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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Calculating Limits Mathematically

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 (9)

2 answers

Last reply by: sorin dragon
Fri Sep 15, 2017 1:26 PM

Post by sorin dragon on September 13, 2017

hello professor Raffi. I have a quick question. How works the method you used for factoring the numerator in Example 1: Calculating Limits? Thank you.

1 answer

Last reply by: Professor Hovasapian
Thu Feb 11, 2016 11:05 PM

Post by Avijit Singh on February 10, 2016

Hi Professor Raffi. I am a student self studying for the AP Calculus AB exam in May 2016, and I just wanted to thank you for your great videos. They really explain the concepts well and give me confidence to study for this exam in the comfort of my home.

I was wondering if you have any advice as to how I could best study for AP Calc. I am hoping to get through the AB course really soon (I have graduated from high school and am just taking this course to prepare for university level calculus). I do have a lot of time on my hands so I am only focusing on AP Calc. Do you think it is a good idea to progress to AP Calculus BC if I manage to complete the AB course in the next month? Thanks very much.

1 answer

Last reply by: Professor Hovasapian
Thu Jan 7, 2016 11:06 PM

Post by Sohan Mugi on January 6, 2016

Hello Professor Hovasapian. Just had a quick question about this lecture. Will we have to possibly face a problem like the last one in this lecture during the AP Calculus AB Exam? Thank you.

1 answer

Last reply by: Professor Hovasapian
Wed Nov 25, 2015 12:35 AM

Post by peter alabi on November 23, 2015

Hi professor Raffi. I don't know how you derived 2x^2 from X^2 - X^2, i just want to know if it a simple mistake or am tripping.
second. last lecture you obliviously mentioned infinitesimal, am just wondering if you can be more elaborate for me, the definition of infinitesimals and hyper real numbers? My professor seems to care so much about it but i don't get it. Despite the minor request great lecture, i never knew math can be so explicitly explain thanks.

Related Articles:

Calculating Limits Mathematically

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
  • Plug-in Procedure 0:09
    • Plug-in Procedure
  • Limit Laws 9:14
    • Limit Law 1
    • Limit Law 2
    • Limit Law 3
    • Limit Law 4
    • Limit Law 5
    • Limit Law 6
    • Limit Law 7
  • Plug-in Procedure, Cont. 16:35
    • Plug-in Procedure, Cont.
  • 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
    • Limits Theorem 2: Squeeze Theorem
  • Example VI: Calculating Limits Mathematically 49:26

Transcription: Calculating Limits Mathematically

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

Today, we are going to talk about calculating limits mathematically.0004

Let us jump right on in.0009

We have been using graphs and we are going to be using tables of values.0012

Now we are going to do it analytically.0015

Let us go to blue here.0018

We have been using graphs and/or tables to find limits.0026

How do we do it analytically?0045

How do we do it analytically, mathematically?0051

If there are some technique that we can use to evaluate this limit.0060

For example, what if we have something like the limit as x approaches 2 of x³ + x² - 4/ 6 - 4x.0066

What if we are faced with some limit like that?0090

How do we deal with this?0092

The short answer is the following.0095

The short answer is plug 2 into f(x) and see what you get.0103

I know, that is it, and see what you get.0119

Yes, that is literally it.0122

Just plug in and see what you get.0127

When we put 2 in here, we end up with 2³ + 2² - 4/ 6 - 4 × 2.0129

8 + 4 – 4/ 6 - 8 = 8/-2 = -4.0145

That is actually your answer.0156

When you plug the value that x approaches into f(x), one of two things happens.0160

If you get an actual number like we did, you can stop.0190

This is your limit.0210

This is the limit.0215

If you do not, if you get something that does not make sense, for our purposes,0222

does not make sense is going to be something like dividing by 0 or 0/0, infinity/infinity, infinity – infinity,0238

things that later on we are going to call indeterminate forms.0249

If you get something that does not make sense, you are going to have to manipulate the expression.0251

