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Derivatives of the Trigonometric Functions

Slide Duration:

Section 1: 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
Section 2: 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
Section 3: 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
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
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
Section 4: 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
Section 5: 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
Section 6: 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
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
Section 7: 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
Section 8: AP Practic Exam
AP Practice Exam: Section 1, Part A No Calculator

45m 29s

Intro
0:00
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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• Transcription

 1 answerLast reply by: Professor HovasapianWed Jun 28, 2017 9:26 PMPost by Peter Fraser on June 28, 2017Example X:I got the same result by doing: y = x.sin(1/x); so y' = 1.cos(1/x); lim x --> inf so y' = 1.cos(0) = 1.1 = 1.  Hope that's right :) 1 answerLast reply by: Professor HovasapianFri Mar 25, 2016 11:11 PMPost by Avijit Singh on March 6, 2016Hi Prof Raffi, I was wondering if it is necessary to simplify (with the trig identities) once we have differentiated y. For example, in example 5 you were able to simplify quite a lot. Will I lose marks on the AP exam if i was to leave it un-simplified? Thanks. 1 answerLast reply by: Professor HovasapianWed Jan 13, 2016 12:01 AMPost by Sohan Mugi on January 10, 2016Hello Professor Hovasapian. For the ladder problem(Example viii), wouldn't the derivative equal -12sin(theta)? Because the derivative of cos=-sin, so it would be -12sin(theta) right? Thank you.

Derivatives of the Trigonometric Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Derivatives of the Trigonometric Functions 0:09
• Let's Find the Derivative of f(x) = sin x
• Important Limits to Know
• d/dx (sin x)
• d/dx (cos x)
• d/dx (tan x)
• d/dx (csc x)
• d/dx (sec x)
• d/dx (cot x)
• 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

Transcription: Derivatives of the Trigonometric Functions

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

Today, we are going to be talking about the derivatives of the trigonometric functions.0004

Let us jump right on in.0008

Let us start off by finding the derivative of sin x.0010

Let me see, I will stick with blue today.0016

Let us find the derivative of f(x) = the sin(x).0024

We have the derivative definition.0040

It is going to be the limit, as h approaches 0 of f(x + h - f(x)/ h).0045

We are going to form quotient and I take the limit as h goes to 0.0055

I’m just going to put little quotation marks here for the limit h approaches 0.0061

I do not want to keep writing it, I hope you will forgive.0064

Our function is sin(x).0068

We have sin(x) + h – sin(x)/ h.0070

And then, when we expand this sin(x) + h, you remember the addition formulas0080

or you remember having seen the addition formulas, it becomes sin x cos h + cos x sin h + sin x, we have all of that /h.0086

What I’m going to do is I’m going to separate these out.0104

I’m going to combine a couple of these.0108

Let me go ahead and put a.0111

This is the limit as h approaches 0 of, I’m making myself a little more room here.0112

We are going to have sin x cos h – sin x/ h + this second part which is cos x sin h/ h.0120

This is going to be, this is going to equal that.0144

This is going to equal the limit as h approaches 0.0147

Let me see, I’m going to take out the sin x.0152

The sin x × this is going to be cos h -1/ h.0156

I’m keeping the x and h together.0166

That is all I’m doing here.0168

+ cos x × sin h/ h.0170

That is it, I just put things together so that I can actually do it.0177

Let me go ahead and write it here.0183

In this case, this is the limit as h approaches 0.0184

This × this + this × this.0188

It is going to be the limit of this × the limit of this + the limit of this × the limit of this.0191

Limit theorems, let me just go ahead and take them one at a time.0198

This actually equal the limit as h approaches 0 of sin x × the limit as h approaches 0 of cos h -1/ h0202

+ the limit as h approaches 0 of cos x × the limit as h approaches 0 of sin h/ h.0220

Here is what happens.0234

As h goes to 0, notice there is no h in here.0235

The limit as h approaches 0 of sin x is just sin x.0238

This is just sin x.0244

This limit right here, let me do this in red.0248

These two limits, this limit and this limit are very important limits.0253

I’m not going to demonstrate how they end up being what they equal but these are two limits that you definitely have to know.0258

