768

I do not care for the symbol myself but again it is ubiquitous in mathematics.0776

We deal with this just like turning everything into fractional exponents.0780

I’m going to rewrite this f(x) as x¹/2 – x¹/4.0786

I can go ahead and apply the power rule to this.0796

Very simple.0799

Again, this comes down, f’(x) is going to equal ½ x and ½ -1 because we subtract 1 from the exponent.0801

This becomes - ½.0812

This, the ¼ comes down, this becomes ¼ x.0815

And then, ¼ - 1 is - ¾.0820

That is it, that is your derivative.0824

You are absolutely welcome to leave it like that.0826

It is perfectly good mathematical symbolism.0829

You can write it as follows, if you want to.0833

This is really going to have to do with your teacher and what it is that they want.0851

You should ask them, is it okay to leave it in this form, do you write in radical form?0855

One of the things you are going to notice, if you have not noticed it, you are going to notice it now that0861

we actually start do taking the derivatives of functions very quickly,0864

especially when we get to product and quotient rule in the next lesson.0867

There are several degrees to which you can simplify a function.0875

At some point, you are just going to have to stop.0878

You are going to have to ask your teacher, where it is okay to stop.0880

If you want, you can write this as of 1/ 2 √x.0884

This x⁻¹/2, bring it down to the denominator, and then put the radical sign back on there.0890

-1/4, this is ¾, this is going to be the 4√x³.0895

This is another way to write it, if you want it to.0905

Example number 3, f(t) = (t/4)³.0912

Let us go ahead and expand this out.0920

We have f of T = t³/ 4³.0922

4 × 4 is 16, 4 × 16 is 64.0930

This is nothing more than 1/64 t³.0933

We can go ahead and treat it, this is the constant, this is the exponent.0937

Therefore, f’(t) is equal to 1/64 × 3t².0942

We end up with 3/64 t² or 3t²/ 64.0952

However it is that you want to write it out.0962

I have always like the constant to be separate but that is just my own personal taste.0966

Example number 4, x³ + 3x² + 2x + 3/ √x.0975

This one again, in the next lesson we are going to be doing something called the quotient rule.0983

In this particular case, we notice that there is only one thing in the dominator.0990

We can actually put each of these terms over that dominator and see if it simplifies, which it does.0995

I can rewrite f of x as x³ / x¹/2, because √x is x ^ ½ + 3x²/ x¹/2 + 2x/ x¹/2 + 3/ x¹/2.1003

This becomes 3 - ½, this is now the same base x, I can just subtract the exponents.1028

I get 3 - ½ is 2 ½, this is going to be x⁵/2 + 3x - ½ is 1 ½, 3x³/2.1038

2x¹/2, 1 – ½ + 3x⁻¹/2.1055

Now I have x to all these different coefficients.1063

I just differentiate, f’(x) is equal to 5/2x.1067

5/2 – 1 is 5/2 – 2/2 which is going to be 3/2 + 3 × 3/2x.1076

3/2 – 1 is ½, ½ + 2 × ½, I bring this down.1088

I subtract 1 from that ½ -1 is -½.1100

And then of course, we have 3 × -1/2 x – 1/2 -1 is -3/2.1105

My final answer f'(x) is equal to 5/2x³/2 + 9/2x ^½.1120

The 2 and the 2 cancel.1133

We are left with +x ^-½.1136

This is -3/2x⁻³/2.1141

There you go, that is your f’(x).1147

I hope that made sense.1155

F (s) is s² – 2/3√s⁴.1160

We are getting pretty accustomed to dealing with this now.1165

This is just s² -2.1167

I will do s, this 4/3, and when I bring it up, it is going to be - 4/3.1172

Is that correct, yes it is.1185

We have f(s), I got to tell you that it is easy to make mistakes in differentiation.1189

One of the things about calculus is it is not particularly difficult in terms of the application of the technique.1198

It is just sort of keeping track of all the little things.1204

In calculus, it is the details that matter, the individual little details, a - sign.1207

Remembering to subtract one from the exponent.1213

Remembering to add one to the exponent, coefficients, all of these things to keep track of.1216

I can guarantee you, I’m going to be making my fair share of mistakes on this.1221

Vigilance, that is all we can try to do is remain vigilant.1225

This is going to be s⁴/3, when we bring it up -4/3.1231

When we take the derivatives of this, this is going to be 2s -2 × -4/3s.1236

Now, -4/3 -1 is - 7/3.1249

F’ (s) = 2s, - and - becomes a +.1257

2 × 4 is 8, we have 8/3s⁻⁷/3.1264

There you go, this is our derivative.1272

Find the equation of a tangent line to the following function at the given point.1281