Turn it into something else, take the limit again.0256

If you get something that does not make sense, you must try other things.0260

Usually what that means, usually other things means, this means rewriting f(x) through some sort of mathematical manipulation.0281

You are going to turn it into an equivalent expression, but you are just going to manipulate it mathematically.0308

Maybe you are going to simplify something algebraically.0313

Maybe you are going to factor and cancel something out.0316

You are going to try different things.0318

Maybe you are going to rationalize the denominator, rationalize the numerator, whatever it is that you are going to do.0319

You are going to convert it to an equal of expression and then take the limit again, until you get something.0324

Usually this means rewriting f(x) through some sort of mathematical manipulation.0331

Let us see why this plug in technique works.0344

I can leave it and just jump right in, but I think it is important to at least see why it works.0350

Let us go to blue.0357

Let us see why this plug-in procedure works.0366

We are going to begin with two very basic limits, very obvious limits.0383

We begin with two basic limits that are obvious.0391

The limit as x approaches a of a constant c = c.0407

The graph of it looks like this.0416

This is your coordinate system.0418

Let us just say this is f(x) equal c, just some constant.0421

Let us say for example, f(x) = 5.0432

It does not matter what the value of x is.0437

f(x) is always going to equal 5.0438

Clearly, no matter what number you approach, the limit of the constant, the limit as x approaches a of 5 is going to be 5.0442

This is an obvious limit.0451

The limit as x approaches a of a constant is the constant.0452

That is our first obvious limit.0459

The second obvious limit, let me go back to blue, is the limit as x approaches a of x actually equal a.0462

This graph looks like this.0475

The graph of the function x looks this way.0483

Let us say this is a, as x approaches a, this side and this side.0486

This function right here is y = x.0495

Therefore, the y value here is also a.0499

The y value is approaching a, from above the y value is approaching a.0504

This is an obvious limit.0510

The limit of the function x as x approaches a = a, = the number you are approaching.0511

Once again, it is clear that the limit as x approaches some number a of 5 is equal to 5, that one is clear.0521

Hopefully also, this one is also clear that the limit as x approaches some number of the function of x = that number.0535

Now that we have those two basic limits, let us write down our limit laws.0547

Here we will let c be any constant and the limit as x approaches a of f(x) is a.0566

The limit of a function as x approaches a exists and it equals a.0584

We will let the limit as x approaches a also of a function g(x), we will let it equal b.0592

Our first limit law is the following.0605

The limit as x approaches a of f(x) + g(x) = the limit as x approaches a of f(x) + the limit as x approaches a of g(x) = a + b.0609

What this says is that, the limit of the sum of two functions is equal to the sum of the individual limits of the functions.0629

You can think of the limit as distributing over both.0638

The limit of something + something is the limit of something + the limit of the second something.0641

We have two functions, you add them, the limit of the sum is the sum of the limits.0648

Let us try this one.0655

The limit as x approaches a of f – g.0659

That is the same thing, this is the same as that except this is just –g.0667

It is going to be the limit as x approaches a(f) - the limit as x approaches a(g) which is equal to a – b.0670

Because the limit of f(x) is a, the limit of f is a, the limit of g is b.0682

The third, the limit as x approaches a of any constant × f(x).0689

It is equal to the constant × the limit as x approaches a of f(x), it is equal to c × a.0700

The limit of a constant × that, you can pull the constant out and put in front of the limit.0708

Number 4, the limit as x approaches a of f(x) × g(x).0715

It is exactly what you think, the limit of the product is equal to the product of the individual limits.0724

= the limit as x approaches a of f(x) × the limit as x approaches a of g(x) which is equal to a × b.0730

We have 5, the limit as x approaches a of f(x)/ g(x) = the limit as x approaches a of f(x)0745

divided by the limit as x approaches a of g(x).0761

That is it, nice and straightforward.0772

The limit of the sum is the sum of the limits.0776

The limit of the constant × the function is the constant × the limit.0778

The limit of the product is the product of the limits.0781

The limit of the quotient is the quotient of the limits.0784

Provided it exists down here, and we said that it does.0786

Let us do a 4’, the limit as x approaches a of f(x) raised to the n power is equal to the limit as x approaches a of f(x).0793

