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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Linear Approximations & Differentials

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
Thu Jul 7, 2016 7:10 PM

Post by Haleh Asgari on July 6, 2016

Hello Professor Raffi,

I really enjoy your lectures but I do not get what you did with the absolute value at the end of Ex 2. How did you get < (5-x)^(1/3) + 0.1 and > (5-x)^(1/3) + 0.1?

Thanks in advance,
HA

2 answers

Last reply by: Professor Hovasapian
Thu Apr 7, 2016 2:02 AM

Post by Acme Wang on April 1, 2016

Hi Professor,

I am little confused in Example IV. Why dr means the error in radius? And why dA is the error in Area?

Also, What is the graph of L(x)? Is L(x) the tangent line at x sub0?

P.S: I really appreciate your classes and they are really really super useful! Thank you very much!

Sincerely,

Acme

Linear Approximations & Differentials

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

Transcription: Linear Approximations & Differentials

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

Today, we are going to talk about linear approximations and differentials.0004

Let us jump right on in.0008

Essentially, what we are going to do is we are going to have some function f(x).0011

Instead of f(x), we are going to come up with something called a linearization of x0023

which we are just going to call l(x).0027

It is going to allow us to use this, instead of this, when we are not too far away from a particular point whose value we do know.0028

Let us jump right on in.0039

I’m going to start off by drawing a picture here.0040

Let me go like this and something like that.0044

Let us see we have got some curve, some function, and then, we have the tangent line.0049

This is going to be our x0 and let us say over here is our x.0059

If we go up, we got this point and this point.0066

This point is going to be our x0, y0.0072

This point over here, this is going to be our xy.0080

It is going to be the y value at another point along the curve.0083

This is our f(x), it is the actual f(x).0088

Now this line here, this line, it is y - y0 = m × x - x0.0093

m is the slope of the line, the derivative.0110

y0 and x0 those are the points that it actually passes through.0112

We know what this is.0117

This height right here, this height is just y0.0120

y is the same as that right there.0128

This height, this is m × x - x0.0131

x - this difference right here, this is x - x0.0138

If I multiply, I have a bigger triangle here.0142

If I'm here, if I move a distance x - x0, the height, this part is nothing more than the slope × the distance.0149

I hope that makes sense.0162

Remember from the slope is Δ y/Δ x =, the Δ y,0163

in other words how high you move is nothing more than the slope × the Δ x.0170

The slope × this is your Δ x, you have your slope.0176

This height right here.0180

Let me write this down now.0183

The whole idea of the derivative and the tangent line which this is to a curve,0185

is to be able to approximate the curve, approximate values along the curve.0208

I will explain what I mean using the picture, after I finish writing this,0222

is to be able to approximate the values along the curve by using the tangent line instead.0226

As long as we do not move to far away from our x0.0244

If we know some point on the curve, this tangent line gives me an approximation.0261

I know if I do not move too far away from x0, either in this direction or in this direction,0266

in this case, let us just worry about this direction.0271

If I do not move too far away, the values of x along the curve are going to be all these values right here.0273

But if I do not move too far away, then this value right here is a pretty good approximation to this value right here.0283

It gets better and better, the closer I actually stay to x0.0290

That is what this is all about.0295

The whole idea of the derivative is to approximate a complicated curve near a certain point.0297

If I want the value at a point near a point that I know, I do not have to use the function itself.0304

I can just go ahead and use the derivative of the function.0310

It gives me an approximation and that is what this is.0312

Instead of finding this value, I find this value.0316

This value comes from this.0321

If I just move this y0 over to this side, I get this y value is actually equal to y0 + this height.0324

That is what this equation is actually saying.0334

If I rearrange this equation, rearranging the equation of the tangent line0336

gives my y value is actually equal to my y0 value + m × x – x0.0354

This is the value that I really like, but I’m going to approximate it by that.0364

y0 is this height, m × x - 0 is this height.0370

If I add those two heights which is what this says, I get this y value which is a pretty good approximation to that.0376