This one is going to be 0.0266

It is sin x × 0 + the limit as h goes to 0 of cos x is just cos x.0268

This limit right here, the limit of the sin of h/ h as h goes to 0 is actually equal to 1.0275

Let us see, therefore, the derivative of sin x = cos x, this term goes to 0.0285

Once again, these two limits are very important to know.0299

You just have to know them, very important to know.0309

First one, the limit as x approaches 0 of the sin(x)/ x that is equal to 1.0320

The other limit, the limit as x approaches 0 of cos x -/ x that is equal to 0.0332

We will be using these limits to evaluate other limits, until the time that we learn something called L’Hospitals rule.0348

And then, we would not have to worry about these limits anymore.0355

But again, they are very important limits, they do tend to come up.0357

As far as the formulas are concerned, let us go back to black here.0361

The derivative with respect to x of the sin(x) = the cos of x.0371

Going with the same process of the different quotient, taking the limit.0378

Again, we will not go through the pool for each one, we are just going to list them because again,0381

what we want to do is we just want to develop technique.0385

We want to be able to differentiate quickly.0390

In this part of the course, this is all we are doing, finding ways to differentiate all this different functions.0392

The derivative with respect to x of cos x is equal to -sin x.0399

Make sure you watch your sign.0406

Derivative of sin is cos, derivative of cos is –sin.0407

The derivative of the tangent function, the tan of x = sec² x.0414

The derivative of cosec of x = -cosec x cot x.0424

The derivative with respect to x of sec x is equal to sec x tan x.0436

The derivative of the cotangent function, cot x is equal to –cosec² x.0448

These are our 6 trigonometric functions and their derivatives.0456

The rest is now just examples.0460

Let us jump right onto the examples and see what we can do.0464

Again, really straightforward, nothing particularly complicated.0467

Just a question of memorizing via derivative formulas and doing a bunch of problems.0471

Differentiate f(x) = x² – cos(x), very simple.0478

F’(x), if you want you can use the dy dx symbolism as well, it does not really matter.0484

You are going to see both.0496

The derivative of this is 2x 4 cos x, it is going to be -, this – stays and this is going to be -4 sin x.0500

The 4 is just a constant.0513

Let us go ahead and take the constant to be consistent.0516

-4 × the derivative of cos x which is –sin x.0520

Our f’(x) is equal to 2x, - and -, +4 sin x.0528

There you go, nice and straightforward.0536

You just have to use the differentiation formulas.0538

All we are doing is practice, practice, practice.0541

Example 2, differentiate f(x) = x⁵ tan x.0546

Here we have a product rule.0551

This is for our first function, this is our second function.0553

We have product rule, in this case one of them is a monomial and the other one is a trigonometric function.0556

It is going to be the derivative of this × that + the derivative of this × that.0563

F’(x) is equal to the derivative of this is 5x⁴ × this × the tan(x) + the derivative of this × that + sec² x × x⁵.0569

We just want to go ahead and bring this x⁵ in front.0593

You do not have to, you can leave it like this, it is not a problem.0596

Traditionally, we tend to bring everything forward and leave the trigonometric function at the end.0599

We have 5x⁴ tan(x) + x⁵ sec² x.0604

That is it, nice and simple.0614

Do you want to simplify it more, do you want to express it in terms of sin and cos, you can if you want to.0618

Simplification is really going to be, it is a totally individual thing how far you want to simplify.0623

You remember from product rule, quotient rule, in the previous lessons.0629

Where do you stop, at some point, you just have to stop.0634

Ultimately, it does not really matter where you stop.0638

You want to simplify as much as reasonable but it is going to be subjective.0640

What is reasonable to you might not be reasonable to somebody else.0645

Ultimately, you just have to ask what is reasonable to your teacher.0648

That takes care of that one.0653

Differentiate x(x) = cos x/3 + sin x.0657

Now we have a quotient rule and some trigonometric functions.0661

You remember the quotient rule, if you have f/g and if you take the derivative, you are going to have g f’ – f g’/ g².0665

This is f and this is g, f is cos x, 3 + sin x is g.0680

Therefore, we have f’(x) is equal to this × the derivative of that.0685

We have 3 + sin x × the derivative of cos x which is –sin x, - this cos x × the derivative of this.0691