The limit of the function f(x) raised to the n is equal to limit of f(x) all raised to the n.0813

It is just f(x) multiplied a certain number of times, that is all it is.0822

That equals the limit raised to the n.0827

Now applying 4’ to the function f(x) = x.0834

We get the limit as x approaches a(x) ⁺n is equal to the limit as x approaches a(x) ⁺n.0849

We said that the limit is x approaches a(x) is equal to a.0867

It is just equal to a ⁺n.0871

That is it, very nice.0874

The last limit, the limit as x approaches a of n √f(x) = the n √of the limit as x approaches a of f(x).0878

You just pass the limit through.0895

Our limit procedure, our plug-in procedure is summarized as,0900

given a polynomial function f(x),0931

if a is in the domain of f, then the limit as x approaches a of f(x) = f(a).0941

I just wrote this down for the sake of writing it down.0964

From a practical standpoint, I would not worry about this.0965

The truth is whether a is in the domain or not, we are just going to plug in a.0970

We are going to see what we get.0974

If it is in the domain, it will work out just fine, we will get a number.0977

If something else happens, like we said before, you are going to have to manipulate it to see what you can do with it.0980

I would not worry about what it is that I just wrote.0986

The idea is basically just plug in and see what you get.0990

Let us go back to that limit that we started off with.0994

Sorry, was it 2 or was it 5?1002

I think it was 2.1006

The limit as x approaches 2 of, we had x³ + x² - 4/ 6 - 4x.1009

The shortcut was just plugging in and see what you get.1019

The break down based on the limit laws is the following.1023

I just want you to see the breakdown as follows.1028

The limit as x approaches 2, the limit of this function.1038

This is a quotient, this equals the limit as x approaches 2 of x³ + x² – 4 divided by,1048

because the limit of the quotient is the quotient of the limit.1062

The limit as x approaches 2 of 6 - 4x.1065

This is equal to the limit of the sum is the sum of the limits = the limit as x approaches 2 of x³1072

+ the limit as x approaches 2 of x² - the limit as x approaches 2 of 4 divided by the limit as x approaches 2 of 6 - this is 4x.1080

I’m going to pull the 4 out, -4 × the limit as x approaches 2(x).1099

Now that you have broken down into its limit laws, now we just plug in 2.1106

You are going to get 2³ + 2² – 4.1115

The limit of the constant is the constant.1126

4 × the limit of 2, the limit as x approaches 2(x) is 2.1129

It is -4 × 2, you get -4.1134

The idea is just put the 2 in, plug it in, see what you get.1137

If you get a number, you are done, you can stop.1140

If you do not get a number, if you get something that is nonsense, you have to manipulate it.1143

We said earlier, as we just said a moment ago, if you plug in and evaluate and you get an actual number, you can stop.1156

You are done, this number is your limit, is your answer.1190

If when you plug in, if you get something unusual, I will put unusual in quotes.1207

If you get something unusual, then you must manipulate f(x) and try the limit again.1218

Let us do some examples.1248

It will make perfect sense, once you see the examples.1249

Let us go ahead, over here, we have got an example.1252

We want to calculate the limit as x approaches 1 of x³ + x² - 17x + 15/ x – 1.1259

Let us plug in.1276

When you put 1 in here, up here is not a problem but you cannot put 1 in here, because 1 - 1 is 0.1277

1 is not a domain of this function.1285

You get something unusual, we have to manipulate this.1289

When we plug in 1, we get 0 in the denominator.1292

Let us manipulate by seeing if we can factor this thing.1311

Let us see what we can do.1316

Let us manipulate this expression, by seeing if we can factor it.1317

You have to understand at this point, there is no way for me to know1334

what manipulation I'm going to have to do, in order to make this work.1337

Factoring is the first thing that I try, simply because I see a rational function.1342

I figured just to go ahead and do the long division.1346

We will divide the bottom and the top, and we will see what we get.1348

That might not work, we might have to try something else.1351

There is no way of knowing beforehand.1355

There is no single algorithmic procedure that you can follow to solve these limits.1356

