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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Continuity

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 (6)

2 answers

Last reply by: Professor Hovasapian
Fri Mar 25, 2016 8:30 PM

Post by Acme Wang on March 12, 2016

Hello Professor Hovasapian,

In example II, I just confused why the function is continuous in the interval (-?, 14/4]. We just got the answer that the function is continuous at x=2 ONLY right?

Also, if f(x) and g(x) are both continuous, is f-g also continuous?

And for example VI, from my point of view, since the f(4) is positive and f(6) is negative, definitely f(x) would have only one real root (f(x) would pass through x-axis only ONCE :). I don't understand why there are more than one root between this interval?

Thank you for your time to answer my question.

Sincerely,

Acme

2 answers

Last reply by: Professor Hovasapian
Fri Mar 25, 2016 8:16 PM

Post by john lee on March 9, 2016

Professor Raffi Hovasapian,
Why f(a) cannot equal to f(b) in intermediate value theorem?

Thanks for the answer!

Continuity

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
  • Definition of Continuity 0:08
    • Definition of Continuity
    • Example: Not Continuous
    • Example: Continuous
    • Example: Not Continuous
    • Procedure for Finding Continuity
  • Law of Continuity 13:44
    • Law of Continuity
  • 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
  • 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
    • Jump Discontinuity
    • Infinite Discontinuity
  • Intermediate Value Theorem 40:58
    • Intermediate Value Theorem: Hypothesis & Conclusion
    • Intermediate Value Theorem: Graphically
  • Example VI: Prove That the Following Function Has at Least One Real Root in the Interval [4,6] 47:46

Transcription: Continuity

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

Today, we are going to talk about continuity.0004

Let us jump right on in.0006

Let me work in blue today.0014

We mentioned several times already that the limit as x approaches a of f(x) and the value of the function at a,0016

f(a), are independent of each other.0040

We saw already that you can have a limit exist, as you approach a certain point.0044

But that limit is not necessarily the same as the value of the function, at that point.0049

It could be, it could not be.0055

They are independent of each other.0057

The limit as x approaches a of f(x) might or might not equal f(a).0075

When it does, that is very special.0087

For that, we say that the function is actually continuous there.0092

Continuous means there is no gap.0094

It is just if you take a pencil, drop it on a piece of paper.0097

With one swing of your pencil, without lifting it up, you have the graph that means every single little point is accounted for.0099

When it does, we say the function is continuous at that point.0108

The function is continuous at a.0121

Our definition, just to be reasonably formal about it.0129

I just repeated what I just wrote.0138

If the limit as x approaches a of f(x) = f(a), then f(x) is continuous at a.0141

If we are talking about a particular interval, say from 0 to 5,0171

and it is continuous at every point in that interval, we said it is continuous over that interval.0175

That is it, pretty straightforward.0181

Let us put some graphs to get a feel.0187

Let us look at some graphs to get a feel for this notion of continuity.0190

Graphs to get a feel what continuity is.0201

Again, it is a very intuitive notion.0208

Something that you know implicitly, you know intuitively.0211

Let us look at some graphs to get a feel for what continuity is.0215

Our first graph, let us go ahead and just do something like that.0231

Let us say we have this and it goes like that.0237

Let us call this 5 and let us say that this is 2.0241

The limit of this function, this is our f(x), the graph.0246

The limit as x approaches 2, which means from below and from above of f(x), it = 5.0253

As we approach 2, the graph, the y value approaches 5.0264

As we approach it from above, the y value approaches 5.0269

The limit is actually equal to 5.0274

Now f(2) is not defined.0276

There is not even a value for f(2), that is why there is a hole there.0282

This is not continuous.0286

That is all continuity means.0294

It is not connected.0296

Let us do another graph here.0303

Now we will just notice nice single sweep of the pencil.0304

Let us say we have a point here.0310

Let us say the y value of this point is 6.0312

Let us say the x value is 5.0314

Here the limit as x approaches 5, again when we do not specify, it is from both ends.0317

The limit of f(x) as x approaches 5 is equal to 6.0324

f(5), f at 5, is actually equal to 6.0330

This is continuous because the left hand limit, the right hand limit, and the value of the function at the point equal each other.0336