The derivative of this is 0 +, this is just cos x/ 3 + sin x².0705

Let us go ahead and distribute this out.0719

Let us see.0724

You know what, I think I made a mistake on my paper so I will do it here, just as is.0732

Sin x cancels, this is going to be -3 sin x – sin² x – cos² x/ 3 + sin x².0736

Let me see, I have got -3 sin x.0762

I’m going to write this part -.0767

I’m going to write this as sin² x + cos² x.0772

The – distributes giving me the two negatives, /3 + sin x².0778

This right here is the identity.0790

This is just 1.0793

Our final answer -3 sin x -1/ 3 + sin x².0797

Let us go ahead and factor out that negative.0814

Let us write it as -3 sin x + 1.0819

No, that is not going to simplify anything.0830

I thought maybe that something might cancel but it is not.0832

You know what, we can stop there, not a problem.0835

There you go, quotient rule, trigonometric functions.0839

Reasonable, very straightforward.0843

Now we have differentiate e ⁺x tan x – sec x.0848

For these two, we have product rule.0855

We have e ⁺x and tan x – sec x, we just differentiate straight.0857

F’(x) = this × the derivative of that = e ⁺x × sec² x + the derivative of this × that.0862

I will leave it in the front.0877

It is the derivative of this × that + the derivative of,0883

Wait a minute, what am I doing here, let us start again.0889

The derivative of this × that.0895

We have e ⁺x × tan x, there we go, + the derivative of this × that +, I will leave this in front, e ⁺x sec² x.0896

And then, -the derivative of the sec x which is sec x tan x.0912

That is fine, you can go ahead and leave it like that.0923

You have a tan x tan x, sec² x sec² x, e ⁺x.0926

You might as well leave it like that, I do not think it will make it any easier.0931

That was nice and simple, good.0935

Again, we have another quotient rule.0940

Let us see what we can do with this one.0943

F'(x) is equal to this × the denominator × the derivative of the numerator.0947

We have cot(x) × the derivative of this which is going to be -cosec x cot x – this0954

which is cosec x – 4 × the derivative of the cot x which is –cosec² x/ the cot² (x).0971

If you want you can write this cot(x)², not a problem.0990

It is just we tend to put these squared on the cot.0996

Whatever is easier for you, you will see in a little bit that I actually do write that sometimes.0999

In the next lesson, when we talk about something called the chain rule, it is often easier.1006

At least in the beginning, when you are getting used to it.1011

To write something like the cos² (x), it is actually easier to write it as cos(x)².1013

It tends to make the differentiation a little bit easier.1021

Again, as you sort of get accustomed to the idea of seeing the squared here.1023

Let us see what we can do with this.1029

Let us go ahead and multiply this out, I guess the numerator.1030

We have –cosec x cot² x.1037

This is – × -, we have + cosec³ x.1046

This is – × - × -, it stays -4 cosec² x/ cot² x.1056

cosec and cosec, I’m going to pull out a cosec.1072

-cosec x × cot² x – cosec² x.1074

The – and – gives me the +.1098

And then, this is going to be a -, so it is going to be a +.1101

-cosec, that is going to be -, it is going to be +.1110

This is -, that is -, this is going to be +4 cosec² x.1114

It is always not that easy to keep track of all these signs and things.1122

By all means, if I make a mistake, it is definitely going to happen.1125

That is the nature of the game.1129

You have all these symbols floating around.1131

Mistakes are going to happen.1133

/cot² x.1135

Let us see, I think I got this right.1141

Let me double check.1143

-cosec, I pulled out a negative.1144

This becomes -, - × - is + goes to +.1147

This is -, so -.1151

Yes, I think we are good.1152

This becomes, I’m going to write this –cosec as -1/ sin(x).1155

Here, cot² – cosec², if you remember from your trigonometric identities, it is equal to 1.1162

This is going to be 1 + 4 cosec² x.1169

The cot² x, I’m going to write as cos² x/ sin² x.1177

When I do that, I’m going to go ahead and cancel this sign and one of these signs.1186