It is all going to comedown to mathematical ingenuity, mathematical insight, luck, try this and try that.1362

Literally, that is what it is going to come down to because things are becoming a lot more complex.1369

It is not just a straight single shot where we see the goal and we know that1374

we are going to have to take this step to get to that goal.1379

If we are faced with another situation, it is not going to be the same step.1383

It is never going to be the same steps twice.1386

Each problem is individual.1388

You have to pull back and get in the habit of not looking to solve a problem immediately, based on what you already know.1389

The reason that calculus is actually called analysis is precisely for that reason.1398

You have to stop and analyze the situation.1403

Take a look at each situation as it arises.1405

When you plug in 1, you get a 0 in the denominator, that is unusual, it does not make sense.1411

We are going to see if we can simplify this, find an equivalent expression by dividing or seeing if we can factor it.1415

Factoring this, it is a cubic equation.1422

I’m just going to do the long division.1424

That is how I’m going to do it.1425

I go ahead and I do x – 1.1428

It is going to be x³ + x² – 17x.1433

Let us write it so they are legible here.1438

My notoriously illegible writing, I apologize for that.1442

Let us go ahead and see if we can do this division.1449

This is going to be x², x² × x is going to be x³.1452

This is going to be - x².1457

I'm going to change that sign and change this sign.1461

I’m going to cancel that and I'm going to get 2x² - 17x.1465

This is going to be +2x, this is going to give me 2x².1473

This is going to give me -2x.1479

Change that sign, change that sign.1482

That cancels and I'm left with -15x + 15.1484

This is going to be -15.1492

It is going to be -15x + 15.1495

Change that sign, change that sign.1500

I’m left with 0, perfect.1505

Now I get a factorization of x - 1 × x² + 2x – 15.1506

This is really great.1518

It looks like x² + 2x – 15, it looks like I can actually factor that too.1535

That will be x - 1 × x + 5, if I'm not mistaken, × x – 3.1540

x² 5x - 3x gives me my +2x, 5 and 3 gives me my -15.1552

My top is actually factorable.1559

I get, the numerator of the function x³ + x² – 17x + 15 factors.1566

We get x - 1 × x + 5 × x - 3/ the denominator which was x – 1.1582

That cancels, worked out well.1597

We can take the limit again.1602

The limit as x approaches 1, not 2.1604

Now we plug in 1 to here, we end up with 6 × 1 - 3 is -2 – 12.1609

We got our actual answer.1621

The original limit is -12.1623

In this case, we factor the numerator.1626

This is our function, just in factored form of the numerator.1629

It turned out that those actually ended up canceling.1632

This expression and the original expression are equivalent.1636

We take the limit of what is equivalent.1641

This time we ended up with a finite number.1643

We could stop, our answer is -12.1645

We got some nonsense, we fiddled with it, and we came up with an answer.1650

Let us try another example here.1657

Let us let our f(h), this time our variable will be h.1663

Let us let it equal 3 + h² -,1667

I’m sorry, 3 + h³ - 27/ h.1673

We would like you to find the limit as h approaches 0 of this, of f(h).1683

The limit as h approaches 0 of f(h).1697

F(h) is this thing.1700

It is 3 + h³ - 27/ h.1701

When I plug in 0 here, I get 0 in the denominator.1708

I cannot do anything with this directly.1711

I’m going to have to manipulate it.1714

We notice that the numerator is not completely simplified.1717

Let us see if we simply it, in other words, expand the 3 + h³.1742

Let us see if that turns it into something where we actually can plug in 0 for h and it will give us something that makes sense.1747

f(h) is equal to 3 + h³ - 27/ h, that =,1758

3 + h³, 1, 3, 3, 1, those are the coefficients of the expansion for an exponent of 3, Pascal’s triangle.1771

