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INSTRUCTORS Raffi Hovasapian John Zhu
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Raffi Hovasapian

Raffi Hovasapian

Overview & Slopes of Curves

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 (17)

1 answer

Last reply by: Professor Hovasapian
Mon Jul 10, 2017 6:10 PM

Post by Mohamed E Sowaileh on July 10, 2017

Hello Dr. Hovasapian,
I hope you are very well,

I am a student who is extremely weak in math. In order to be very strong in math, specially for engineering field, could you provide me with sequential order of mathematical topics and textbooks. With what should I begin so that I can master big topics like calculus, statistics, probability ... etc.

Your guidance is precious to me.

Thank you so much.  

1 answer

Last reply by: Professor Hovasapian
Wed Oct 26, 2016 6:43 PM

Post by Peter Fraser on October 26, 2016

Thanks, that lecture was great!  It looks like the point-slope equation, y - y1 = m(x - x1), will work really well with the derivative to find the equation of the line, y = mx + d, tangent to any point of a non-linear function, because d, the y-intercept, can be found by setting x of the point-slope equation to 0 and because m of the same equation is effectively the slope of the tangent to the coordinates of the chosen point of the function.  So, for point (7?/6, -½), the equation for the line tangent to this point will be found from y - (-½) = cos(7?/6).(0 - 7?/6); y + ½ = (-?3/2)(-7?/6); y = (-?3/2)(-7?/6) - ½ ~ 2.674.  So the approximate equation for the tangent line for the derivative of sin (7?/6) is y ~ (-?3/2)x + 2.674.  Generally, y = f’(x1)(-x1) + y1, right?

1 answer

Last reply by: Professor Hovasapian
Fri Mar 25, 2016 10:44 PM

Post by Eric Liu on March 18, 2016

Hello Mr. Hovasapian,

I love your lectures, is there a time table for when AP Calc BC will come out?

Thanks!
Eric Liu

2 answers

Last reply by: Professor Hovasapian
Wed Jan 27, 2016 4:05 PM

Post by Harold Snook on January 13, 2016

Mr. Hovasapian,
 The equation which I found on the Internet a couple of weeks ago has disappeared and the paper I wrote it down on, along with symbol designation, has also disappeared. While searching for the old one, I found another,
V=h[r^2 arc cos(r-d/r)-(r-d)sq rt (2dr-d^2). d=depth of fuel, r=radius, h=height of tank when standing on end.
 Of course, I am not sure either is accurate.
 I am not nearly as concerned about finding a formula that will work as I am about how to come up with an equation for solving the problem.  I don't even know how to start.
 It has me stumped.
Thanks,
Harold

2 answers

Last reply by: nathan lau
Mon Jan 11, 2016 4:55 PM

Post by nathan lau on January 9, 2016

and one more question, when you take the derivative of a function, is the answer not directly related to the slope, but rather the x value of the point that we plug into the f'(x) to get the actual slope? That sort of confuses me, because i thought the derivative of a function was just the slope of tangent line at that point. And if the problem you were dealing with was all symbolic, than how would you end up getting a numerical value for the slope by plugging the newly found value for x into f'(x)? For example, if the original function was f(x)=x^3+x^2+2, than f'(x) would be 3x^2+2x+0. Than to find the numerical value for the slope, i believe you would have to plug the derivative back into f'(x), that would make the function needed to find the numerical valve for the slope f'(3x^2+2x+0), but wouldn't that give you the 2nd derivative, or is the 2nd derivative just written that way to make things easier to look at? sorry, i know this is really wordy and i may be looking at things wrong, i just want to get my understanding straight. thanx :)

1 answer

Last reply by: Professor Hovasapian
Mon Jan 11, 2016 1:33 AM

Post by nathan lau on January 9, 2016

hey, so i have already learned all of semester 1 of AP calculus ab. i have a final on monday, do you know how long this course is up to the second half(integration)? i just want to know if it is even possible to listen to all the lectures in time. i know everything pretty well, but i get allot of anxiety while taking tests, especially because i have some gaps in my knowledge. i just need to stultify all my knowledge and the rules to feel fully confident to do well, thanx! :)  

