CC2 Exponential v.s. Log Functions

Transcription

1 CC2 Exponential v.s. Log Functions

2 CC1 Mastery Check Error Analysis tomorrow Retake? TBA (most likely end of this week) *In order to earn the chance for re-assessment, you must complete: Error Analysis for your Mastery Check Defense Assignment Oct 10th

3 From geometry: Reflecting across the line y = x Pre-Image Image F(-3, 0) F (0, -3) S I I(4, 0) I'(0, 4) F I S(4, -9) S'(-9, 4) H F H(-3, -9) H'(-9, -3) H S What do you notice about the x and y coordinates of the pre-image and image points?

4 In the same way, the inverse of a given function will undo what the original function did. For example, let s take a look at the function: f(x) = x 2 f(x) f --1 (x) x 2 x

5 Find the inverse of a function Example 1: y = 6x - 12 f 1 ( x) Step 1: Switch x and y: x = 6y - 12 Step 2: Solve for y: x x y 12 y 6 x 2 y

6 Example 2: Given the function : y = 3x find the inverse: Step 1: Switch x and y: x = 3y Step 2: Solve for y: x 2 3y x 2 3 y 2 2 x 2 y 3

7 Pair Work: Practice on Inverse function

8 Oct 11 th : Error Analysis : You need to identify all your mistakes on the test. Give yourself a grade based on the Rubrics on your syllabus. Let s say I tolerate about 10-15% of DOK2 computational mistakes. So if you get 100% on DOK1 + 85% DOK2 it s a 2- But not misconceptions.

9 If I am giving you an opportunity to retake prior to Progress report: Most likely this Friday. Maximum problems out of DOK1 +2: will include a Rational Function Graph and an AROC or IROC What do you have to do to earn it? Error Analysis in details Extra practice assignment

10 1 f ( x) x 1 find f ( x) domain : x 1 2 f x f x x 1 1 ( ) ( ) 2

11

12

13

14 Given Parent Graph f ( x) 2 x Sketch : gx ( ) w( x) 1 2 x 3 x 2 x x 3 ux ( ) 2 3 s( x) 2 2

15 Do you remember all the steps of transformation? Can you find the Asymptotes?

16

17

18

19

20 How about? y 2 x and x 2 y

21

22 What do you think?

23

24 Like absolute value function here, quadratic has an issue when performing inverse operations

25 So don t forget to eliminate the part of the function that will make its inverse function a Non-function

26 One-to-One Function Test (Horizontal Line Test)

27 This goes for all the even functions.

28

29 Concept Category (unit) 2 Learning Targets (sections): Inverse Functions Exponential v.s. Log Functions (including graphing & transformation) Evaluate/Solve Exponential, Log Functions The Three Properties of Log Applications

30 Exponential Equation: y = b x (b > 0, b 1) But what about solving the same equation for y? You may recall that y is called the logarithm of x to the base b, and is denoted log b x. Logarithm of x to the base b y = log b x if and only if x = b y (x > 0)

31

32

33 Oct18th Recall yesterday:

34 Exponents, Radicals, and Logs For example) Radical (Root) = Rational exponent Log

35 More Examples:

36 Solving Log Equations: First thing ALWAYS: rewrite them into exponential equations. Then use different methods to solve for the missing variable: Power? Base? Or the Result? a) Log 27 9 b) Log 64 6 w 3 c) log 25 R 2

37 Power Base Result a) Log 27 b) Log 64 6 c) log R 9 w E 9 27 w E E 3 2 * 2 * 2 * 2 * 2 * ( 3 ) 3 w (5) E R

38 Try these ones please

39

40 Try these two problems: 1 2Log 3 Log Log Log 27 Log 4 Log

41

42 Solve x: Try this problem:

43

44 Your calculator:

45

46

47 Know your Composition

48

49

50 Proof (examples) Log 5 3? 5? log2 5 2? How to simplify Log 5? 2

51 Common Log Formula: to solve for missing exponent when we don t have same base Log R 2 b Log R Log b So : Log 5 E 2 5 But 2 and 5 do not have the same base. Log 5 E? 2? Log 2 E E Check your work

52 Try these problems: Log 16 Log 8 Log Log

53 Oct 21 So far we have covered: Inverse Functions Translate Log to Exponential Equations Solve for the missing elements of Exponential Equations: Power, Base, and Result Common Log Formula: when we are solving for power but we do not have the same base, for example 3 E 5 solve E

54 The 3 Properties of Log: Consider these cases a) Log x Log 3x 2 Solve x 2 2 b) Log x Log 4 3 Solve x c) Log 64? When converting individual Log expressions is no longer possible or simple

55 Basically Sometimes if we do not compress multiple Log expressions down to a single one, we cannot actually convert them to an exponential expression, thus the 3 properties as a tool to simplify.