Let us see, that is going to give me.1193

Now I have f’(x) is equal to,1201

Let me write that part over, actually.1209

-1/ sin(x) × 1 + 4 cosec(x).1211

I think that was what is on top.1220

Here we have cos² x/ sin² x.1222

We cancel the sin and we can flip this.1226

This sin ends up coming to the top as sin x, the – stays.1230

Here we have 1 + 4 cosec x.1237

Over here, we have cos² x stays on the bottom.1243

Then, let me write this as sin(x) × 1 + 4/ cosec is sin(x)/ cos² (x).1254

Again, you do not really need to simplify all this much.1275

But to solve the identities, let us take it as far as we can take it.1277

This is going to be –sin(x), when I distribute this.1284

This, the sin x are going to cancel so it is going to be -4/ cos² (x).1289

I think that should take care of it.1298

I think that is as far as we want to take that one.1303

There you go.1307

Find an equation of the line tangent to y = x² sin² x at x = π/6.1310

We are looking for an equation of the tangent line.1318

We are definitely going to be using y – y1 = m × x – x1.1321

We have x, that is π/6, it is going to be our x1.1331

Let us go ahead and find the y1 first.1335

Y(π/6) is going to equal what we just put in there.1339

It is going to be π/6² × sin(π/6).1346

π/6² is π²/36 × sin(π/6), sin(30 degrees) is 0.5 or ½.1352

We end up with π²/72.1363

We know our point.1370

Now, π/6 and π²/72, that is tan.1372

Tan which means find the derivative.1383

Y’, this is a product rule.1386

It is going to be the derivative of this × this.1391

We get 2x × sin(x) + the derivative of this × this + cos(x) × x².1393

Y’ at π/6, we are trying to find the slope.1406

It is going to equal to × π/6 × sin(π/6) + cos(π/6) × π/6².1414

What do we have here?1433

Let us just go ahead and write it all out, it is not a problem.1436

This is going to be, let me do it over here.1440

2 × π/6, sin(π/6) is ½, cos(π/6) is √3/2, × π²/ 36.1443

Here we have, the 2 and 2 cancels.1458

We are left with π/6 + π² √3/72.1460

This is going to be our slope.1473

That is our slope.1477

Now we put it into the equation.1481

We get y – y1 which is π²/72 = the slope which is π/6 + π² √3/72 × x – π/6.1484

There you go, that is your equation.1505

Again, depending on the extent to which your teacher wants you to simplify that, you can go ahead and do so.1509

Nice, nothing particularly strange.1515

For what values of x does the graph of the function x + 3 cos(x) have a horizontal tangent?1520

Horizontal tangent means that the derivative is 0.1527

Horizontal tangent that implies that f’(x) is equal to 0.1536

Let us go ahead and take the derivative of this.1544

If f(x) = x + 3 × cos(x), that means that f’(x) is going to equal 1.1548

The derivative of cos(x) is – sin x, we have -3 sin x.1560

We are going to set that equal to 0.1566

1 – 3 sin x = 0, that means that 3 sin(x) is equal to 1.1569

That means sin(x) is equal to 1/3.1577

Therefore, x is equal to the inv sin(1/3).1581

This is a positive number.1588

The sin is positive in two quadrants.1592

We are going to get two answers.1594

We are going to get an angle of this quadrant.1595

We are going to get an angle of this quadrant.1597

The angle that we are taking, the first one is going to be this one.1599

The second one is going to be that one.1602

When I do this, when I put this into the calculator, I get x = in radians, 0.3398 radians.1607

That is this first angle.1619

The other angle, that is going to be 0.3398, that is going to be this angle over here but we do not measure it from here.1623