2 answers

Last reply by: Professor Hovasapian
Thu Jan 7, 2016 11:55 PM

Post by Harold Snook on December 31, 2015

Mr. Hovasapian,
I am an "older" student who enjoys learning. I find your lectures very well done, challenging and enjoyable.
This is not a question about this lecture. I write it here because I cannot find a better place to include it on Educator and I was told by Katie that you are very knowledgeable in mathematics.
I am attempting to find how a formula is derived. Recently, I needed a formula to calculate the volume of a cylindrical fuel tank, which is laying on its side, by measuring the height of the fuel when sticking a ruler through the inlet. I found an algebraic formula on the Internet but cannot find how it was derived.
A=pi*a^2/2-a^2*arcsin (1-h/a)-(a-h)*sqrt(h(2a-h)).
Would you be able to show me the derivation? Or is there another equally good formula with an easier derivation? I realize this would probably be lengthy, so if this is not practical, do you know of a web site or book  that would give me the derivation?
Thanks,
Harold Snook

Related Articles:

Overview & Slopes of Curves

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
  • Overview & Slopes of Curves 0:21
    • Differential and Integral
    • Fundamental Theorem of Calculus
    • Differentiation or Taking the Derivative
    • What Does the Derivative Mean and How do We Find it?
    • Example: f'(x)
    • Example: f(x) = sin (x)
    • General Procedure for Finding the Derivative of f(x)

Transcription: Overview & Slopes of Curves

Hello, welcome to www.educator.com, welcome to the first lesson of AP Calculus AB.0000

I thought what I would do is take about 5 or 10 minutes to give a nice overview of the course as a whole,0007

so that we have a sense of where we are going.0012

Then, we are going to launch right into the calculus proper.0014

Let us get started and welcome.0016

Calculus is basically going to be about two things, two processes.0025

We are going to spend the first half learning how to do something called differentiating a function.0030

We are going to spend the second half learning how to do something called integrating a function.0033

That is essentially it.0037

Each one of those tools is going to yield different applications.0039

That is all we are going to be doing.0043

Let me work in blue here.0046

Given a function f(x), we will be spending our entire time doing two things to this function.0051

The first thing is we are going to be differentiating it.0087

It also called taking the derivative.0099

Both of those terms, we will use interchangeably.0106

The second thing we are going to be doing to this function,0110

it is just going to occupy the second half of calculus, is we are going to be integrating it.0114

These are the two major rivers of calculus.0123

These are the two great rivers of calculus.0128

The differential and the integral, two basic set of tools that allow us to solve certain types of problems with the differential calculus,0139

certain other types of problems with the integral calculus.0154

Essentially, what is going on is this.0158

We have some function f(x).0160

One of the things that we can do to it is differentiate it.0163

When we differentiate a function, we are going to get another type of function.0166

We are going to symbolize that with f’(x).0171

The other thing that we can do to it is we can integrate this function.0175

And then, when we integrate a function, we are going to get yet another type of function0179

which we will symbolize normally with F(X).0183

This is a new function and we call this function the derivatives.0187

That is called the derivative because it is derived from the original function.0195

This one is also a new function.0199

We call this one the integral.0206

That is essentially what calculus comes down to.0209

It is finding ways to find derivatives of functions, applying it in certain cases, and then finding integrals of functions.0212

Applying the integral in certain cases.0218

Let us move on to the next page here.0228

These two tools namely differentiation and integration are very powerful.0229

It will allow us to solve the most extraordinary problems, that is what beautiful about the calculus.0256

So far in your mathematical studies from elementary school, all the way through high school,0274

there has been a pretty steady increase, in terms of the level of difficulty, the techniques that you develop.0278

Some or more problems that you can solve.0284

That is essentially what it is, it is just, you are making the class of problems that you can solve bigger and bigger.0285

Now the calculus, it is a huge jump.0291

It is not a little jump, it is not a stair step.0294

You are going to be introduced to these tools, differentiation and integration.0296

They are very powerful tools.0300

It is going to take you up here, in terms of mathematical sophistication and the types of problems that you can solve.0301

It is really quite extraordinary.0308

Everything that we enjoy in the modern world, I do mean everything,0311

It is worth writing down.0331

I do mean everything.0332

Let us capitalize this one.0339

I do mean everything is made possible with calculus.0342

It really is the single tool, these two things, differentiation and integration.0358