Graph both the function and the tangent line on the same screen.1286

We have our function f(x) = x² + √x.1290

We want to find the equation of a line that is the tangent line.1296

In other words, that is the derivative.1302

I’m sorry, no, the equation of the tangent line through this point.1305

The slope of that line is going to have is the derivative of this function at that point.1311

Because that is what the derivative is, the derivative is the slope of the line of the tangent line to the graph at that point.1318

That was one of our interpretations of the derivative.1327

The derivative is a rate of change.1330

It is the rate at which y is changing, when I make a small change in x.1331

It is also the slope of the tangent line to the graph at that particular x and y value, at that particular point.1336

Let us go ahead and find the derivative first.1345

We have f(x), let us rewrite it as x² + x¹/2.1350

We have f’(x) which is going to be 2x + ½ x⁻¹/2.1358

In this particular case, because I’m going to be putting values in, I’m just going to go ahead and write it in a form that is more familiar.1370

+1/2 × √x.1378

This is the formula for the derivative.1385

When we put this x value into there, we actually get the slope of the line.1388

We want to find f’(2).1394

That is going to equal 2 × 2 + ½ √2.1399

When I work that out, I get 4.35355.1407

This is our slope, it is the slope of the tangent line.1415

When I put the x value into the derivative function that I get, it gives me the slope of the tangent line at that point.1423

I go ahead and find the line itself.1430

The line is going to be 1 -y1 = the slope m × x - x1.1433

I have x and I have y, that is the point that it passes through.1441

I get y -5.414 = 4.35355 × x-2.1446

I’m going to go ahead and leave it in this form.1460

It is up to your teacher if they want you to actually multiply this out, simplify fractions that might show up.1462

However is it that they want to see it.1468

Perhaps, they want to see it in ax + by= c form, personal choice.1470

That is it, nice and straightforward.1476

Let us go ahead and take a look at what this actually looks like.1480

This right here, this is f(x), that is the function.1486

That point right there is our 2 and 5.414, that is the scales, the scales are not the same for the x and y axis.1491

I sort of expanded them out.1502

This is the equation of the tangent line.1503

This is the tangent line, the equation that we got.1507

This is the equation for the tangent line.1512

This is not f’(x).1520

F’(x), f’, in this particular, is f’(2).1523

It is the slope of the tangent line.1531

The derivative is the slope.1533

The derivative is not the line itself, very important.1535

The slope changes, if we hit peak to different point, it is going to be a different slope.1539

If we hit to peak a point up here, it is going to be a different slope.1544

Keep repeating it over and over again.1548

The derivative is the slope of the tangent line not the equation of the tangent line.1552

Find both the first and the second derivatives of e ⁺x -∛x.1561

Very simple, we are just going to take the derivative twice.1566

Let me rewrite f(x).1570

F(x) = e ⁺x -, I will write this as x¹/3.1580

We have f’(x) =, the derivative of the exponential function is the function itself.1588

That is just e ⁺x – 1/3x⁻²/3.1595

Bring this down, 1/3 – 1, the exponent – 1 becomes -2/3.1602

That is f’, now I will go ahead and do f”(x).1608

We take the derivative of the derivative, the first derivative, this becomes the second derivative.1614

The derivative of the exponential function is e ⁺x.1619

This is -1/3, now we take the derivative of this.1623

We bring this down, x-2/3 – 1- 2/3- 3/3 is - 5/3.1629

Our f’(x) and another notation for that is d ⁺2y dx², we have seen that notation before, =e ⁺x.1641

- and - is +, 2/9x⁻⁵/3.1653

Nice and straightforward.1663

S = 3t³-6t² + 4t + 2 is the equation of motion of a certain particle, with s in meters and t in seconds.1672

This is the position function.1681

This is the position function, in other words at any time t, let us say t is 5 seconds, I put T in there.1686

What I will end up getting is where the particle is, along the x axis tells me where it is.1692

We want you to find the following.1702

Find the velocity and the acceleration functions as functions of time,1704

the acceleration after 3 seconds, the acceleration when the velocity is 0.1707

Graph the position of the velocity and acceleration graphs on the same screen.1714

Let us go ahead and find the velocity and acceleration functions as functions of time.1721

Again, we have dealt with these before.1725

Whenever you get in the position function, the velocity function is the first derivative.1727

The acceleration function is the second derivative.1731

Velocity = ds dt or s’, however you want to do it, that is equal to, 3 × 3 is 9.1736

We dropped the exponent by 1, 9t².1748

2 × 6, the 2 comes down, 2 × 6 is 12.1753

It becomes -12t and +4 because 1 × 4 is 4 and t¹-1 is t0.1758

It ends up going away.1768

This is the velocity function.1769

The acceleration that = the derivative of the velocity with respect to t, that is equal to the second derivative of the position function d² s dt² .1772

