Exponential and logarithmic functions

Transcription

1 Exponential and logarithmic functions, حصه الرومي الطالبات.. ندى الغميز, الزهراني مالك المسعد خديجه, األستاذة.. يسرى

2 Modeling with Exponential and Logarithmic Functions

3 Exponential Growth and Decay Models The mathematical model for exponential growth or decay is given by f (t) = A 0 e kt or A = A 0 e kt. If k > 0, the function models the amount or size of a growing entity. A 0 is the original amount or size of the growing entity at time t = 0. A is the amount at time t, and k is a constant representing the growth rate. If k < 0, the function models the amount or size of a decaying entity. A 0 is the original amount or size of the decaying entity at time t = 0. A is the amount at time t, and k is a constant representing the decay rate.

4 Population (millions) Example The graph below shows the growth of the Mexico City metropolitan area from 1970 through In 1970, the population of Mexico City was 9.4 million. By 1990, it had grown to 20.2 million Year Find the exponential growth function that models the data. By what year will the population reach 40 million?

5 Example cont a. We use the exponential growth model A = A 0 e kt in which t is the number of years since This means that 1970 corresponds to t = 0. At that time there were 9.4 million inhabitants, so we substitute 9.4 for A 0 in the growth model. A = 9.4 e kt We are given that there were 20.2 million inhabitants in Because 1990 is 20 years after 1970, when t = 20 the value of A is Substituting these numbers into the growth model will enable us to find k, the growth rate. We know that k > 0 because the problem involves growth. Use the growth model with A 0 = 9.4. When t = 20, A = Substitute these values. A = 9.4 e kt 20.2 = 9.4 e k 20

6 Example cont Isolate the exponential factor by dividing both sides by / 9.4 = e k 20 Take the natural logarithm on both sides. Simplify the right side by using ln e x = x. Divide both sides by 20 and solve for k. ln(20.2/ 9.4) = lne k / 9.4 = 20k = k We substitute for k in the growth model to obtain the exponential growth function for Mexico City. It is A = 9.4 e 0.038t where t is measured in years since 1970.

7 Example cont b. To find the year in which the population will grow to 40 million, we substitute 40 in for A in the model from part (a) and solve for t. This is the model from part (a). Substitute 40 for A. Divide both sides by 9.4. Take the natural logarithm on both sides. Simplify the right side by using ln e x = x. A = 9.4 e 0.038t 40 = 9.4 e 0.038t 40/9.4 = e 0.038t ln(40/9.4) = lne 0.038t ln(40/9.4) =0.038t Solve for t by dividing both sides by ln(40/9.4)/0.038 =t Because 38 is the number of years after 1970, the model indicates that the population of Mexico City will reach 40 million by 2008 ( ).

8 Text Example Use the fact that after 5715 years a given amount of carbon-14 will have decayed to half the original amount to find the exponential decay model for carbon-14. In 1947, earthenware jars containing what are known as the Dead Sea Scrolls were found by an Arab Bedouin herdsman. Analysis indicated that the scroll wrappings contained 76% of their original carbon-14. Estimate the age of the Dead Sea Scrolls. Solution We begin with the exponential decay model A = A 0 e kt. We know that k < 0 because the problem involves the decay of carbon-14. After 5715 years (t = 5715), the amount of carbon-14 present, A, is half of the original amount A 0. Thus we can substitute A 0 /2 for A in the exponential decay model. This will enable us to find k, the decay rate.

9 Text Example cont After 5715 years, A = A 0 /2 Solution Divide both sides of the equation by A 0. A 0 /2= A 0 e k5715 1/2= e kt5715 Take the natural logarithm on both sides.ln(1/2) = ln e k5715 ln e x = x. Solve for k. ln(1/2) = 5715k k = ln(1/2)/5715= Substituting for k in the decay model, the model for carbon-14 is A = A 0 e t.