That is the whole idea, this is the linearization.0386

Now let me do this in terms of f(x).0389

Let us do it again.0392

I'm going to change the labels but it is still the same thing.0395

I have got a graph, I have a tangent line.0402

This is my x0, this is my x.0409

This point right here, this is x sub 0, f(x) sub 0.0418

Let me write this a little bit better.0426

Instead of y, I’m going to call it f(x) sub 0.0427

This point is x sub 0, f(x) sub 0.0431

This point right here, this is nothing more than x f(x).0441

Now this height, let me go ahead and go this height.0453

This is nothing more than f(x0), right.0461

If I go straight across that means this height is nothing more than f(x0).0467

This height is f’ at x0 × x - x0 because the equation of this line is f(x) - f(x0) = f’ at x0 × x – x, y - y0 = m × x - x0.0480

I just replaced y, y0 with f(x) and f(x0).0508

Therefore, this point which is an approximation to this point is this height + this height gives me an approximation.0522

Now we have f(x) rearranging this, moving this over to that side, = approximately f at x0 + f’ at x0 × x - x0.0540

That will take me to that point right there.0559

It is almost equal to that point right there.0562

Again, this is greatly magnified.0565

We call this the linear approximation.0568

We call this the linear approximation of f(x) at x sub 0.0575

It is symbolize with the l.0592

The linear approximation is equal to f(x0) + f’ at x0 × x - x0.0595

It is also called the linearization of f(x) at x0.0607

It is also called the linearization of f(x) at x = 0.0611

When you are given an x0, you are going to find the f(x0), you are going to find the f’.0625

At x0, you are going to put those numbers here and here.0630

You are going to form this thing.0633

Now that you have a linearization, now you can replace f(x).0636

You can replace, now l(x) replaces f(x).0642

Instead of using f(x), you can use l(x) instead.0656

That is what is going on here.0660

When you create this linearization, this function, which is going to be some function of x,0661

you can use that instead of f(x) directly.0667

It is an approximation.0671

Instead of dealing with the function directly, you are going to deal with the linear approximation to the function.0674

You are going to deal with the equation of the tangent line, that is what is happening here.0679

Let us do an example, I think it will make sense.0685

Example 1, use linear approximations to find the natlog of 5.16.0689

What it is that is actually happening here?0716

I will draw it out real quickly.0718

We know that the natlog function is something like that, it crosses at 1.0719

We want to know what the natlog is at 5.16.0726

Here is 5, let us just say that 5.16 is right there.0730

This is 5.16.0735

We want to know this value.0738

We will use linear approximation not the actual function itself.0743

5.16 is close to 5, we go to 5.0747

We draw a tangent line at 5.0752

I’m going to use this equation, the linearization.0756

In other words, the equation of the tangent line to find that value.0759

If I magnify this area, it is going to look something like this.0764

The curve is going to be like that, then it is going to be like this.0768

Here is where the actual point is, I’m going to find this one that is really close to it.0773

Let us go ahead and do it.0779

Let us recall our linearization is equal to f(x0) + f’ at x0 × x - x0.0780

5.16 is close to 5, our x0 is 5.0792

Our x that we are interested in is 5.16, we have that.0798

Again, it needs an x0.0802

Our x0, in this case is 5, just based on what the problem is asking.0805

5.16 is closest to 5.0809

The function itself is the natlog of x because the natlog function is what we are dealing with.0815

The derivative f’(x) = 1/x.0824

f(5) is equal to 1.609, f’ at 5 = 1/5.0831

Therefore, the linear approximation, the substitution that I can make for f(x) for the natlog is f(x0).0845

X0 was 5, f5 is 1.609 + f’ at x0.0855

F’x0 is 5, f’ at 5 is 1/5 + 1/5 × x – x0.0863

There you go, this is my actual function of x that I can substitute now for the log of x.0871