We take all of our measurements from the +x axis.1630

We are going to take π which is 180 - 0.3398.1632

π - 0.3398 rad.1638

There you go, once again, we draw this out.1650

This angle is x1, the 0.3398 it happens to be 19.47°.1653

The other one s 19.47°, in the second quadrant but we measured it from here.1666

This is x2, this is the π - 0.3398.1675

It happens to equal what is going to be 180 - 19.47.1680

We have got 160.52°.1684

There you, these are our two answers.1689

Of course, every 2π after that, it comes around again.1693

It is a periodic function.1699

Let us see, word problem, nice.1704

A ladder 12ft long rest against the wall, if the bottom of the ladder slides from the wall,1710

how fast does the height of the ladder against the wall change with respect to the angle that the ladder makes with the wall,1715

as the ladder is being pulled away at the base?1723

Let us go ahead and draw a picture here.1726

We have a wall here and we have our ladder.1729

Let me just go ahead.1736

We have our ladder that way.1740

They want to know, this ladder is 12ft long.1742

The bottom of the ladder slides away.1747

Now, this ladder is going to be moving this way which means that the top of the ladder is going to be moving that way.1748

How fast does the height of the ladder against the wall change?1755

How fast does this height changing?1758

Let us call this height y, let us call this x.1761

With respect to the angle that the ladder makes with the wall.1766

The angle that the ladder makes with the wall, let us call it θ.1769

They want to know how fast does the height of the ladder against the wall change, with respect to the angle that the ladder makes.1776

They want to know dy dθ.1782

They want to know how fast this changes, y with respect to the change in the angle θ.1788

That is dy dθ, that is it.1797

What that means is we need to find an equation y = some function of θ and we need to differentiate it, y’.1802

Y’ is dy dθ.1813

If this is what you are looking for, you need to find an equation of this equal to some function of this.1819

And then, you differentiate it and that will give you what this is.1825

You have a relationship here between y and θ and 12.1829

We have that the cos(θ) is equal to 1/12 which means that y = 12 × cos(θ).1834

Y’ which is equal to dy dθ is equal to -12 × cos(θ).1850

We are done, we have what we wanted.1860

The rate of change of y is equal to -12 × cos of whatever θ happens to be at that moment.1865

If θ happens to be some number that means y is changing.1874

Notice, it is negative, it s getting smaller.1880

The ladder is falling, it is going this way.1884

This negative sign confirms the fact that it is getting smaller.1886

If I were to do dx dθ, I would find that this is actually positive because it is getting bigger.1891

All of this is confirmed.1897

Once again, we are looking for dy dθ.1899

How fast does y change, when I change θ?1902

That means I have to find an equation of this, in terms of this, y as a function of θ.1905

I found it, now I just differentiated it and I have what I want.1911

Nice and simple.1915

If they gave us a specific θ, you would put it in and solve.1916

For example, if they said π/3.1920

Cos(π/3) is ½ so -12 × ½, it is going to be -6.1924

The rate of change of y when θ actually hits π/3, it is going to be changing by 6ft/s, something like that.1932

There you go, very important set of problems, this whole idea of the rate of change.1941

That is what we are doing, the rate of change.1951

These word problems in calculus are going to be profoundly important.1952

Again, the difficult part is not the actual differentiation, that is the easiest part of the whole problem.1958

The difficulty with these word problems is going to be taking a look at this and constructing what is going on,1963

and extracting an equation among the variables that you are interested in.1968

Keeping track of what it is that you want.1974

Very important.1976

When they say how fast does the height of the ladder against the wall change with respect to the angle?1978

That is dh dθ or in this case I chose y as my height, dy dθ.1985

I need an equation y, in terms of θ, that is just automatic.1991

That is what is going on, this is what you want to extract from these word problems.1995

Let us take a look at example number 9.2003

Evaluate the limit as x approaches 0 of tan(7x)/x.2005

Again, now we are going to use those limits that we saw.2012

The limit of sin x/ x and limit of cos x – 1/x.2015

The limit as x goes to 0.2020

We are going to use them to evaluate these limits.2021

Let us see what we can do.2025

In this particular case, we have 7x/ x.2027

This is not tan(x), this not sin(x)/x.2029

We need to sort of fiddle with this.2034

Let us see what we can do.2037

Let us see if we can somehow transform this and turn it into something that looks like something we already know.2039

What I'm going to do, I’m going to work with tan(7x)/x.2047

I’m going to leave the limit off for the time being.2052

I will come back to it at the end, when I simplify it.2055

I’m going to multiply this by 7/7.2058

I’m just multiplying by 1.2060

I’m not actually changing anything.2061

This equals 7 × tan(7x)/ 7x.2064

This is nice, now I have the argument of the trigonometric function and the denominator the same.2072

