Chapter 3 Exponential Functions Logistic Functions Logarithmic Functions

Transcription

1 C Chapter 3 Exponential Functions Logistic Functions Logarithmic Functions ` Mar 17 9:06 PM Exponential Functions Equation: y = ab x a is the initial value b is the growth factor and r is the rate of growth (1 + r) = b when b > 1 exponential growth (r > 0) when 0 < b < 1 exponential decay (r < 0) Verbal Representation Exponential Growth Decay Compound Interest Radiactive Decay Population Growth Half Life Appreciation Depreciation Mar 17 9:12 PM 1

2 Mar 17 9:45 PM Explain how you know that this table of values is exponential Mar 17 9:46 PM 2

3 Mar 17 9:10 PM Mar 17 9:10 PM 3

4 Finding the equation of an Exponential Function Given a starting point and the percent of growth. Ex: The population in Mexico grows at a rate of 2.6% per year. In 1980 there were million people. Growth Factor is b = Let t=0 in 1980, then a = P=67.38(1.026) t Mar 29 8:23 PM On August 2,1988 a US District Court imposed a fine on the city of Yonkers NY for defying a federal court order involving housing desegregation. The fine started out as $100 for the first day and doubled daily until the city chose to obey the court order. What was the daily percent growth rate of the fine? (200%) Find a formula for the fine as a function of t, the number of days since August 2, (2) x If Yonkers waited 30 days before obeying the court order, what was the fine? 100(2 30 )=10,737,418,240 Mar 29 9:08 PM 4

5 Class work/discussion p , 25 to 30, 31 to 34 p Homework p 270 # p 279 # 21,29,31,32 Mar 17 10:28 PM p 270 #11 12 x f(x) g(x) / /8 4/ (0,3) (2,6) (0,2) (1, 2/e) Apr 10 9:15 AM 5

6 0 p ) (0,4) (5,600.25) 29) (475,000) 3.75% each year When 1 million 31) yr > 6250 people growth 2.75% yr 1915 yr? 50,000 yr ) yr > 4200 rate2.25% Yr 1930 yr? 20,000 yr 1945 Apr 10 9:21 AM Goal: Another exponential function Discovering e What is the number e? Compound interest limit of compounding Mar 29 9:04 PM 6

7 Mar 17 9:56 PM e = lim (1 + 1/x) x x-> Mar 17 10:05 PM 7

8 Mar 17 10:00 PM Mar 17 10:01 PM 8

9 Mar 17 10:02 PM The number e is a famous irrational number. (almost as important as π) It is found in many applications of mathematics. The first few digits of e are It is often known as Euler's number after Leonhard Euler Mar 29 9:19 PM 9

10 lim(1+1/n) n = e n-> Mar 29 9:33 PM Any exponential function f(x) = ab x can be rewritten as f(x) = a e kx, for an appropriately chosen real number constant k. If a>0 and k > 0, f(x) = a e kx is an exponential growth function If a>0 and k < 0, f(x) = a e kx is an exponential decay function Mar 17 10:19 PM 10

11 The Natural Base e Function F(x) = e x Family Exponential Domain all real numbers Range (0, ) Continuity Vertical Asymptote None Horzontal Asymptotes y=0 increasing for all x Bounded below by x axis End behavior x -> y-> Symmetry NO Decreasing No local extrema x -> - y-> 0 Mar 17 10:02 PM Apr 15 10:35 AM 11

12 Find the equation given two points Find the equation of the exponential function that goes through the point (-2,45/4) and (1, 10/3) Step 1 Write two equations using each of the points in the form y = ab x 45/4 = ab -2 10/3 = ab 1 Step 2 Divide the equation so the a drops out. and solve for b 27/8 = b -3 or 8/27 =b 3 b = 2/3 Step 3 Use b and one of the points to solve for a 10/3 = a(2/3) 1 5 = a Step 4. Write equation with a and b y = 5 (2/3) x Mar 29 8:30 PM Classwork FMC Green Book p 117 # 4-7 Precalc: p 126 # 21,25, Homework Precalc Book p 271 # 45-48, 55, 56 FMC p117 #13,15 Mar 29 8:41 PM 12

13 Apr 12 11:12 AM (-3, ) ( 2, ) Apr 15 10:37 AM 13

14 Goal: Logistic Growth Exponential and Logistic modeling In many growth situations there is a limit to the possible growth. The growth often begins as an exponential manner, but the growth eventually slows and levels out. Ex: Population in a fixed enviroment (fish in an aquarium) Plant growth Mar 17 10:33 PM Logistic Growth Function Let a,b,c and k be positive constants, with b < 1. A logistic growth function in x is a function that can be written in the form f(x) = c or f(x) = c 1 + ab x 1 + a e -kx where the constant c is the limit to growth. Mar 17 10:39 PM 14