Now I just take the derivative of the velocity.1785

2 × 9 is 18, that is 18t-, I take the derivative of this.1788

There we go, this is my velocity function, this is my acceleration function.1793

At any time t, I just plug it in to get the velocity and the acceleration.1798

At t = 2, the position is this, it is going this fast and it is accelerating with that acceleration.1803

Nice and straightforward.1810

Part B, they want you to find the acceleration after 3 seconds.1814

That is nice and easy, they just want the acceleration at 3.1820

We use this, 18 × 3 – 12.1824

I hope to God that I did my arithmetic correctly.1829

I’m notorious for making arithmetic mistakes but I think the answer should be 42 m/s².1831

That is the unit of acceleration.1838

Velocity is in meters per second.1841

The position is in meters, velocity is in meters per second.1843

Acceleration is in meters per second which is m/s².1845

Part C, they say they want the acceleration when the velocity is 0.1852

We have to set the velocity function to 0.1865

Find the values of t at which the velocity is 0, then plug those t values back in the acceleration equation.1868

Let us do exactly what it says.1874

We set the V equal to 0.1878

We have to find t, then use the acceleration function.1884

The velocity as a function of t is equal to 9t² -12t + 4.1893

I set that equal to 0 and I solve for t.1903

In this particular case, I’m not going to go through the particulars of this quadratic equation.1907

I hope that is not a problem.1911

I'm presuming that you have your calculator at your disposal or perhaps you just want to do it by hand,1913

by completing the square or using the quadratic formula.1918

In this particular case, I do not think it could be factored, whatever it is that you need to do by all means.1921

T ends up being 0.6667, one answer.1927

When a quadratic equation has only one answer not two, that means it just touches the graph, that touches the x axis.1935

Our acceleration that we are going to be looking for is the acceleration 0.6667.1945

That is equal to 18 × 0.6667 -12.1952

Our acceleration ends up being 0.1959

Let us go ahead and take a look at the graph of all three on the same screen, on the same page.1964

Here is our s(t), this is our position function that was the cubic equation.1973

This one right here, the blue, this is our S‘, this is our velocity function v(t).1980

This one right here, I think it is purple or magenta, this is S", this is the acceleration function.1990

Notice, positive slope, velocity is positive.1999

The velocity is positive, it is above the x axis.2009

The particle is moving to the right.2011

The position, the derivative actually goes to 0.2018

The slope goes to 0 that is why it hits that and it becomes positive again, and it increases.2028

The derivative follows the slope of the tangent line along that.2035

The acceleration, the acceleration is the derivative of the derivative.2042

Now, this parabola that we have, the velocity function is the function.2046

The derivative of that is going to be a straight line.2052

The slope is negative but it is increasing, negative but it is increasing.2058

At this point, the slope hits 0 and then the slope become positive.2064

That is why you have from negative to positive.2070

The function, its first derivative, the velocity function, its second derivative, the acceleration.2074

Nice and straightforward, I hope.2084

Let us go ahead and try this one.2091

A little bit longer, a little bit more involved.2095

Again, it is not like you have a lot at your disposal.2097

What you have at your disposal right now is this idea of the derivatives, we go with that.2101

Find the cubic function, ax³ + bx³ + cx + d, whose graph has horizontal tangents at -2,8 and 2,0.2107

In this particular case, we need to find a, we need to find b, we need to find c, and we need to find d, four variables that we need to find.2118

Let us go ahead and write this out.2128

F(x) = ax³ + bx² + cx + d.2130

We have 4 variables.2142

We know really only one way to find 4 variables, when we are given this little information.2145

We need to find 4 equations.2150

We know that if we have 4 equations and 4 unknowns, theoretically, we can find all of the unknowns.2152

In this case, a, b, c, and d.2157

We are going to be looking for 4 equations.2159

Where those equations are going to come from.2161

We know f(-2), we have it right here f(-2) is 8.2164

We just put -2 in for x and we put 8 equal to 8.2175

When I put -2 in for here, we get -8a + 4b, I’m putting -2 in for x, + 4b -2c + d = 8.2184

That is my first equation.2205

We also know f(2), f(2)=0.2207

Again, I can put 2 in for x and get something at a, b, c, and d, that is equal to 0.2210

We know f(2), f(2) that becomes 8a + 4b + 2c + d = 0.2220

This is our second equation, we are doing well.2232

They also tell us something else.2237

They tell us that the graph actually has horizontal tangents at these points.2238

Horizontal tangent means that the graph itself, no matter what the graph looks like, still the slope is 0.2243