It is going to be 3³ × 3² h × 3h² h³.1784

That takes care of the expansion, that is -27/ h.1797

That = 27 + 27h + 9h² + h³ - 27/ h.1804

The 27 goes away, here I’m going to factor out an h.1819

h × 27 + 9h + h²/ h.1824

The h cancels, I'm left with f(h) = 27 + 9h + h².1839

This is the same function as we had before.1853

It is just simplified and we use algebra to simplify it.1856

Now we take the limit again.1860

Now we take the limit as h approaches 0 of f(h) which is this.1863

27 + 9h + h².1877

When we plug in 0, this one goes to 0, this one goes to 0.1882

I’m left with an answer of 27, that is my limit.1885

That is it, it does not work, simplify it, manipulate it.1889

Do whatever you need to do until it works.1893

If it does not work the second time, you try it again.1896

Welcome to calculus.1901

Let us try another example.1906

The limit as x approaches 0 of x² + 16 - 3/ x².1909

Again, we have a problem that if we plug 0 in, on the top is fine but we are going to have 0 in the denominator.1924

We have to do something to this.1932

Again, plugging 0 in gives us a 0 in the denom.1938

Let us manipulate, this time, we are going to rationalize the numerator.1957

You are accustomed to rationalizing the denominator.1962

You can do the numerator.1964

It actually does not matter.1965

You are just going to multiply the top and bottom by the conjugate of the numerator.1966

It is that simple.1969

Let us manipulate by rationalizing the numerator.1971

It is going to look like.1984

This is going to be x² + 16, under the radical sign, -3/ x² × x² + 16, all under the radical sign, + 3.1984

That is the conjugate of the numerator.2000

x² + 16 + 3.2003

I’m sorry, did I do this wrong?2009

I think this is actually a 4 not a 3.2011

This is a 4, this is a 4, and this is a 4.2019

I think that is all I have.2029

Let us see what we have got, when we actually do the multiplication.2033

When we multiply this and this, this and this, we are going to get x² + 16.2037

This and this cancel.2047

-16/ x² × √x² + 16, under the radical, + 4.2051

That and that go away.2062

We are left with x²/ x² × √x² + 16 + 4.2065

That goes away, we are left with the 1/ x² + 16, under the radical sign, + 4.2078

Now we simplify this as much as possible.2087

It is the same function.2090

All we have done is multiplied by something and change the way it looks.2091

We have affected it cosmetically.2094

Now we can take the limit again.2097

Now the limit as x approaches 0 of this function 1/ x² + 16 + 4.2099

We plug 0 in for here, this goes to 0.2111

The square root of 16 is 4 = 1/ 4 + 4, our answer is 1/8, an actual number.2114

I think you get the idea.2125

Let us do some more examples.2129

Let us try something a little bit more complicated.2137

The limit as x approaches 0 of the absolute value of 3x/ x.2145

Absolute values scares the hell out of everyone, including me.2152

I just gotten used containing to containing my fear at this point.2157

They are daunting, like how do you handle it.2160

You remember that the absolute value, it actually consists of two things.2164

We have to find two limits here.2167

Again, once you put 0 in for x, you are going to get a 0 in the denominator.2171

That does not make sense.2175

We are going to have to manipulate this somehow.2176

The limit of as x approaches 0 of the absolute value of 3x/ x.2180

Remember our absolute value sign, the constant inside the absolute value can come out.2188

This is actually equal to 3 × the limit as x approaches 0 of the absolute value of x/x.2193

We have to deal with this limit actually, and whatever we get we multiply by 3.2203

Let us recall what absolute value means.2209

The absolute value of x is two things.2212

It is equal to x, when x is greater than 0.2216

It is equal to –x, when x is less than 0.2219

We have to do two separate limits.2228

We have to deal with this absolute value of x/x, as two separate limits.2236

We have to do one, when x is bigger than 0.2249

Remember, we are approaching 0 here.2257

0 is what we are approaching.2259

When x is bigger than 0, approaching 0 when x is bigger than 0 means we are approaching it from the right.2261