15 Basic Logistic Function Function F(x) = e -x Family Exponential Logistic Domain all real numbers Range (0,1) Continuous Concaved up Concaved Down (-,0) (0, ) Vertical Asymptote None Horzontal Asymptotes y=0, y = 1 increasing for all x Bounded below and above End behavior x -> y->1 Symmetry NO Decreasing No local extrema x -> - y-> 0 Mar 17 10:44 PM Graph the function f(x) = 20/(1+2e -3x ) Determine the horizontal asymptotes Y-intercept Mar 17 10:58 PM 15

16 p270 #49 Logistic Function Function F(x) = e -2x Family Exponential Logistic Domain Range Continuous Concaved up Concaved Down Vertical Asymptote Horzontal Asymptotes increasing Bounded End behavior Symmetry x -> y-> Decreasing local extrema x -> - y-> Mar 17 11:05 PM graph y-intercept Mar 17 11:11 PM 16

17 Ex 1 Suppose a radioactive substance decays at a rate of 3.5% per hour. What percent of the substance is left after 6 hours. Mar 17 11:11 PM Ex 2 A new franchise of fast food restaurants is expected to grow at a rate of 8% per year. There are presently 200 restaurants. What is the expected number of restaurants in 15 years? Mar 17 11:14 PM 17

18 Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated. How many players remain after 5 rounds? Mar 17 11:18 PM In the movie the blob, a substance doubles every 3 hours. The initial mass of a substance is.6 grams. What is the hourly growth rate? How large is the substance after 24 hours? after 3days? Mar 17 11:24 PM 18

19 Classwork p 280 #24,26,28, 34(Use e), 43 Group discussion; Homework p280 # 23,25, 27,33, 39, 44 Mar 17 11:18 PM Apr 16 12:48 PM 19

20 Apr 16 1:10 PM Solve the equations 5 x+1 = x+1 =7 3x-2 Apr 16 9:58 AM 20

21 Solving equation without logs Rewrite the equation in the same base Work with only the exponents Example 2 (X+5) = 8 (2x+1) Apr 15 11:38 AM 3 2x-1 =27 x 5 3x-8 =25 2x What about 5 x =7 Can you do this the same way? Apr 16 9:52 AM 21

22 What's my inverse? What undoes x+3 x-6 5x x/3 x 2 x 3 1/x 2 x e x Mar 17 11:24 PM The inverse of an exponential function f(x)=b x is the logarithmic function with the base of b. If f(x) = b x then f -1 (x) = log b (x) If x > 0 and 0 < b 1, then y=log b (x) iff b Y =x Apr 6 11:07 PM 22

23 log 2 8 = 3 because 2 3 =8 log 3 3 = log 5 (1/25) log 4 1 log 7 7 Apr 6 11:14 PM Basic Properties of Logarithms log b 1 = 0 log b b = 1 log b b y = y b log b x = x Common log is base 10 log 1 = 0 log 10 = 1 log10 y = y 10 logx = x Apr 6 11:24 PM 23

24 Because of their special calculus properties, logarithyms with the natural base e are used in so many situations that they have a special notation for a log with base e. THE NATURAL LOG LN LN without a subscript is understood to mean log e x =ln x Basic Properties of Natural Logarithms Let x and y be real numbers with x > y ln1 = 0 because e 0 =1 ln e = 1 because e 1 =1 lne y = y because e y =e y e ln x = x because ln x =ln x Apr 6 11:37 PM Classwork p all Homework p p 324 2,4,9,10 Apr 6 11:47 PM 24

25 What does the graph of a logarithmic function look like. To draw a logarithmic graph first draw the function e x, now draw the feflection over y = x line Apr 6 11:56 PM The Natural Log (LN) Function F(x) = ln x Family Inverse function of Exponential e x Domain (0, ) Range all real numbers Continuous Vertical Asymptote x=0 Horzontal Asymptotes none increasing for all x Decreasing Bounded on the left by y axis No local extrema End behavior x -> y-> x > 0 + y -> - Symmetry NO Apr 6 11:59 PM 25

26 Class work p Apr 7 12:02 AM Apr 7 12:02 AM 26

27 Apr 7 12:02 AM 27