This is the linearization function.0882

Now I want to solve it.0884

I want log of 5.16.0886

Therefore, the l(5.16) is equal to 1.609 + 1/5, x is 5.16 – 5.0888

When I solve this, I get 1.6414, that is my answer.0906

I identify my function, log(x).0915

I identify my x0, I have the linearization equation.0918

I calculate the f at x0, I calculate f’ at x0.0925

I put it in, now I have my linearization function.0929

This is a substitute for the original function log(x).0932

l(x) replaces f(x).0937

l(x) approximates, that is why we can replace it, approximates f(x).0943

This function is an approximation.0950

When I'm close to 5 for ln(x).0952

That is what is happening here.0958

By the way, the actual value of the natlog of 5.16 is equal to 1.6409.0963

Not bad, 6409, 6414, that is very good approximation.0975

As long as we do not deviate too far from x0, either in this direction or this direction,0982

the tangent line is a good approximation to the curve.0987

Clearly, if that is your curve, as you get further and further away from the x value,0995

it is going to deviate from the curve.1006

Now you are here and here as opposed to here and here.1008

The idea is to stay as close as possible.1011

Choose your x0 wisely.1013

Here are our f(x), once again, is the natlog of x.1019

The linearization was 1.609 + 1/5 × x – 5.1029

We asked for the natlog of 5.16 which is the l(5.16).1041

In this case, the absolute value of the linearization, the actual f(5.16) – l(5.16),1058

the actual value - the linear approximation.1077

This is the actual - the linearization was 1.6409 - 1.6414, actually gave us 5 × 10⁻⁴.1082

The error was on the order of 10⁻⁴.1101

In general, what you can do is, if you know what error you want to be within, you can adjust your choice of x.1106

Here we just happen to pick 5.16.1116

We said it is not that far from 5, let us choose x0 = 5.1120

If you want your error to be within a certain number, if you want to keep your error,1124

let us say to a 0.1 or 0.01, 0.056, whatever it is that you want, you can actually adjust and1128

it will tell you how far you can deviate from the x0 that you choose.1134

In general, if you want to, I should say if you want,1139

the error between f(x) and l(x) to be within a certain bound,1152

then you have to solve the following.1177

You have to solve that.1182

You have to put in, in this case it would be ln(x) - 1.609 + 1/5 x – 5 under the absolute value sign.1188

You have to solve the certain bound b.1197

I should say a certain bound b.1203

You have to solve this equation either analytically or graphically.1208

As you will see in a minute, graphically is probably the best way to do this.1212

Let us go ahead and do an example, I think it will make sense.1216

Our example number 2, we are going to let f(x) equal the 3√5 - x.1224

The question is for what values of x will the linearization at x sub 0 equal 1 be within 0.1?1237

We are saying the x is equal to 1.1266

How far can I move to the right of 1 and to the left of 1, to make sure that the error,1269

the difference between the actual value of the function and the linear approximation of the function stays less than 0.1?1275

Now I’m specifying the error that I want, now I need to know how far I can move away.1284

Let us go ahead and do it.1290

f(x) is equal to the 3√5 - x which is equal to 5 - x¹/3.1293

l(x), we know that l(x) is f(x0) + f’ at x0 × x - x0.1305

Let us find f(x0) first.1320

Let me go ahead and do this in red.1324

I have got f(x0).1326

x0 is 1, f1 is equal to 5 - 1 is 4, 4¹/3.1328

It is going to be the √4 which is going to equal 1.5874.1340

I have taken care of that, now I need to find f’ at 1.1348

f’ at x is equal to 1/3 × 5 – x⁻²/3 × -1.1358

When I put in 1, I will write it this way, -1/3 × 5 – x⁻²/3.1368

Therefore, f’ at x0 is f’ at 1.1387

I put 1 into this and I end up with, -1/3 × 5 – 1⁻²/3 = -0.1323.1392

I have taken care of that.1404

I will go back here, I put this number and this number into here and here.1407

Now I have my linearization, my l(x) is equal to 1.5874 - 0.1323 × x - x0 which is 1.1413