It is that simple.0345

Let us look at another graph.0351

Let us say we have got this function right here.0356

Let us say we have got something that looks like that.0360

Let us say this is the point 3, let us say this is 4.0364

Let us say the y value of this one is 1.0367

Now we have the limit as x approaches 3 from below, we are approaching it this way, of f(x), that is equal to 4.0371

The limit as x approaches 3 from above, the value of the function, the limit is 1.0388

f(3) = 4, that is the solid dot, right there.0399

In this case, even though the value of the function, f(3) happens to equal one of the limits,0407

in this case, the left hand limit, they are not all equal.0413

The left hand limit has to equal the right hand limit, has to equal the value of the function.0416

This is not continuous, we do say that it is left continuous.0420

If the function were defined down here at 1, we would say it is right continuous.0424

We do differentiate, just like we have left hand derivative, right hand derivative, left hand limit, right hand limit.0431

We have left continuity and right continuity, but this function is not continuous.0437

Here left hand limit does not equal the right hand limit.0445

We know that the limit does not exist.0462

I do not need to write all these, it is not a problem.0470

You know what, I do not even need to write any of this.0483

These do not agree.0487

It is that simple, we do not need to label the point with silly formalities.0488

Once again, the left hand limit does equal f,0502

Let me write this out.0512

The limit as x approaches 3 from below of f(x) does equal f(3).0517

That is it, we just call this right, it is left continuous.0526

f(x) is left continuous.0532

That is it, it is continuous from the left, that is all that means.0537

Over all, it is not continuous at 3.0546

You can see this is not continuous.0547

I have to do this then I have to stop.0549

I have to pick my pen up, start again over here.0550

That is the intuitive notion of continuity, one nice notion.0555

It does not need to be a smooth curve motion.0561

It can be something like this.0565

This is still continuous because I have not lifted my pencil off the paper but these are sharp points.0572

The graphs confirm your intuition or should I say your intuitive notion of continuity.0587

In other words, without breaks in the graph.0612

That is it, that is all it means, without breaks in the graph.0615

In other words, when you put your pencil to paper to draw the graph of a continuous function,0623

you can draw it without ever stopping and lifting your pencil.0663

I should not say stopping because we can stop, without ever lifting your pencil.0677

That is all continuity is.0694

Once again, the limit as x approaches a of f(x) is equal to f(a).0706

One, you find the limit if it exists.0718

Two, the second thing you do is you find f(a).0732

The third thing that you check is do they equal each other.0740

If they equal each other, the function is continuous at that point.0752

If they do not equal each other, the function is not continuous.0754

You are doing two things.0757

You are taking the limit, you are taking the two, and you are checking to see if they are equal.0758

That is the question mark.0763

This is the definition.0765

When they give you a definition, when you see a definition that involves inequality,0766

what that means is essentially what you have to do is you have to check the left side, check the right side.0772

If the equality is satisfied, then the definition is satisfied.0777

Then it is that thing, whatever the definition says.0781

Those of you that go on in mathematics, when you take linear algebra,0785

you are going to be talking about something called a linear function, which is not exactly what you think it is.0788

There is a certain mathematical definition to it.0794

A linear function is defined by equality.0797

There is something equal to something else.0800

What you are going to have to do is you are going to check the left hand side of the equality,0802

check the right hand side of the equality, and then you have to confirm that they are equal.0805

If they are equal, the function is linear.0809

If they are not equal, the function is not linear.0811

That is what equations in definitions mean.0814

The laws of continuity correspond to the laws of limit.0820

We will write them down.0826

The laws of continuity correspond to the limit laws.0828

They are entirely obvious.0845

We will say let f(x) and g(x) both be continuous.0854

They can be continuous over an interval or continuous at a given point.0868

Then, the sum of the two functions is continuous.0872

The product of two continuous functions is continuous.0881

The quotient of functions is continuous, provided, of course, g does not equal 0.0887

Any constant times a continuous function is continuous.0894

Most functions you deal with are going to be continuous.0901

If they have a discontinuity, it is going to be a very few places like for example,0903

a place where the function is not defined.0909

In other words, a vertical asymptote.0910

It is going to be at a place of discontinuity.0912

Those are really the only two cases where you are going to deal with something that is discontinuous.0916

Most functions you will deal with are continuous.0920

Functions are great, they behave very nicely.0936

Let us do a few examples.0940

We have an example, it says it is going to be example 1.0955

We are going to draw a graph and we are going to ask you to tell us where the function is discontinuous.0964