10 Exponential Functions

11 The Exponential Function w/ base a for a > 0 f x a x

12 f x a x * For a 1, - Domain: (, ) - Range: (0, )

13 f x a x for a > 1

14 f x a x for 0 < a < 1

15 Natural Exponential Function f x e x with base e

16 Compound Interest t r n A P( 1 ) nt -A(t) = amount after t years -P = Principal -r = interest rate -n = # of times interest is compounded per year -t = number of years

17 Continually Compounded Interest A t Pe rt -A(t) = amount after t years -P = Principal -r = interest rate -t = number of years

18 Ex 1: If $350,000 is invested at a rate of 5½% per year, find the amount of the investment at the end of 10 years for the following compounding methods: - Quarterly - Monthly - Continuously

19 Exponential Growth n t n o e rt n(t) = population at time t n o = initial size of population r = rate of growth t = time

20 Logarithmic Functions

21 The Log Function is the inverse of the Exponential Function, so If fx a x Then f 1 x log a x

22 And for fx log a x Domain: (0, ) Range: (, )

23 logarithmic form exponential form log a x y a y x base exponent base exponent a is positive with a 1

24 Common Logarithm (base 10) log x log10 x

25 Natural Logarithm (base e) ln x log e x ln x y e y x

26 Properties of Logs 1) log a 1 0 2) log a 1 a 3) 4) log a x a a loga x x x

27 Properties of Natural Logs 1) ln1 0 2) 3) 4) ln e 1 ln e x x e ln x x

28 Exponential and Logarithmic Functions Exponential Functions

29 Exponential Functions These functions model rapid growth or decay: - # of users on the Internet 16 million (1995) 957 million (late 2005) - Compound interest - Population growth or decline

30 Comparison - Linear Functions Rate of change is constant - Exponential Functions Change at a constant percent rate.

31 The Exponential Function y = ab x b is the base: - It must be greater than 0 - It cannot equal 1. x can be any real number

32 Identify Exponential Functions * Which of the following are exponential functions? y = 3 x yes y = x 3 no y = 2(7) x yes y = 2(-7) x no

33 Identify the Base * Identify the base in each of the following. y = 3 x y = 2(7) x y = 4 x - 3 y = 3a x

34 Evaluate Exponential Functions y = 3 x for x = 4 y = 2(7) x for x = 3 y = -2(4 x ) for x = 3/2

35 y Graph Exponential Functions (b > 1) * Graph y = 2 x for x = -3 to 3 x y Graph of y = 2^x -3 1/ / / x

36 y Graph Exponential Functions (0< b < 1) * Graph y = (1/2) x for x = -3 to 3 x y -3 8 Graph of y = (1/2)^x / / / x

37 Summary y = ab x - x can be any value - The resulting y value will always be positive. - The y-intercept is always (0,1) - When b > 1, as x increases, y increases. - When 0 < b < 1, as x increases, y decreases.

38 Practice * Using Microsoft Excel: - Graph the function y = 3 x for x = -3 to 3 (in 0.5 increments) - Graph the function y = (1/3) x for x = -3 to 3 (in 0.5 increments)

39 Practice: Graph using Excel - Result: Graphing Exponential Functions

40 Exponential Functions

41 Example Suppose you are a salaried employee, that is, you are paid a fixed sum each pay period no matter how many hours you work. Moreover, suppose your union contract guarantees you a 5% cost-of-living raise each year. Then your annual salary is an increasing function of the number of years you have been employed, because your annual salary will increase by some amount each year. However, the amount of the increase is different from year to year, because as your salary increases, the amount of your 5% raise increases too. This phenomenon is known as compounding.

42 Example Assume your starting salary is $28,000 per year. Let S(t) be your annual salary after full years of employment. Therefore, S(0) is interpreted to mean your initial salary of $28,000. How can we evaluate S(1), your salary after 1 year of employment? Since your salary is increasing by 5% each year, this means S(1) is 5% more than S(0). In other words, S(1) is 105% of S(0). Thus, we can evaluate S(1) as shown here, by changing the percentage 105% to a decimal number: S(1) = 105% of S(0) = 1.05 S(0) = S(2) = 105% of S(1) = 1.05 S(1) = S(3) = 105% of S(2) = 1.05 S(2) = S(4) = 105% of S(3) = 1.05 S(3) = S(5) = 105% of S(4) = 1.05 S(4) =

43 Graph of Exponential Functions

44 Graph of Exponential Functions

45 Exponential Functions Exponential functions have symbol rules of the form f (x) = c bx b: base or growth factor -- must be positive real number but cannot be 1, i.e. b > 0 and b 1 c: coefficient greater than 0 the domain of f is (, ) the range of f is (0, )

46 Natural Exponential Function f (x) = ex f (x) = e x

47 Example

48 Example

49 Example

50 Find the exponential function graph as shown below. whose

51 Find the exponential function graph as shown below. whose

52 Logarithmic Functions

53 Logarithmic Functions Consider the exponential function f shown here with base b = 2 and initial value c = 1 Suppose we want to find the input number for that matches the output values 8 and 15, in other words, we want to solve the equation

54 Logarithmic Functions Let's introduce a new function designed to help us express solutions to equations like the two shown here, which are solved by finding particular input numbers for the exponential function f. We give this new function a special label:

55 Logarithmic Functions helps us express inputs for the function f. Thus, for example, we evaluate, because f(3)=. Likewise, we evaluate In general, That is exponential function and logarithmic function are inverse of each other.