This function can now replace this function, if I'm not too far from 1.1426

If x is 1.1, 1.2, 1.3, whatever.1436

If I’m not too far away, I can replace this function with a linearization.1439

It is saying, how can I keep this error between the two?1443

How can I keep the error less than 0.1?1449

We have to solve f(x) – l(x) less than 0.1.1451

We want the absolute value, we want the difference.1463

We do not care about the positive or negative.1466

That is why there is an absolute value sign.1467

What we want is this, f(x) is 5 - x¹/3 - 1.5874 - 0.1323 × x – 1.1471

We want that to be less than 0.1.1491

This is the same as 1.5874 - 0.1323 × x - 1 less than greater than 5 – x¹/ 3 + 0.1, 5 - x¹/3 - 0.1.1494

I hope that makes sense why that is the case.1524

We have an absolute value sign, get rid of the absolute value sign.1526

You put a -0.1 here.1530

Then, I just move this function over that side, move this function over to that side.1531

I think the best thing to do is graph this.1540

You are going to graph the function, you are going to graph the function -0.1.1546

You are going to graph the function + 0.1.1550

You want this thing to be between those two graphs.1554

This is a straight line, this is the equation of the tangent line to this function at the point 1.1560

What you get is the following.1572

When we graph this, we get this.1575

The middle graph is the actual graph itself.1586

Let me write that down.1590

The middle graph that this is one, that one.1592

The middle graph, that is the function itself, 5 – x¹/3.1598

The lower graph, that one, that is 5 - x¹/3 - 0.1.1607

The upper, exactly what you think, it is 5 - x¹/3 + 0.1, that is that one.1615

The line, this, that is the linearization, that is the tangent line at the point 1.1624

Notice it touches the graph at x = 1.1634

The black line is the linearization.1640

The black line is l(x).1645

In order for the difference between l(x) to be between 0.1 above and 0.1 below,1649

we need to make sure that this tangent line,1661

we see where it actually touches either the upper or the lower graph and we read off the x values.1664

As long as the linearization stays between the upper and lower which comes from what we just did in the previous page,1671

remember we solve the absolute value, we rearranged it.1679

We put 0.1, -0.1, we move the function over.1684

That inequality has to be satisfied, that inequality is this graph.1688

Instead of solving analytically, just do it graphically and you can just see1692

where it touches the upper and lower, and read off the x values.1698

Let us write that down.1702

Wanting the absolute value of f(x) – l(x) to be less than 0.1 which is what this graph says means1703

we want the tangent line which is l(x) between the upper and lower curves, the upper and lower graphs.1722

Just read off the x values.1746

When you read off the x values, you have -2.652.1755

That means as long as x, here is 1, this is our x sub 0.1762

As long as x goes that far and goes this far, as long as x is between there and there,1769

my linear approximation will give me an approximation to the actual function itself, the 5 – x³, to an error of 0.1.1779

That is what this means.1792

As long as I stay within this and this, the tangent line itself does not go past 0.1.1793

By specifying an error, I solve this thing.1800

Graphing this thing, I see where the tangent line touches and I read off the x values, less than 3.398.1803

This is there.1818

As long as x is in this interval, the difference between f(x) and the actual linearization is less than 0.1.1819

I hope that made sense, that is what you are doing.1828

Now let us talk about differentials.1833

Essentially the same thing except in calculus notation.1835

It is really simple.1842

Let y equal x³.1846

We know that dy/dx is equal to 3x².1849

Let us move this over, this is just a number, a small number.1854

dy = 3x² dx, this is the differential.1858

The differential of x³ at a particular x, at a particular x sub 0 is 3 × x sub 0² dx.1865