56 Properties of Logarithms CONDENSED EXPANDED 1. log a MN log a M log a N log a log a M N M r log a r M log a log M a N Note: Rule 1 and 2 only work if all log expressions have the same base

57 Oct 21 First 25 minutes Error Analysis * First, check your final answers. Are they right? * Then mark your mistakes: CO conceptual error CA calculation error Turn in your corrected QC. No detailed correction = no grade. Then some note-taking

58 Consider these cases From yesterday: a) Log x Log 3x 2 Solve x 2 2 b) Log x Log 4 3 Solve x c) Log 64? When converting individual Log expressions is no longer possible or simple

59

60 Solution

61

62

63 Solution for b. Log 2 x 4 x 3 x 4 x x 4 2( x 3) x 4 2x x x 2 x 1

64 Verify : Log Log ( 2 4) Log ( 2 3) (2) Log (1) E 2 2 E

65 a) Log (15x 1) Log ( x 1) b) 2 Log x Log ( x 8) Watch out for the Goal Problems

66

67

68

69 Expand the following equations

70

71 When solving log equations: compress them into single log equation, then convert it to exponential equation When solving exponential equations with Like Bases : your answers should be in integers or fractions Different Bases : you need a calculator and the Common Log Formula

72 a ) 3 2x b) 2(3 5x 1 ) 10 2

73 Exponential Equations with Like Bases Example: Solve 2x x x x 1 2 2x 1 x 1 2 Isolate the exponential expression and rewrite the constant in terms of the same base. Set the exponents equal to each other (drop the bases) and solve the resulting equation.

74 Exponential Equations with Different Bases Solve: 5x 1 2(3 ) x 1 2(3 ) 8 5x Log 4 5x Log 3 5x x Isolate the exponent, since x is part of the exponent Use Common Log Formula to solve for the exponent

75 You Try: Solve the exponential equation But instead of using Log key, use Ln key also to find the x Exp Ln R Ln B v.s. Exp Log R Log B

76

77 Natural logarithm e is used because the exponential is often used when doing interest/growth calculation. If you are in continuous time and that you are compounding interests, you will end up having a future value of a certain sum equal to f () t Ne rt (where r is the interest rate and N the nominal amount of the sum). Since you end up with exponential in the calculus, the best way to get rid of it is by using the natural logarithm and if you do the inverse operation, the natural log will give you the time needed to reach a certain growth.

78 Base e and Natural Logs (ln) Natural Base (e): e Natural Base Exponential Function: y e x x (inverse of ln) Natural Logarithm (ln): log x ln x (inverse of e) e

79

80 log e ln

81 Solve : 2x 4e 16 e 2x ln (3x 9) 21

82

83 ln 2x (3x 9) 21 e

84 Use Factoring or Cross-Mult. as tools a) b) 10 c) 2 x 1 e 5 ) e 3 x d

85 Solution a)

86 Solution c)

87 Solve:

88

89

90

91

92

93

94

95 Example of Compound Growth

96

97

98

99

100 Example

101

102

103

104 Always have your notes with you Anything missing? Ask a peer first what did you miss? Do you all have the same question? Then ask the teacher * Might need to update notes*

105 Sketch a graph given the following description: lim f 0 x lim f 2 x 1 lim f 1 x 1 lim f 2 x 0 f (0) 2 lim f 2 x 1 f (1) 3 lim f 3 x 2 f (2) 3 lim x 2 f lim f 0 x

106 lim f 2 x 1 lim f 3 x 2 f (2) 3 lim f 1 x 1 lim f 2 x 1 f (1) 3 lim f 0 x lim f 0 x lim f 2 x 0 f (0) 2 lim x 2 f

107 lim x 1 f lim x 3 f lim x 4 f lim f 0 x lim f 0 x f (3.5) 2.5 lim x 1 f lim x 3 f lim x 4 f

108 15 minutes: Use the answer key Are your answers right or wrong? With peers, use your notes where di your work go wrong?

109 Use the solution sheet, and your notes, And you can still work with your peers: Analyze all your errors (categorize them)

110

111

112 Going back to our continuously compounding interest problems... A $20,000 investment appreciates 10% each year. How long until the stock is worth $50,000? Remember our base formula is A = Pe rt... We now have the ability to solve for t A = $50,000 (how much the car will be worth after the depreciation) P = $20,000 (initial value) r = 0.10 t = time From what we have learned, try solving for time

113 Going back to our continuously compounding interest problems... $20,000 appreciates 10% each year. How long until the stock is worth $50,000? A = $50,000 (how much the car will be worth after the depreciation) P = $20,000 (initial value) r = 0.10 t = time

114 Example 7 Base e Applications kt y ae [A] Suppose you deposit $700 into an account paying 6% compounded continuously. i) How much will you have after 8 years? ii) How long will it take to have at least $2000? (round to the tenth of a year) y y 700e (.06 8) $1, e (.06x) (.06x) 2000/700 e. 06x ln(20/7) x 17.5 years

115 Example 7 Base e Applications kt y ae [B] Suppose you deposit $1000 into an account paying 5% compounded continuously. i) How much will you have after 10 years? ii) How long will it take to triple your money? (round to the tenth of a year) y y 1,000e (.0510) $1, e (.05x) 3 e ln 3. 05x (.05x) x 22.0 years

116

117

118

119

120