It means approaching 0 from there.2268

We have to do one for x less than 0.2273

We have to do when x approaches 0 on the left.2277

We are approaching 0.2281

We have to do it this way, that is it from the positive end, from above.2283

We have to approach 0 from the negative end, from below.2288

Let us see, let me actually change colors here.2292

Let me go to purple.2298

You know what, purple is nice but I think I like blue better.2302

Let us do for x = greater than 0, for that one.2308

For x greater than 0, the absolute value of x is x.2311

3 × the limit as x approaches 0 of the absolute value of x/x.2323

It says the absolute value of x = x for x greater than 0, I just plug in x for here.2330

That is equal to 3 × the limit as x goes to 0 of x/x.2335

x/x is just 1.2342

The limit equals 3.2353

If I’m approaching 0 from the right, when x is bigger than 0, my limit of this thing is +3.2354

Now let us go x less than 0, let us approach 0 from the negative numbers.2364

For x less than 0, the absolute value of x = -x.2375

3 × the limit absolute value of x/x = 3 × the limit of x approaches 0.2385

Absolute value of x is –x.2394

-x/x is -1 = 3 × -1 = -3.2403

There you go.2411

The limit from the right is 3.2414

The limit from the left is -3.2417

3 does not equal -3.2421

In other words, the right hand limit does not equal the left hand limit.2425

This means that the limit as x approaches 0 of the absolute value of 3x/ x does not exist.2433

That is how you handle absolute values.2444

You actually have to separate it into x being positive and x being negative.2446

That is it, I hope that made sense.2452

Let us do another example.2455

f(x) = combined function, x² – 25, under the radical.2467

When x is bigger than 5 and it equals 20 - 4x.2479

Sorry, combined function.2486

Less than 5 should probably be a little bit more mathematical precise, than I usually am, forgive me.2489

We see the 5 is the dividing point.2495

When x is bigger than 5, we use this function.2498

When x is less than 5, we use this function.2501

We want the limit as x approaches 5 of f(x).2508

That is what we want, the limit of this function.2517

5 is the dividing point.2521

Clearly, we are going to have to do an x approaches 5 from above and use this function.2523

As an x approaches 5 from below and use this function.2528

We evaluate the limit as x approaches 5 from above, by using that function.2539

The limit as x approaches 5 from below, by using this function.2551

We already know that we have to do the left and right anyway.2557

Because it does not specify whether it is left or right.2560

We have to actually do left and right, whether they are separate functions or not.2562

Let us see what this gives us now.2570

Let us do the left hand limit.2581

The limit as x approaches 5 from below, we are going to use the one for when x is less than 5.2585

Our function is 20 - 4x.2592

Just plug 5 in.2598

You are approaching it from below, that from below part has nothing to do with the number itself.2601

It just means you are approaching it from below.2606

You are still approaching 5.2607

In order to find out what happens, plug 5 in.2609

It is going to be 20 - 20 = 0.2612

The left hand limit = 0.2616

Let us remind ourselves what f(x) is.2619

It is 20 - 4x and it is x² – 25.2623

This is for when x is greater than 5.2627

This is for when x is less than 5.2629

Now the right hand limit, the limit as x approaches 5 from above, this function.2632

It is going to be x² – 25.2640

Plug it in, you are going to get 25 - 25 under the radical = 0.2644

Here the left hand limit = the right hand limit.2652

Therefore, which implies that the limit exists and the limit approaches 5 of f(x) = 0.2656

Let us write down a couple of theorems that may actually help in the evaluation of limits.2683

Some theorems that may help.2692

The first theorem, if f(x) is less than or equal to g(x) near a point a and the limit as x approaches a(f)2701

and the limit as x approaches a(g) both exist, then the limit as x approaches a of f(x)2727

is less than or equal to the limit as x approaches a of g(x).2746

Basically, f(x) is less than or equal to g(x), you already know that what you do to the left side,2755

if you do it to the right side, any operation that you take, retains the relation.2760

If I multiply f(x) by 5, I multiply g(x) by 5.2769

5 f(x) is less than or equal to 5 g(x), because f(x) is less than g(x).2773

It is the same thing.2780

f(x) is less than or equal to g(x).2781

Therefore, the limit of f(x) is less than or equal to the limit of g(x), provided both limits exist.2782