It starts to look interesting.2077

I’m going to express tan(7x), this is 7.2079

The tan(7x) is equal to sin(7x)/ cos(7x).2088

I’m going to write this as 7 × sin(7x)/ 7x × cos(7x).2097

That is it, all I have done is change the tangent to this.2107

I’m going to take this and this, and then this, separately.2110

This is equal to 7 × sin(7x)/ 7x × 1/ cos(7x).2120

Now I’m going to take the limit of that.2133

The limit as x approaches 0 of this thing 7 × sin(7x)/ 7x × 1/ cos(7x).2136

The limit as x approaches 0 of 7 is just 7.2150

The limit as x approaches 0 of sin(7x)/7x that is where we use that limit that we have.2153

The limit as x approaches 0 of sin(a)/a is equal to 1.2160

Here, as x approaches 0, cos(7) × 0 is cos(0).2168

Cos(0) is 1, 1/1 is 1.2173

Our final answer is 7.2176

I have transformed this thing, tan(7x)/ x by just manipulation into something that I actually recognize, using what is able to find that.2178

This limit is easy, this limit is easy, I get 7.2191

I hope that make sense.2195

Evaluate the limit as x approaches infinity, sin(1)/ x.2200

Let us take this x × sin(1)/ x.2208

We have x × sin(1)/ x.2220

I’m going to multiply by 1, in the form of 1/x/ 1/x.2224

That is equal to 1/x × x × sin(1)/x / 1/x.2233

I will just multiply 1/x and 1/x down below.2248

This and this cancel.2253

I’m left with the sin(1)/x/ 1/x.2254

I have this thing.2261

Now we take the limit, but this time the limit is saying as x goes to infinity of this thing.2269

I do not like doing that.2278

Sin(1)/x/ 1/x, I like it to be vertical.2279

This and this = a.2286

As x goes to infinity, as this goes to infinity, this whole thing goes to 0.2293

A is going to 0.2301

This is the same as the limit as x goes to 0 of sin(a)/a.2304

We know what this limit is already, it is equal to 1.2317

That is it, mathematical manipulation.2320

The question that most kids ask is that how do you know how to do this?2326

You do not, there is no way of looking at a problem and knowing exactly what steps you are going to take.2329

You just try.2335

What you saw here is the finished product.2337

What you did not see was the 5 or 6 different things that we try to do,2341

manipulations that did not go anywhere, that we ended up hitting a wall.2346

That is what is going to happen.2350

What you want to do in calculus from this point forward in your math work and your science work,2352

you want to disabuse yourself of this notion that you are supposed to look at a problem2359

and just all of the sudden know exactly what to do.2364

There are too many steps at this point in higher math and higher science,2366

for you to actually be able to see the entire path in your head.2371

Sometimes you will but sometimes you just have to start.2374

Especially, when it involves some form of mathematical manipulation to make the problem a little bit more practical.2377

You just have to start and you see where you go.2382

Maybe you would have tried x/x, that did not do anything.2384

You just try and if that does not work, if you hit a wall, you go back and you start again.2388

In this case, it turned out to be 1/x and 1/x.2393

It fell out, it fell out, perfect.2396

Trust the mathematics.2400

You want to trust the mathematics and you want to trust your intuition.2404

But do not trust the fact that just because you do not see the entire path, that you know how to solve the problem.2408

You need to be able to work a couple of steps at a time because the solution might be so far ahead that you cannot actually see it.2415

The solution might be, there might be a curve, it might be along the curve.2422

You cannot see the curve ahead but if you take a couple of steps toward it, then you can see the curve ahead.2426

That is how this works.2431

Just do that, do not worry about it.2434

It is not like we look at these problems and automatically know what to do.2437

Most of the time, we do not, particularly with the things that we deal within our research or anything else.2441

We do not know, when we try things, most of the time we actually fail.2447

What you are seeing is the actual define of success.2452

It makes it look like this is really simple, let us just do this, no.2456

Anyway, I hope that helps.2461

Take good care and I will see you next time.2464

Thank you for joining us here at www.educator.com, bye.2466

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