They have allowed us to absolutely enjoy everything that we enjoy, computers, cell phones, cars, you name it.0364

If there is anything in the modern world that you take for granted, that you enjoy, it is because of the calculus.0372

Once again, we are going to spend the first half learning and applying differential calculus.0380

And then, we are going to spend the second half, roughly, learning and applying the integral calculus.0384

It is going to turn out that these two independent techniques, the differentiation and the integration,0391

are in fact so deeply related, that we call this relationship the fundamental theorem of calculus.0420

You will often see it as FTC.0461

Differentiation and integration, they are reasonably independent techniques.0464

There is no reason to believe that one is actually related to the other.0469

They solve different set of problems.0472

Yet, there is a very deep relationship that exists between them.0474

This relationship is what we call the fundamental theorem of calculus.0477

What is really interesting about the fundamental theorem of calculus,0481

something that many of you who go on to higher mathematics will discover,0483

is that this relationship that exists between differentiation integration is actually true, not just in one dimension.0487

In other words the real line, which is what we have essentially been dealing with ever since elementary mathematics.0495

We stayed on with the real numbers.0499

It is true in any number of dimensions, dimensions 2, 3, 4, 5, 6, 15, 34, or 147.0502

It is true in any number of dimensions.0508

It is a very profound relationship.0509

We will actually be getting to that, when we are getting to integration.0517

We will study the fundamental theorem of calculus.0521

Now the nice thing is, I can already tell you now what this relationship is.0523

You have an idea of where is that we are going.0526

I can tell you now what this relationship is.0532

Later, we will explore this relationship.0535

I can tell you now what this relationship is.0540

Again, there is nothing here, this is just overview.0547

Just giving you an idea of what it is that we are in for, so that you are not going into this blindly.0549

So you just have some sense of why we are doing what we are doing.0554

I can tell you now what this relationship is.0557

The relationship is, each of these processes namely differentiation and integration,0564

each of these processes, dif and int is the inverse of the other.0573

In other words, if I start with some function f(x), we said that I can differentiate it.0592

You know what I’m going to do actually here, I will go ahead and write dif down here.0619

I said that I could differentiate it to come up with a new function, the derivative.0627

If I want to go back, I integrate this function.0632

If I integrate, it will take me back to my original function.0638

Or if I integrate, it will give me some other function which we call the integral.0642

If I want to go back to the original, I just differentiate.0648

That is the relationship between the two.0653

Two entirely independent techniques, for the most part, because they solve entirely different set of problems.0654

Yet, the relationship is one is the inverse of the other.0661

The same way that if you will take a number and if you take the logarithm of that number, you end up with some other number.0663

If you want to go back, you exponentiate this thing and it takes you back.0670

The logarithm and the exponential are inverse processes.0675

The sin and the inverse sin are inverse processes.0681

Cos, inverse cos, are inverse processes.0685

Differentiation or integration are inverse processes.0688

This is extraordinarily deep, extraordinarily beautiful.0691

Again, there is no reason to the world to believe that they are connected and yet they are.0694

Let us write that last part.0701

There is no obvious reason why they should be connected, at least cosmetically.0704

There is no reason to just sort of look and say that this is related to that.0711

There is no obvious reason why differentiation and integration should be related, but they are.0715

This relation has consequences that go further and deeper than you can imagine.0746

One of the beautiful things about mathematics is that, you will have different people0779

or perhaps the same person will investigate different areas of mathematics, to solve a certain type of problem.0783

When you take a look at this set of mathematics that you develop for this problem0792

and this set of mathematics that you develop for this problem,0796

when you realize that there is actually a connection between those two mathematics, they come together, you unify that.0799

That relationship that exists between the various areas of mathematics takes you to a deeper level,0806

a deeper understanding of reality, a deeper understanding of how the physical world works.0813

This is what we strive for, we strive for unification.0819

This is what makes it beautiful, things that should not be related, at least, as far as our intuition is concerned,0823

they end up being not only related but very deeply related.0829

The consequences of those relationships are profound.0833

Anyway, this is really beautiful stuff and it begins right here, with your first course in calculus.0838