Example 1, give the points on the following graph. I decided to draw the graph by hand, where f fails to be continuous.0969

There is your graph.1033

Here is a vertical asymptote.1036

The function is going off to infinity there.1040

Where are the points of discontinuity, perfectly obvious.1043

It is discontinuous here.1047

At x = 1, it is discontinuous here.1052

Clearly, at x = 2, it is continuous here.1053

Even though it is a shell point, it is still continues.1057

I did not have to lift up my pencil to continue again.1059

It is discontinuous here.1062

This goes off to infinity, never touches this, and never touches this.1065

They are not connected.1068

Those are your three points of discontinuity, 1, 2, and 6.1070

Very intuitive, very obvious.1074

Example 2, show that the following function is continuous at a given point.1081

Also give the interval over which the function is continuous.1085

Now we do not have a graph to help us out.1090

If you have a graphing utility, if the particular test you are taking, you are allowed to use a calculator to,1095

or graphing calculator or utility, to help you out, by all means use it.1101

But again, the whole idea of the calculus is to learn to do things analytically so that we do not have to rely on the graph.1106

If we have a graph to help us out, that is fine.1112

But there are times you are not going to have a graph.1114

We want to develop these notions that are always true.1117

Not have to rely on geometric notion of a graph.1121

Now we have to do this analytically, show that it is continuous at x = 2.1123

We know what the definition of continuity is.1128

The definition says the limit as x approaches in this case 2, of this f(x), has to equal f(2).1130

That is what we are going to do.1141

First, we are going to find the limit as x approaches 2 of this function.1142

And then, we are going to evaluate the function.1146

We are going to see if they are equal to each other.1147

If they are, we are done, it is very simple.1149

First, let us evaluate the limit.1153

The limit as x approaches 2 of the function x³ - 14 - 4x.1157

Again, do not let the symbolism intimidate you.1167

It is just a basic limit.1168

What you do with a basic limit, what you do to all limit, you put the value in first to see what happens.1172

If you get an actual finite number, you can stop.1177

You are done, that is your limit.1180

If you get something that does not make sense, you try to manipulate the expression and take the limit again.1181

That is all we have been doing, that is all you have to do.1189

We put 2 in, this is going to be 2³ - √14 - 4 × 2.1192

2³ is equal to 8 - the square root of,1203

What is 14 – 8, it is 6.1211

8 – √6, is 8 - √6 a finite real number?1213

Of course it is, yes.1216

That is our limit, very simple.1219

Now we evaluate the function at 2.1220

f(2) = same thing, 2³ - 14 - 4 × 2 = 8 -√6.1227

They are equal to each other.1242

Yes, this function is continuous.1243

You might be saying to yourself, wait a minute, we just did the same thing twice.1252

Yes, we do the same thing twice but under two different auspices.1256

One, we are evaluating the function at.1261

Here we are evaluating the limit but we know from our previous work with limits,1264

that the way you evaluate a limit, in the case of a function like this, is to actually plug it in.1268

Even though you are doing the same thing, you are doing it for different reasons.1275

I know it seems a little weird but that is what is going on.1281

It is actually two different processes.1284

√14 - 4x, we see that this, our domain is going to be,1291

Now I give the interval over which it is continuous.1298

We know it is continuous at 2.1301

Over which numbers is it actually continuous, besides 2?1304

This implies that 14 - 4x has to be greater than or equal to 0, because you cannot take a square root of a negative number.1308

Our domain is restricted.1318

Therefore, 14 has to be greater than or equal to 4x.1319

Therefore, 14/4 has to be greater than or equal to x.1324

The domain of definition is, the domain of f is negative infinity to 14/4 exclusive.1333

It is continuous over this entire interval.1346

There is no point in this interval where it is not continuous.1351

The function is defined and it is perfectly valid there.1355

Is the following function continuous at a given point.1364

f(x) = e ⁺2x, for x less than or equal to 0.1366

It is equal to x², for x greater than 0.1372

Is it continuous at x = 0?1375

Here we have a piece wise continuous function.1380

It is one function to the left of, 0 it is another function to the right of 0.1383