56 Common and Natural Logarithms A common logarithm is a logarithm with base 10, log10. A natural logarithm is a logarithm with base e, ln.

57 Properties of Logarithms

58 Graphs of Logarithmic Functions

59 Graphs of Logarithmic Functions

60 Graphs of Logarithmic Functions

61 Laws of Logarithms

62 Change of Base

63 Compound Interest If P is a principal of an investment with an interest r for a period of t years, then the amount A of the investment is

64 Modeling with Exponential and Logarithmic Functions

65 Exponential Growth Model A population that experiences exponential growth increases according to the model where population at time t initial size of population relative rate of growth time

66 Radioactive Decay Model If m0 is the initial mass of a radioactive substance with half-life h, then the remaining mass of radioactive at time t is modeled by where

67 Total differential of Q using logs

68 68 dl L dk K da A Q dq dl L dk K da A Q dq L d K d A d Q d L K A Q L AK Q 1 1 ln ln ln ln ln ln ln ln

69 Exponent example 69 * * 1 1 and K for L Solve 0 5) 0 4) 3) 5 2) 1) pp ) 5, (Example firm a of decisions Input 11.6(c) P-r K L α π P-w K L α π -wl-rk K L P π. α K L Q PQ-wL-rK R-C π α α K α α L α α α α

70 Maximization conditions

71 71 25 years % ) let ( ). ( t r r t r t r t r t bottle $148.38/ 1218 bottle $12.18/ bottle $1/ ) let ( V e. $ V Ae V e A(t) e A(t) k ke A(t) ) )( (. rt. ) )(( (. rt t ½

72 10.6(c) Timber cutting problem plot of.5(t) -.5 ln(2)=r, r=.05, t=48 72

73 10.7(b) Rate of growth of a combination of functions; Example 3 consumption & pop. v v u u v u v u v u u z z z z r s r s r v u v r v u u r vr t g ur t f t f t f r if t g t f v u r t g t f dt d v u r v u dt d v u r v u dt d z dt d r v u z t g v t f u v u z ) '( and, ) '( then, ) ( ) '( ) ( )) '( ) '( ( 1 )) ( ) ( ( 1 ) ( 1 ) ( ) ln( ) ln( ) ln( ) ( ) ( t f t f dt t f d ln

74 Trigonometric, Logarithmic, and Exponential Functions

75 In this tutorial, we review trigonometric, logarithmic, and exponential functions with a focus on those properties which will be useful in future math and science applications. Trigonometric Functions Geometrically, there are two ways to describe trigonometric functions:

76 Trigonometric Functions Geometrically, there are two ways to describe trigonometric functions: Polar Angle x=cos y=sin Measure in radians: =radiusarc length For example,180=rr= radians Radians=180degrees

77 (0< b < 1) Graph Exponential Functions

78 y Graph y = (1/2) x for x = -3 to 3 : x y Graph of y = (1/2)^x /2 1/4 1/ x

79 Summary y = ab x x can be any value The resulting y value will always be positive. The y-intercept is always (0,1) When b > 1, as x increases, y increases. When 0 < b < 1, as x increases, y decreases.

80 Text Example cont Solution A 0 /2= A 0 e k5715 After 5715 years, A = A 0 /2 1/2= e kt5715 Divide both sides of the equation by A 0. ln(1/2) = ln e k5715 Take the natural logarithm on both sides. ln(1/2) = 5715k ln e x = x. k = ln(1/2)/5715= Solve for k. Substituting for k in the decay model, the model for carbon-14 is A = A 0 e t.

81 Text Example cont. Solution A = A 0 e t This is the decay model for carbon A 0 = A 0 e t A =.76A 0 since 76% of the initial amount remains = e t Divide both sides of the equation by A 0. ln 0.76 = ln e t Take the natural logarithm on both sides. ln 0.76 = t ln e x = x. t=ln(0.76)/( ) Solver for t. The Dead Sea Scrolls are approximately 2268 years old plus the number of years between 1947 and the current year.

82 Finally we hope you can understand the logarithmic function, and we hope you like our project =)