This says, if I change my x value by a small amount, and the small amount is dx,1875

then the y value changes by this small amount.1902

The y value changes by this small amount, in terms of graphs.1912

Same thing that we did, except now we are talking about really tiny motions.1927

We have our function, we have our tangent line, this is our x0.1931

Now this distance is dx, this distance is dy.1940

We know what the distance dy is.1953

It is just of the slope × dx.1955

There you go, the slope at a given point.1961

X0 is that, the slope is the derivative.1964

This is just notation, that is all it is.1967

It is the exact same thing.1970

There you go.1972

If you want to know how the y value changes when you make a small change in x,1974

just move this x over and multiply it by the slope, the derivative at that point.1980

From a given point, your change in y is going to be this much, if you change x by this much.1984

That is all this says, the differential.1992

If I said what is the differential of the function x³ at x0 = 2, I will in plug 2 to here.1995

2 × 2 is 4, 3 × 4 is 12.2007

We have got dy = 12 dx.2010

If I move away from 2, a distance of 0.1, y is going to change by 12 × 0.1.2015

That is what this is saying.2025

If I change x by this much, how much is that going to change?2027

It is the same thing here right.2030

It is a rate of change.2032

If I change x by a certain amount, how much is y going to change?2033

Except now I have a dx here, all I have done is actually move it over.2037

There is nothing different than what it is that we are doing.2041

We are just looking at it slightly differently.2044

Let us do an example, very simple.2049

Example 3, what is the differential of f(x) = e ⁺x³ + sin x?2055

Very simple, just take the derivative.2075

This is just y = e ⁺x³ + sin(x), dy dx = e ⁺x³ + sin(x) × 3x² + cos(x).2078

Therefore, the differential dy is equal to 3x² + cos(x).2103

I just decided to move it over to the left, × e ⁺x³ + sin(x) dx.2110

For a particular x value, at a particular x value, from that x value, from that x0,2122

I should say from that particular x0 value, if I move away to the left or to the right by dx,2128

the value of my function, the vertical movement is going to be that.2135

That is all this is, the differential.2140

If I have a differential movement in x, what is my differential change going to be in y, that is all this is.2142

We actually call that the differential, when we have moved the dx over.2148

Let us try another example here.2157

Let us try example 4, use differentials to evaluate e⁰.02.2162

Linear approximation, differential, it is essentially the same thing.2180

We are just different notation.2183

-0.02 is close to 0, I’m going to take x0 = 0.2188

That is going to be my base point.2200

The function that I’m interest in is, clearly y = e ⁺x.2202

Dy dx = e ⁺x.2209

Therefore, dy = e ⁺x dx.2214

At the point it is equal 0, we get the differential is equal to e⁰ × dx.2223

e⁰ is 1, dy = just 1 dx.2233

Our dx from 0, our 0 point, we are at -0.02, we are moving this way.2241

Now we are at 0.02.2249

Our dy is equal to 1 × -0.02.2252

Therefore, e⁰.02 is equal to e⁰ + dy + the differential from that point.2265

I’m going to evaluate it at that point and if I change by dx, dy is going to change this much.2292

From 0, if I go to 0.2, it is going to be e⁰ +, in this case I calculated that the differential is -0.02 =, e⁰ is 1 + -0.02 = 0.90.2299

That is it, nothing particularly strange about this.2316

I hope that makes sense.2323

Let us finish off with a nice little problem here.2326

The radius of a circular table is measured to be 30 inches with a maximum error and measurement of 0.4 inches.2330

We have a table and measure the radius to be 30.2338

At a possible error 0.4 inches which means it could be 29.6 or could be 30.4.2346

It might actually be a slightly smaller or slightly bigger.2355

What is the possible error in the calculated area of the table between the 30 + 0.4, the 30 - 0.4?2362

What is that possible error?2371

In other words, how much of a difference, what is the outside extra area or if it is smaller what is the inside extra area?2373

What is the error in the actual area, if I have a possible error of 0.4 away from 30?2385