We need the left hand limit to equal the right hand limit.1388

We need it, in order for them to be continuity, we need it.1397

First, the limit as x approaches 0 from below is this one.1400

We use that function of e ⁺2x.1415

As x goes to 0, 2 × 0 is going to be 0.1421

It is going to be e⁰ and this can equal 1.1429

Now the right hand limit.1434

Now it is going to be the limit as x approaches 0 from above of x².1439

When I put that in, that is just going to be 0.1445

In this particular case, we can if we want to.1451

Let us just do it for the heck of it.1456

Let us just do f(0).1458

f(0), here it is less than or equal to 0.1462

We use this function.1465

This is going to be e⁰ = 1.1468

We see that the left hand limit and the value of the function are equal.1473

The function is left continuous, but these three are not equal.1477

It is not continuous at 0, it is not continuous at x = 0.1481

In this particular case, you do not even have to do this.1493

Basically, the left hand limit and the right hand limit are not the same, the function is discontinuous at the point.1497

You did not really need to check the value of the function at the given point, at a.1503

But in this case, we did, it is not a problem.1509

It turns out to be left continuous.1512

But overall, not continuous.1513

Let us go ahead and do a theorem for composite functions.1525

Again, this is mostly just a formality.1538

Let us recall what we mean when we say composite function.1545

If we are given f(x) and if we are given g(x), two functions of x.1551

When we form f(g), that is equal to f of g(x).1556

When you form the composite, that is what you are doing.1561

If g(x) is continuous at point a and f(x) is continuous at g(a),1566

then f(g) is continuous at a.1597

Let us do an example.1615

Let me go to red.1619

I think this is example 4, if I’m not mistaken.1623

The question here is, the cos of x³ + natlog of x is continuous at,1627

I’m not going to write it all out, I’m just going to say cont.1643

Is it continuous at x = π/2, that is our question.1647

Using this theorem that we just did, here our f is equal to cos(x).1658

Our g is equal to x³ + ln x.1666

That is the composite function, it is going to be f(g).1673

Cos of x³ + ln x.1676

f(g) is equal to the cos of x³ + ln x.1682

Let us see what we can do here.1692

If g(x) is continuous at a and f is continuous at g(a), then f(g) is continuous at a.1697

We ask ourselves, is g(x) continuous at π/2.1706

The limit as x approaches π/2 of g(x) is equal to the limit as x approaches π/2 of x³ + ln of x = π/2³ + ln of π/2.1719

That is the limit.1753

Now let us check to see the actual value.1755

g(π/2) = π/ 2³ + ln of π/2.1759

This and this are equal.1771

It is continuous.1772

Yes, g(x) is continuous at our point of interest which is π/2.1775

That takes care of the first one.1783

The second part is, is f(x) continuous at g(a).1785

Here is f(x) continuous at g(π/2).1791

We said that g(π/2) is equal to this thing.1811

Is f(x) continuous at g(π/2)?1824

g(π/2) is equal to π/2³ + natlog of π/2.1828

Therefore, what we want to check, for continuity we are checking to see that the limit = the value.1841

We are going to calculate the limit as x approaches g.1849

What we are saying is f(x) continuous at g(π/2).1855

It is going to be x approaching g(π/2) of f(x) which = the limit as x approaches the value π/2³ + ln of π/2 of the cos(x).1860

That is f, what we separated, that is why we wrote cos(x), instead of cos(x³) + ln x.1893

We ask ourselves, is g(x) continuous? Yes.1902

Now, is f(x) continuous at g(π/2)?1906

We are going to do this.1909

This limit =, we just put this here.1911

It equals the cos of π/2³ + ln of π/2.1916

This is a perfectly valid number.1925

We also evaluate f(x) which is f of π/2³ + natlog of π/21927

= the cos of π/2³ + natlog of π/2.1951

This and this are equal.1962

Yes, that is true.1964

Therefore, now we can conclude that fg which is equal to cos(x)³ + ln x is continuous at π/2.1969

Those are long process.1988

The basic idea is that a composite function is generally going to be continuous.1990

If you have f of g(x), if g(x) is continuous and f is continuous at g(x), then your f(g) is going to be continuous.1998

You can treat it individually like that or you can basically take a look at the whole function, f(g).2011

You can form the function f(g) as a single function of x and check the continuity of that function.2016

You can do it either way, it is not a problem.2022

A lot of these things pretty straightforward, they are pretty intuitive.2026

Sometimes you just break them down so much, that it tends to complicate them more than necessary.2029

That is exactly what this problem was.2036

It was just an excessive complication of something that is reasonably straightforward.2037

Let us see, what value of a will make the following function continuous at every point in its domain?2049

