*Finding the tangent line at a point P boils down to finding the slope of the tangent line at point P.

Transcription

1 The Derivative & Tangent Line Problem *Finding the tangent line at a point P boils down to finding the slope of the tangent line at point P. 1

2 The Derivative & Tangent Line Problem We can approximate using a secant line through the point of tangency and a second point on the curve. second point on the curve (c, f(c)) is the point of tangency Slope of secant line (Difference Quotient) 2

3 second point on the curve (c, f(c)) is the point of tangency 3

4 The Derivative & Tangent Line Problem **The slope of the tangent line to the graph of f at point (c, f(c)) is also called the slope of the graph of f at x=c. 4

5 The Derivative & Tangent Line Problem Example 1: Find the slope of the graph of when c=2 5

6 The Derivative & Tangent Line Problem Example 2: 6

7 The Derivative & Tangent Line Problem Example 2: 7

8 The Derivative & Tangent Line Problem The derivative is also used to find the instantaneous rate of change of one variable with respect to another. Differentiation: the process of finding the derivative of a function. A function is differentiable at x when its derivative exists at x and is differentiable on an open interval (a, b) 8

9 The Derivative & Tangent Line Problem Example 3: Find the derivative of definition of derivative. using the 9

10 The Derivative & Tangent Line Problem HW 2.1 (p. 103) #1 4, 5 13 odd, odd, 25, 27, odd 10

11 11

12 The Derivative & Tangent Line Problem Example 4: 12

13 The Derivative & Tangent Line Problem Example 5: 13

14 The Derivative & Tangent Line Problem Differentiability & Continuity The existence of the limit in this form requires that the one sided limits exist and are equal. The one sided limits are the derivatives from the right and left f is differentiable on the closed interval [a,b] when it is differentiable on (a, b) and when the derivative from the right at a and the left at b both exist. 14

15 The Derivative & Tangent Line Problem Example 6: Is the following function is continuous at x=2? Differentiable at x=2? 15

16 The Derivative & Tangent Line Problem Example 7: Is the following function is continuous at x=0? Differentiable at x=0? 16

17 The Derivative & Tangent Line Problem 17

18 Basic Differentiation Rules & Rates of Change The Constant Rule: The derivative of a constant is 0. If c is a real number, then Example 1: Find the derivative of the following functions: a) b) c) d), k is a constant 18

19 Basic Differentiation Rules & Rates of Change The Power Rule If n is a rational number, then the function differentiable and is For f to be differentiable at x=0, n must be a number such that is defined on an interval containing 0. *When n=1: 19

20 Basic Differentiation Rules & Rates of Change Example 2: Find the derivative of the following functions a) b) c) **Rewrite, Differentiate, Simplify** 20

21 Basic Differentiation Rules & Rates of Change Example 3: Find the slope of the graph of value of x. for each a) b) c) 21

22 Basic Differentiation Rules & Rates of Change Example 4: Find an equation of the tangent line to the graph of when 22

23 Basic Differentiation Rules & Rates of Change 23

24 Basic Differentiation Rules & Rates of Change Example 5: Find the derivative of the following functions 24

25 Basic Differentiation Rules & Rates of Change Constant Rule & Power Rule Combined 25

26 Basic Differentiation Rules & Rates of Change **Rewrite, Differentiate, Simplify** Example 6: Find the derivative of the following functions a) b) c) d) 26

27 Basic Differentiation Rules & Rates of Change The Sum & Difference Rules Example 7: Find the derivative of the following functions a) b) c) 27

28 Basic Differentiation Rules & Rates of Change Derivatives of Sine and Cosine Functions Example 8: Find the derivative of the following functions a) b) c) d) 28

29 Basic Differentiation Rules & Rates of Change Position Function: A function s that gives the position (relative to the origin) of an object as a function of t. If over a period of time, amount: the object changes its position by the Remember: Average Velocity: Change in distance Change in time = 29

30 Basic Differentiation Rules & Rates of Change Example 9: 30

31 Basic Differentiation Rules & Rates of Change *Velocity is the derivative of position* Speed is the absolute value of velocity (since speed can't be negative) Example 10: Refer to example 9 and find the instantaneous velocity of the billiard ball when t=1 sec 31

32 Basic Differentiation Rules & Rates of Change The position of a free falling object (neglecting air resistance) under the influence of gravity can be represented by the equation: acceleration due to gravity On Earth, or initial height initial velocity 32