Once again, welcome, and let us get started.0845

We are going to spend the first half talking about differentiation.0856

We are going to put integration on the shelf, for the time being.0859

We will come back to it later.0861

We are going to begin with taking derivatives, differentiation.0861

We begin with differentiation.0865

Actually, let me write something here.0880

Differentiation or also called taking the derivative.0883

I will just write it over here, taking the derivative.0890

In other words, starting with some f(x), performing the differential operation on it,0896

and ending up with a new f’(x), a new functions that is going to give us other information.0904

Either by the situation, or it is going to give us information about the original function, whatever it is.0910

But it is a new function that we have derived.0915

Two questions, the obvious questions.0920

Two questions, what does the derivative mean and how do we find it?0926

Given this, how do we find the derivative and once we have the derivative, what does the derivative mean?0933

What does it give us, what does it tell us, what problems does it solve?0939

Two questions, what does the derivative mean?0943

Two, how do we find it?0958

How do we find it given some f(x)?0961

How do we derive and get?0967

The answers are as follows.0971

The answers, the answer to number 1, what does a derivative mean?0974

The derivative is the slope of a function curve.0978

A function is just some curve that you draw on the xy plane at a given point on the curve.0999

We will explain what that means more, in just a second.1011

That is pretty much it.1015

A derivative is a slope of a curve at a particular point.1017

The strange thing is you have been doing derivatives for many years now.1024

If you have the function y = 3x + 4, you know that is the equation of the line.1027

The curve itself is a straight line but we call it a curve.1037

In general, it is a line in space.1040

What is the slope of this line, it is 3.1043

The function is 3x + 4, what is the derivative of that function?1046

The derivative is 3 because no matter where you are on that line, the slope is 3.1049

That is what you have been doing, you have been finding derivatives.1056

Now in calculus, we are not just going to find the slopes of straight lines.1058

We are going to find the slopes of curves.1062

What is that slope, the slope there, slope there.1066

The slope is going to change as you move along the curve, that all we are saying.1069

We are just giving you a fancy name and calling it the derivative, that is all.1074

Number 2, let us go back to blue here.1080

The answer to the question, how do you find it?1083

Here is how you find, f’(x), in order to find f’(x), here is what you do.1087

Limit as h goes to 0, you form f(x) + h, given whatever f is, you subtract from it the original f(x).1094

You divide it by h and then you subject it to this process called taking the limit as h goes to 0.1106

We will be discussing what this means, how does one do this.1114

It is actually quite simple, just algebraically tedious.1119

How do we find it? We find it like this.1124

What does it mean, it means it is a slope.1126

We are going to start with number 1.1131

As far as number 2 is concerned, the how, I’m going to leave that for a future lesson.1133

For the next couple of lessons, I’m going to be talking about what the derivative is, slope of curves,1139

getting ourselves comfortable with the idea of a slope of a curve, as opposed to just the slope of a straight line.1145

And then, once we have a reasonably good sense, once we feel comfortable with that,1150

we will talk about how to find this so called f’(x), the derivative.1154

Let us go over here, I will stick with blue.1163

Let us start with number 1, in other words, the meaning, what does it mean?1170

The derivative f’(x) is a function which gives us the slope of the curve or graph.1194

You know what, maybe I will just call it a graph, which gives us the slope of the graph of the original function at various values of x.1232

Here is what this means.1260

Let us take a look at let us say the sine functions.1262

Our function f(x), original function is sin(x), also y = sin(x).1271

We are going to be working in the xy coordinate system.1280

We are never going to be moving out of that.1282

Whenever you see f(x), you can just replace it with y, it is the same thing.1284

We know what the sine function looks like, it looks like this.1289

This is 0, this is going to be our π, this is going to be 2 π.1302

Over here at π/2, it is going to hit a value of 1.1306

Over here at 3π/ 2, it is going to hit a value of -1, standard sine function like that.1310

The slope of the curve is, basically, what you are doing is you are finding the line that touches the curve at a given point.1322

What we call the tangent line.1335

The slope of that line is the slope of the curve, at that particular value of x.1338

At π/2, think of it as just some tangent line that is following the curve along.1343

You have that slope.1351

At this point over here, the slope is that.1353

At this point over here, the slope is that.1356

At this point over here, the slope is that.1359

This point over here, the slope is that.1362

Here, the slope is that.1365

You can see that the slope changes depending on where you are.1366

That is what f’(x) gives.1372

F’(x), you have the original function, you do something to it, which we will talk about later, how to find the derivative of it.1374