Here we have ax² + 7x, 4x less than 4, and x³ - ax for x greater than or equal to 4.2064

4 is the dividing point.2075

4 is our dividing point.2079

We are basically going to be taking limits of the function as x approaches 4, from below and from above.2085

In order for this to be continuous, for continuity,2094

we need the limit as x approaches 4 from below of f(x) to equal the limit as x approaches 4 from above f(x).2105

We need that equal to f(4).2122

We need all three things to be equal.2127

Let us calculate.2130

The limit as x approaches 4 from below, that is this one.2134

We are going to use this function of ax² + 7x =, we put x in for there.2140

We get a × 4² + 7 × 4.2148

You end up with 16a + 28.2155

That is our left limit.2161

Our right hand limit.2164

What is the limit as x approaches 4 from above?2167

From above, we are going to use that function.2170

It is going to be x³ - ax = 4³ - a × 4.2174

This one is going to give us 64 - 4a.2185

f(4) less than or equal to, we are going to use that right there.2195

That = x³ – ax, it is going to be 4³ – a × 4 = 64 – 4a.2202

We see that this, that this is the same as this.2215

It is right continuous.2219

We want, again, we said that we want the left hand limit to equal the right hand limit to equal f(4).2222

It already turns out that those two equal each other.2231

We want this equal to that.2234

We want this equal to these.2237

We are going to set 16a + 28 is equal to 64 - 4a.2246

Let us rewrite 16a + 28 = 64 - 4a.2266

You are going to get 20a = 36.2275

a is equal to 9/5.2282

That is it, it is that simple.2288

Whenever you need to set some piece wise continuous function,2292

you need there to be continuity at the particular point where they are piece wise separate.2295

You need to set the left hand limit and the right hand limit equal to each other,2301

and it has to equal the value of the function there.2304

Just use the definition of continuity which says that the limit as x approaches a of f(x) must equal f(a).2308

This left hand limit must equal the right hand limit.2335

A couple of nomenclature issues, just words that seem to come up.2359

I will just say, by the way.2364

Discontinuities are given different names.2368

I do not why, but okay.2371

That is just the way it is.2376

What if we have a situation like this.2377

We call this removable discontinuity.2380

The reason I call it removable discontinuity is because basically at a point where it is discontinuous,2383

whatever that is, let us say a, you are going to define it anyway you want.2392

You can just say f(a) = either that point or another point.2396

You can give any definition.2400

It is a discontinuity that is removable, we can remove it.2402

Here this type of discontinuity, we call this a jump discontinuity.2407

Again, these names are completely ridiculous.2416

They are completely meaningless.2421

People use them but for all practical purposes, it is continuity and discontinuity.2423

We do not need to know what type of continuity it is or discontinuity it is.2428

Of course, there is this one, you have some sort of an asymptote and you have a function going this way and a function going this way.2432

This is called an infinite discontinuity.2440

Just to let you know that.2444

You will see these terms, they do not mean anything.2453

The most important and practical theorem regarding continuity,2459

there are several of them but for our purposes,2479

is something called the intermediate value theorem.2487

The intermediate value theorem.2492

Here are the hypotheses, the if section.2503

If then, hypothesis, or hypotheses, conclusion.2508

The hypotheses, there are few of them.2513

Hypotheses are if a is less than b on the interval, if f(a) is different than f(b),2517

if they are not equal, if f(x) is continuous on the closed interval ab.2530

If all of these three are satisfied then the conclusion, the then statement, the conclusion is there exists,2543

I will write it out, there exists, I will put it in parentheses.2556

The symbol for there exists, the logical symbol is a reverse e.2563

There exists a number c somewhere between a and b, such that f(c) is equal to some number a,2567

or a is between f(a) and f(b).2600

This is a formal statement of the theorem.2607

We have the hypothesis which is your if clause and we have your conclusion which is your then clause.2609

Let us go ahead do what this actually means.2616

Pictorially, it looks like this.2619

I have got something that looks.2628

We have our axis.2631

We have a nice continuous function.2636

We have our point, let us say a is here that means f(a) is there.2640

Let us say that b is over here.2649

This is f(b) is over there.2653

It says that this is some number c and this is the number a.2661

Basically, what this is saying is that if a is less than b on the x axis, here we have a definitely less than b.2674