33 Basic Differentiation Rules & Rates of Change Example 11: At time t=0, a diver jumps from a platform diving board that is 32 feet above the water. Because the initial velocity of the diver is 16 ft/sec, the position of the diver is: where s is measured in feet and t is measured in seconds. a) when does the diver hit the water? b) what is the diver's velocity at impact? 33

34 34

35 Product & Quotient Rules & Higher Order Derivatives The Product Rule Example 1: Find the derivative of 35

36 Product & Quotient Rules & Higher Order Derivatives Example 2: Find the derivative of 36

37 Product & Quotient Rules & Higher Order Derivatives Example 3: Find the derivative of 37

38 Product & Quotient Rules & Higher Order Derivatives The Quotient Rule 38

39 Product & Quotient Rules & Higher Order Derivatives Example 4: 39

40 Product & Quotient Rules & Higher Order Derivatives Example 5: 40

41 Product & Quotient Rules & Higher Order Derivatives 1. Draw an example of a function that is continuous but not differentiable at x = 2 2. Determine the point (if any) at which the graph of the function has a horizontal tangent line. a) b) 41

42 Product & Quotient Rules & Higher Order Derivatives Friday October 26th Example 6: Differentiate the following functions (You need to rewrite them prior to differentiating) 42

43 The Product Rule The Quotient Rule 43

44 Product & Quotient Rules & Higher Order Derivatives Example 7: Determine the derivative of tan using the quotient rule 44

45 Product & Quotient Rules & Higher Order Derivatives Derivatives of Trig Functions 45

46 Product & Quotient Rules & Higher Order Derivatives Example 8: Find the derivative of the following 46

47 Product & Quotient Rules & Higher Order Derivatives Example 9: 47

48 Product & Quotient Rules & Higher Order Derivatives Position Function: Velocity Function: Acceleration Function: 48

49 Product & Quotient Rules & Higher Order Derivatives Example 10: 49

50 Chain Rule The chain rule allows us to differentiate composite functions The Chain Rule: 50

51 Chain Rule 51

52 Chain Rule Example 1: Decompose the following functions. i.e. determine the outer and inner function and rewrite. 52

53 Chain Rule Example 2: Find for 53

54 Chain Rule General Power Rule 54

55 Chain Rule Example 3: Find the derivative of 55

56 Chain Rule Example 4: 56

57 Chain Rule Chain Rule Worksheet Answers

58 58

59 Chain Rule Example 5: 59

60 Chain Rule Example 6: 60

61 Chain Rule Example 7: Find the derivative of 61

62 62

63 Chain Rule Example 8: Find the derivative of 63

64 Chain Rule Trig Functions & the Chain Rule 64

65 Chain Rule Example 9: Find the derivative of 65

66 66

67 Chain Rule Monday Nov. 6 Do # 9 10 on your Chain Rule Worksheet... you will have to apply product or quotient rule in addition to the chain rule 67

68 Chain Rule Monday Nov. 6 Find the derivative of the following functions a) b) c) d) e) 68

69 Chain Rule Find the derivative of the following function 69

70 Chain Rule 70

71 Chain Rule 71

72 Implicit Differentiation 72

73 Implicit Differentiation Example 1: Differentiate with respect to x a) b) c) d) 73

74 Implicit Differentiation 74

75 Implicit Differentiation Example 2: 75

76 Implicit Differentiation Implicit Differentiation Worksheet 76

77 Monday Nov. 13th Differentiate with respect to x. Rewrite, differentiate, simplify! Differentiate with respect to x. Find dy/dx

78 Implicit Differentiation 78

79 Implicit Differentiation Example 3: 79

80 Implicit Differentiation Example 4: 80

81 Implicit Differentiation Example 5: 81

82 Implicit Differentiation Example 6: 82

83 Implicit Differentiation Example 7: 83

84 Related Rates Imagine water draining out of a conical tank... the radius, height and volume are all functions of time. 84

85 Related Rates In the conical tank above, the height of the water level is changing at a rate of 0.2 ft/min and the radius is changing at a rate of 0.1 ft/min. What is the rate of change in the volume when the radius is 1 ft. and the height is 2ft? 85

86 Related Rates Example 1: 86

87 Related Rates Example 2: 87

88 Related Rates 88

89 Related Rates Example 3: Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet. 89

90 Related Rates Example 4: 90