When you find the derivative, it is going to be another function which we symbolize with f’(x).1382

The different values of x, it gives us some number, when you actually solve that function.1386

That number is the slope of the line that touches the graph, at that particular xy value.1393

As we can see for various values of x, the slope at that point, the slope at the point xy,1401

if this is x, this is the point xy, it is different.1439

We can see it geometrically, if that is the case.1447

It is a straight line, for tangent line it is just going to be touching it at a certain point, the slope is going to change.1449

Once again, given f(x), we differentiate it and it gives us f’(x).1457

This thing, this tells us what the numerical value of the slope is.1465

The derivative itself is a function of x, we do not know.1488

We have to put in different values of x, to see what the slope is.1491

It tells us what the numerical value of the slope is for different values of x.1494

Let us go over here.1522

Let me draw a little bit of a curve, I will draw it like this.1524

I will take a point on that curve.1529

Again, we are looking for the slope of the line that just touches that curve at one point.1531

This is our f(x).1539

This is called our tangent line.1544

This point here is going to be xy, or x, if you prefer f(x), however you want to list it.1552

Once again, the line that touches a curve at a single point is called the tangent line.1564

It is kind of redundant, the tangent line of the curve at that point, I just said that.1606

We will just say, it is called the tangent line.1611

I think it is perfectly clear what we are talking about.1613

It is called the tangent line.1616

The slope of the tangent, it is the specific numerical value of the derivative of the function at that point.1619

It is the derivative at that point.1644

Again, the point itself is a point xy in the plane.1666

When we find f’(x), let us say we have some f(x), we find f’(x), that is the derivative, it is the x value of the point.1674

It is the x value of the point that we put into f’(x).1697

Notice, it is a function of x not a function of y.1706

If we want to find the particular numerical value of the derivative, if we have f’(x), it is a function of x.1709

We are going to put the x value in there and it is going to spit out some number.1716

That number is the slope of that line.1720

It is the x value that you put in.1723

When we find f’(x), it is the x value of the point that we put into f(x), in order to get a numerical value for the slope.1724

Let us see what we have got, let us do some examples here.1754

Some examples, I think what we will do is we will let f(x), let us go ahead and take the same function y = sin(x).1759

f(x) = sin(x) or y = sin(x).1772

I’m going to go ahead and tell you what the derivative is here.1777

Again, we will talk about how we got this later on.1780

It will turn out that f’(x) actually equals cos(x).1783

The original function is sin(x), its derivative is going to turn out to be cos(x).1798

When I put in different values of x into cos(x),1804

that will give you the numerical value of the slope of the tangent line touching the sine curve, at that point, like this.1807

I think I will work in red for this one, that will be nice, a little change of pace here.1818

Of course, we have that there.1828

Let us go ahead and draw our sine curve again.1830

We said we have 0, we have π, we have 2 π.1834

This is 1 and this is -1.1838

Now let us take the point 0.1841

When x = 0, let us find the y value.1845

We know y is 0 but let us actually find it.1849

f(0) is equal to sin(0) which = 0.1856

Yes, our point is going to be 0,0.1863

Again, sometimes we do not have a graph to work with, which is the reason I went through it analytically here.1866

It is very clear that this is going to be 0,0.1872

It is going to be very clear that this point up here is going to be π/2, 1, 3π/ 2, -1, π 0, 2π 0.1876

We can see it graphically but we would not always have a graph.1884

f’(0) of f’ is cos(x), that is the cos(0).1889

What is the cos(0)?1897

The cos(0) = 1.1898

What that means is that the slope of the tangent line through 0,0 has a slope of 1.1901

What is that tangent line?1926

Let me extend this out a little further so it goes down that way.1930

My tangent line is the line that touches the graph at that point, that is my tangent line.1936

The slope of that line is given 1, because the derivative of sin x is cos x.1944

Let us find what the slope is at π/2.1952

When x is equal to π/2, the f value, the y value, the f(π/2) which is equal to sin(π/2), that is equal to 1.1959

Therefore, I know that the point that I’m talking about is π/2 and 1.1972

I think I will do a little purple.1979

I know my tangent line is there, it is the line that touches the graph at that point.1985