If f(a) and f(b) are different, here f(a) and f(b) are definitely different.2682

And if the function itself is continuous.2689

This function f(x), yes, we can see that it is nice and continuous.2691

If that is the case, then the conclusion I can draw is that there is some number, at least one, there may be more than one,2696

that some number c between a and b such that the f of that number is actually going to fall between f(a) and f(b).2701

That is what the intermediate value theorem says.2714

Is that, if the three hypotheses are satisfied then there is some number between the two,2717

such that when I take the f of that number, the y value is going to fall between f(a) and f(b).2722

All three hypotheses have to be satisfied.2729

a less than b, f(a) not equal to f(b), and it has to be a continuous function.2731

That is the real important one.2737

It has to be continuous.2739

If it is not continuous then there is no guarantee that a value of a actually even exist at all.2740

That is the whole idea because if we were to break the graph like that, if there is a discontinuity in the graph,2744

you might have a number but it is not defined there.2753

There is no guarantee that some number a in between f(a) and f(b) has some c value.2758

Continuity of the graph is profoundly important.2768

Let us write, since f is continuous as x goes from a to b, as x moves along from a towards b, f(x) hits every value.2774

In other words, as we move from here to here, f actually hits every single value between here and here.2806

It hits every single value.2818

It might hit it once, we do not know what the graph looks like.2820

But we know that it hit it at least once.2825

Because f is continuous, as x goes from a to b, f(x) hits every value between f(a) and f(b).2827

In another words, because the function itself is continuous, as we move from here to here,2849

we move continuously from f of f(a) to f(b).2854

There is always going to be some number between them.2858

I hope that make sense.2861

Again, I think intuitively, it absolutely does make sense.2864

Let us do a problem, prove that the following function has at least one real root in the interval for 6.2867

Here we have a function and they want us to prove it has at least one real root.2875

What that means is that, you do not know what the function looks like.2881

We can graph it, if we want to.2887

But again, we do not have a graph at our disposal.2888

We are going to use this theorem, analytical methods,2890

to show that this function actually crosses the x axis at least once, between 4 and 6.2893

It is either going to cross like this way or it is going to across this way, going from negative to positive or positive to negative.2903

We do not know which one first but we want to show that at least it hits the x axis, at least one time.2911

That is what this says, has at least one real root.2917

That means it hits the real axis at least one time.2919

The y value is going to be 0, that is what the root means, in the interval 4 to 6.2923

Let us check our hypotheses.2928

I think i will do this in blue actually.2932

Our hypotheses, the first one is a less than b?2942

Yes, 4 is definitely less than 6.2950

Our second hypotheses, our second hypothesis is, is f(4) less than f(6)?2956

We have to check that.2979

f(4) = 1.643, when I put 4 into there, I get 1.643.2981

Definitely, make sure that your calculators are in radian mode not degree mode.2990

When you put in 4 in here, all the functions in calculus,2996

when you use you calculator to solve trigonometric function, make sure you are in radian mode.2999

You will get a number if you are in degree mode, but the numbers are going to be wrong.3005

f(4) is that, f(6) = -1.497.3008

Yes, this is true.3014

f(4) is different than f(6).3017

Hypothesis three, is f(x) continuous over the closed interval of 4 to 6.3021

Yes, sin(2x) is a completely continuous function.3034

Sin and cos are continuous everywhere.3037

Cos is continuous and we know that the sum of the difference of a continuous function is continuous.3039

Yes, that is continuous, that hypotheses are satisfied.3044

Therefore, there exists at least one number between 4 and 6, such that f of this number, some number, let us call it c.3048

There exists at least one number c between 4 and 6.3089

There is some number here in that interval, such that f(c) is equal to 0, because 0 is between 1.643 and -1.497.3092

f(4) is 1.643, let us just put it right there.3119

f(6) is -0.497, let us just put it right over there.3123

The function is continuous.3128

Because it is continuous, when I'm going from here to here, I cannot lift my pencil.3130

Therefore, somewhere, it does not matter what trajectory it takes, some path,3135

whatever the function looks like, it has to pass through 0.3145

That is what we are saying.3149

That is what the intermediate value theorems says.3150

If a is less than b, if the f values are different from each other and if it is continuous,3154

that there is some number between the x values, such that the f of that number is between the f values of the two numbers.3162

That is it, thank you so much for joining us here at www.educator.com.3174

We will see you next time, bye.3179

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