Express the error as relative error as well.2394

We will do both of those.2397

The area, we know that the area of the circle is π r².2399

The differential, if I change 30 to 30.4, or to 29.6, that 0.4 is my differential, that is my dr.2405

Dr here equal 0.4 + or -.2419

The change in the area, the differential of the area, da dr = 2π r.2424

Therefore da = 2π r dr.2434

I change my area by a certain amount.2440

My area is going to change by a certain amount.2442

This dr is the error in the radius.2446

Da is going to be the error in the area.2450

That is what they are asking, estimate the possible error in the calculated area.2453

We are actually going to express it as a differential.2458

Here r = 30, dr = 0.4.2462

Therefore, da is equal it 2 × π × 30 × 0.4.2469

da = 75.4 in², that is the error.2482

The error is 75.4 in².2494

If I made a measurement of 30, if they are telling me that my error is off by 0.4 inches + or -,2501

that means the area that I calculated, the error in the area is going to be +75.4 or -75.4.2511

That is how big of a difference my error in the area is going to be.2523

It is just using differentials, that is all it is going on here.2527

Relative error, let us go to blue.2530

Relative error is the error divided by the calculated value.2534

A couple of ways that we can do this.2550

The error itself we calculated, that was 2π r dr.2552

The actual value, the area is π r².2558

The π cancels π, the r cancels r.2562

You get 2 × dr/ r.2565

Here the relative error is 2 × dr 0.4/ r which is 30, 0.027.2571

If I express this is a percentage, I will move this over and that would be 2.7%.2587

That means that if the error that I'm making measurement from 30 is 0.4,2592

that means the difference in area is going to be 2.7% of the area calculated at 30.2602

If it is 30.4, that means I'm going to be over by 2.7% of my actual area measured at 30.2609

If I’m at 29.6, if the error is under the 30, that means I'm going to be short by 2.7% of my total area calculated at 30.2618

That is what this means.2629

Relative error is the error itself that you calculate divided by the actual area of the value that you measured.2630

We could have done this directly.2639

Here we actually did relative error.2641

We use da/a, the error/ the actual value.2642

We get it in terms of variables and then we plug the variables in.2647

I could just have done it directly.2651

We could also have just done it directly.2658

In other words, the definition of relative error is equal to the actual error/ the actual value.2669

Sorry, I should say the error/ the actual value, da/a.2679

We found da, we found 75.4.2687

Area is equal to π r², area = π × 30².2692

It is equal to 2827.43.2700

da/a, error/ actual value, error/ actual value = 75.4/ 2827.43 = 0.027.2707

Again, this can give, if I want to speak in terms of percent, 2.7%.2725

That is what a differential does.2731

Anytime you have a given function, go ahead and differentiate, move the dx over.2732

And that tells you, if I change x by a certain amount, how much does y change?2736

That is it, it is really what we have been doing all along.2743

Now we just want to think about it that way.2746

Once again, if I have y = some f(x).2749

I think it would be the best if I just use actual functions here.2761

An example, if y = sin(x), dy dx, the rate of change of sin(x) is equal to cos(x).2768

If I just want to concentrate on how much does y change when I change x, I move the dx over.2779

If I change dx by 0.2 away from the point π,2786

If I’m at the point π, and if I go 0.2 away from π to the left or 0.2 away from π to the right, y changes by cos(π) × 0.2.2794

That is the differential, that is the whole idea of the differential.2808

In this case, the differential actually gives me some error.2811

If I have a measured value, and if I’m not quite sure about that measured value,2815

let us say it is + this way and + this way, my measured value is my x0.2820

The difference positive or negative is my dx.2827

The y gives me the change in the overall function that I'm dealing with, in this case it was area.2832

Area = π r².2838

Here it is cos x dx.2839

This is the dx, this is the dx.2843

The differential itself gives me the total change of whatever it is that I’m calculating.2844

I hope that makes sense.2849

Thank you so much for joining us here at www.educator.com.2852

We will see you next time, bye.2853