Therefore, the slope is the derivative.2253

I take the derivative of this thing to get a new equation and I set that equal to 0 at these points.2256

I plug in -2 and 2 into the derivative of this to get two other equations.2266

Let us write all that out.2272

Horizontal tangents at x = -2 and x =2, that means a slope of 0.2276

This implies that f’ of x at 2 and -2 = 0.2297

A slope of 0 means the derivative is 0.2311

Let us take f’(x).2314

F’(x) is equal to 3ax² + 2bx + c.2316

The derivative of d is 0.2331

Now let us find, let us form the equation f’(-2).2334

F’(-2), -2 we are putting into the x value.2339

-2² is 4, 4 × 3 is 12.2344

We have 12a.2348

-2 and 2 this is -4b + c.2350

We know that the slope is actually equal to 0.2355

We have our third equation.2360

Now let us do f’ at 2.2362

F’ at 2 is going to be 12a + 4b + c.2364

That slope is also equal to 0.2371

There we have it.2374

We have our first equation, we have our second equation, we have our third equation, and we have our fourth equation.2376

4 variables, 4 equations, 4 equations, 4 unknowns.2384

Let us go ahead and work out the rest here.2391

We have 4 equations and 4 unknowns.2393

Our unknowns are a, b, c, and d.2396

Let us rewrite the equations again.2401

We have -8a + 4b -2c + d, that = 8.2405

That is fine, I guess I can do it over here.2420

We have the equation 8a + 4b + 2c + d = 0.2423

We have 12a -4b + c = 0.2435

We have 12a + 4b + c = 0.2441

Again, I’m not going to go through the process of actually solving these equations simultaneously.2448

It just has to do with, you can enter it into your software.2452

You can do it on your calculator.2456

You can do it at any number of online free equation solvers, that is how I did it.2458

Or you can do it by hand, I also did it by hand just to double check.2466

By fiddling around with some equations, solving for one variable here, solving for the same variable here.2469

Putting into these equations, whatever technique that you are going to use.2475

Again, these are techniques from previous mathematics.2479

We are just using them in support of what it is that we are doing now, with just calculus.2482

However you solve this, when you do, you end up with a = 0.1667.2487

You end up with b = 0.2496

You end up with c = -2.2498

You end up with d = 5.333.2501

Our f(x) equation, the equation that we wanted is 0.1667 x³ - 2x is +0x², I will just leave that out, +5.333.2507

There you go, nice and straight forward.2533

Again, we have the functions, we have the points, the x and y values,2535

and they say horizontal tangent so we know that the derivative is 0.2543

We can create 4 equations in those 4 unknowns.2546

Let us see, find the points on the graph of the function f(x) = 1 + 3 e ⁺x-2x2554

with tangent line at such a point is parallel to the line, the 2x – y = 4.2562

We have f(x) that is equal to 1 + 3 e ⁺x – 2x.2570

Let us go ahead and take the derivative of this, point at the tangent line2579

because the slope of the tangent line is going to be the derivative at that particular point.2586

F'(x) = 0, it is going to be 3 e ⁺x-2.2591

They say that the tangent line, the tangent line at such a point is parallel to the line 2x – y = 4.2600

Over here, I have the line 2x - y = 4.2609

This is y = 2x-4.2615

If the tangent line is going to be parallel to this line, it is going to have the same slope.2622

It is going to have a slope of 2.2627

The f'(x) is an equation for the slope.2633

Therefore, I’m going to set this equal to 2.2637

F'(x), we want f'(x), the slope of the tangent line to equal 2 because it is parallel.2643

Let us form that equation.2655

Let me do it over here.2661

We have 3 e ⁺x-2 = 2.2663

We get 3 e ⁺x = 4/3.2672

We take the nat-log of both sides.2678

We get x = nat-log of 4/3.2681

When we do this, the nat-log of 4/3 is 0.2877, that is the x value.2691

We take this x value.2700

They want the point on the graph of the function where the tangent line at such a point is parallel to the line.2703

We found the x value of that point.2709

If we want the y value, we have to put it back into the original function.2712

We get f(0.2877) that is equal to 4.247.2717

Our point is 0.2877, 4.247, there we go.2732

Once again, find the point on the graph of the function f(x) where the tangent line at such a point is parallel to the line.2749

I took the derivative of this function.2755

The derivative of that function at a given point is going to be a slope of the tangent line through that point.2758

They say that the line, the tangent line is parallel to this.2767

The slope of this line is 2.2775

I set the derivative equal to 2 to find the x value.2777

I found this x value, I plugged it back in the original equation to find the y value.2780

It passes through this point and has a slope of 2.2786

Thank you so much for joining us here at www.educator.com.2791

We will see you next time, bye.2793

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