That is the tangent line.1991

What is the slope of that tangent?1992

Graphically, just by looking at it geometrically, I know that the slope is 0.1994

We would not always have a graph, let us do it analytically.1999

Analytically, I know that the derivative is cos(x).2002

f’ at π/2, remember, we put in the x value, is equal to cos(π/2) that is equal to 0.2008

As you can see geometrically, analytically here, the slope of the tangent line through π/2, one has a slope of 0.2021

Let us do one more.2050

Let us take the point 7π/ 6, not quite so easy this time.2054

We will take x = 7π/ 6.2059

f(7π/ 6), f is sin, sin (7π/ 6), we are just trying to find the y value first, to find out where the point is.2064

It is going to be -1/2.2077

Our point is 7π/ 6 is our x value, -1/2 is our y value.2086

You are looking at it on the graph.2096

7π/ 6 is somewhere like right over here.2097

This point right here, that point is our point 7π/ 6, -1/2.2103

If this is going to be 1, that is probably going to be that way.2115

I have not drawn it that great but you get the idea.2119

Let us try this again, shall we, all this crazy writing.2127

This point over here, on the graph it is 7π/ 6 and -1/2, that is the coordinate of it.2132

What about f’, the derivative?2145

f’(7π/ 6) = cos(7π/ 6) because the derivative of sin x is cos x.2148

The cos(7π/ 6) is –√3/2.2160

The slope of the tangent line, that tangent line,2175

the slope of the tangent line through the point 7π/ 6, 1/2 is –√3/2.2181

Geometrically, we can see that it is going to have to be a negative slope because it is going from top left to bottom right.2202

Numerically, analytically, we have to use the formula for the derivative, to find its actual numerical value.2207

The slope is a derivative.2215

When we say find the derivative of a function, we are saying do whatever you need to do to find the derivative of the function,2217

which is going to be another function of x.2223

And then, put in the x value of whatever point on the curve you want, that will give you the slope of the tangent line.2225

That is the derivative.2232

The derivative of 7π/ 6 – 1/2 of sin x is equal to -√3/2.2234

The derivative of the function is cos(x).2241

The derivative of the function, the numerical value.2245

Let us stick with red here.2255

It would be very nice to have a general procedure for finding the derivative of f(x).2258

We have a general procedure.2291

That general procedure says, let me write it a little bit more clearly here.2293

And then later on, we will be a little bit more messy.2302

The limit as h approaches 0 of f(x) + h - f(x)/ h, this is our general procedure.2306

It is a procedure that we are going to address in a later lesson, not right now.2316

I’m going to save the procedure for how to find the how, I'm going to save for another lesson.2322

For right now, I want to concentrate on the y.2327

What does it mean, we want to get a feeling for this.2331

I will start discussing this procedure in a future lesson.2339

The first thing we are going to do is, when we do this, first, we will discuss what this part means, what that means.2361

The second thing we will do, then, we address the whole thing.2382

If you saying to yourself, why does he keep writing this thing over and over again?2397

There is a reason for it, there is a pedagogical reason for it.2400

This is a very important thing.2403

I’m writing it over and over again so that by the time you actually do see it, it will be a sort of like you have seen it before.2406

That is the reason I'm doing it.2412

It is not because I’m obsessive compulsive, over h.2414

When we actually discuss this in a future lesson, the how, I’m going to discuss what limits are first, how to find limits.2419

And then, we will go ahead and address how to take the limit of this particular quotient, which will give us the derivative.2426

For the next few lessons, we will continue with slopes of curves and what derivatives mean.2436

Once again, we want to become familiar with this idea of the slope of a curve.2475

We want to be able to handle a few things, in basic brute force way.2481

We want to know what is going on, how this idea of the slope,2487

how we are going to relate it to what we have done with slope before.2492

We want to get comfortable with it, before we start actually introducing calculus ideas.2495

That is what is going to occupy us, for the next probably three lessons.2500

We are going to spend a couple of lessons discussing what these things mean and2505

we are going to do a lesson on some example problems.2508

We will begin by discussing this idea of a limit of a function.2511

What does this limit as h approaches 0 mean.2516

With that, I will go ahead and stop this first lesson there.2519

Thank you again for joining us, I hope this turns out to be a wonderful experience for you.2522

Thank you and see you next time, take care.2527

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