The next one which is pretty important.2793

Probably we are going to use it, but every once in while it might come up.2799

It is among that is hardest to remember.2802

I think in my entire mathematical career, I think I have used it 4 times.2805

If f(x) is less than or equal to g(x), it is less than or equal to h(x) and2810

the limit as x approaches a of f(x) = the limit as x approaches a of h(x), which happens to equal a,2827

if this relation exists and the limit of f(x) and the limit of f(x), the two flanking functions,2843

if they happen to have the same limit then the limit as x approaches a of g(x) also = a.2852

It makes sense, if the limit of this is 5 and the limit of this is 5, this is in between those two.2864

The limit has to be 5, that is pretty much what is going on.2870

It is called the squeeze theorem.2874

Graphically, it looks like this.2887

Let us say this is our point a, let us say it is over there.2891

Let us say this is our a.2896

Let us say we have some function which is something like that.2900

Then, maybe something like that and something like this.2908

Let us let this be the h(x).2916

Let this be the g(x) and let this be f(x), this is a.2920

Near a, you see that f(x) is less than g(x) is less than h(x).2927

As you get close to a, if the limit of f(x) is a, the limit of h(x) is a,2940

basically g has no choice but to be squeezed in between them.2951

The limit of g(x) is equal to a, that is why they call it the squeeze theorem.2955

Let us go ahead and do an example of one of these.2963

What is the limit as x approaches 0 of 5x² × cos(1/x).2973

Clearly, if we plug 0 in, we cannot because we have 1/0 here.2983

That is not going to work, we have to do something.2988

We do know one thing, we know something about cosine.3008

This fact about sine and cosine, very important fact.3019

Remember this one fact, it will probably save you a lot of grief and3022

make a lot of problems that are otherwise intractable, very easy to solve.3027

We know something about cos(1/x).3031

In other words, the cos(a) whatever a happens to be, some function of x.3037

We know that the cos(1/x) lies between 1 and -1.3042

The sine and the cosine functions, they maximum value is 1 and their minimum value is 1, always.3050

We know that this is true.3057

Since that is the case, watch this.3059

This is a relationship that is true.3062

I’m going to multiply everything by 5x² which means 5x² × -1 is less than or equal to 5x² × cos(1/x) is less than or equal to 5x² × 1.3064

This is just -5x², this is 5x² × cos(1/x) which is our original function.3083

It is less than or equal to 5x².3092

Let us take the limit of this and this function, and see what we get.3096

Again, this is true, limit, limit, limit.3099

If I apply the same operation to everything in this relational chain, the relation is retained.3104

The limit as x approaches 0 of -5x² is going to be less than or equal to the limit as x approaches 0 of 5x²3115

× the cos(1/x) is going to be less than or equal to the limit as x approaches 0 of 5x².3128

Plugging here, we get 0 is less than or equal to the limit as x approaches 0 of 5x² × cos(1/x),3137

less than or equal to this limit.3150

When we plug it in, we get 0.3152

0,0, therefore, our original limit is 0.3154

You are more than welcome to graph it yourself, to actually see that it is 0.3165

There you go, that is calculating limits mathematically, calculating them analytically.3172

Plug in the value that x is approaching and see what happens.3178

If you get a number, you can stop.3182

You are done, that is your limit.3183

If not, you are going to have to subject the function to some sort of manipulation.3185

You are going to be converting it into something equivalent.3188

You are not going to changing it.3190

You are converting it to something equivalent using the various tools that you have at your disposal.3192

Factoring, the squeeze theorem, rationalizing numerators, rationalizing denominators,3197

whatever else that your own peculiar personal ingenuity can come up with.3203

That is the wonderful thing about these, is every year,3209

it amazes me the different ways that kids come up with solving this limits.3212

I mean it is almost infinite, the number of variations that they can come up with, it is exciting.3217

In any case, thank you for joining us here at www.educator.com.3224

We will see you next time, bye.3226

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