Chapter 2 Functions and Graphs

Transcription

1 Chapter 2 Functions and Graphs Section 5 Exponential Functions

2 Objectives for Section 2.5 Exponential Functions The student will be able to graph and identify the properties of exponential functions. The student will be able to graph and identify the properties of base e exponential functions. The student will be able to apply base e exponential functions, including growth and decay applications. The student will be able to solve compound interest problems. 2

3 Table of Content Exponential Functions Base e Exponential Functions Gross and Decay Applications Compound Interest 3

4 Terms exponential function base e exponential functions relative growth rate exponential decay depreciation compound interest interest rate principal P (present value) amount A (future value) compound growth continuous compound interest 4

5 Exponential Function The equations are not the same function. f(x) = 2 x and g(x) = x 2 The function g is a quadratic function. The function f is a new type of function called an exponential function. 5

6 Exponential Function The equation f (x) b x, b 1 defines an exponential function for each different constant b, called the base. The domain of f is the set of all real numbers, while the range of f is the set of all positive real numbers. We required the base b to be positive to avoid imaginary numbers and we exclude b = 1 as a base, since f(x) = 1 is a constant function already considered. 6

7 Riddle Here is a problem related to exponential functions: Suppose you received a penny on the first day of December, two pennies on the second day of December, four pennies on the third day, eight pennies on the fourth day and so on. How many pennies would you receive on December 31 if this pattern continues? Would you rather take this amount of money or receive a lump sum payment of $10,000,000? 7

8 Solution Complete the table: Day No. pennies ^ ^ ^

9 Solution (continued) Now, if this pattern continued, how many pennies would you have on Dec. 31? Your answer should be 2 30 (two raised to the thirtieth power). The exponent on two is one less than the day of the month. See the preceding slide. What is 2 30? 1,073,741,824 pennies!!! Move the decimal point two places to the left to find the amount in dollars. You should get: $10,737,

10 Solution (continued) The obvious answer to the question is to take the number of pennies on December 31 and not a lump sum payment of $10,000,000 (although I would not mind having either amount!) This example shows how an exponential function grows extremely rapidly. In this case, the exponential function f( x) 2 x is used to model this problem. 10

11 Graph of f (x) 2 x Use a table to graph the exponential function above. Note: x is a real number and can be replaced with numbers such as 2 as well as other irrational numbers. We will use integer values for x in the table: 11

12 Table of values y f (x) 2 x x y = 1/2 4 = 1/ = 1/ = 1/ = 1/ = = = 4 12

13 Graph of f( x) 2 x 1 2 x Using a table of values, you will obtain the following graph. x The graphs of f ( x) b and f ( x) b x will be reflections of each other about the y-axis, in general graph of y = 2^(-x) 6 approaches the positive x-axis as x gets large 4 2 passes through (0,1)

14 Basic Properties of the Graph of f (x) b x, b 0, b 1 1. All graphs will pass through (0,1) (y intercept) 2. All graphs are continuous curves, with no holes of jumps. 3. The x axis is a horizontal asymptote. 4. If b > 1, then b x increases as x increases. 5. If 0 < b < 1, then b x decreases as x increases. 14

15 Graphing Other Exponential Functions Now, let s graph f( x) 3 x Proceeding as before, we construct a table of values and plot a few points. Be careful not to assume that the graph crosses the negative x-axis. Remember, it gets close to the x-axis, but never intersects it. 15

16 Preliminary Graph of f( x) 3 x 16

17 Complete Graph y = 3^x Series

18 Other Exponential Graphs This is the graph of f( x) 4 x It is a reflection of the graph of f( x) 4 x about the y axis It is always decreasing. It passes through (0,1). 18

19 Properties of Exponential Functions For a and b positive, a 1, b 1, and x and y real, 1. Exponent laws: a x a y a x y a x y a xy a x a y ax y ab x a x b x 2. a x = a y if and only if x = y a b x ax b x 3. For x 0, a x = b x if and only if a = b. 19

20 Base e Exponential Functions Of all the possible bases b we can use for the exponential function y = b x, probably the most useful one is the exponential function with base e. The base e is an irrational number, and, like π, cannot be represented exactly by any finite decimal fraction. However, e can be approximated as closely as we like by evaluating the expression 1 1 x x 20

21 Exponential Function With Base e x 1 1 x x The table to the left illustrates what happens to the expression 1 1 x as x gets increasingly larger. As we can see from the table, the values approach a number whose approximation is x 21

22 Exponential Function With Base e Exponential functions with base e and base 1/e, respectively, are defined by y = e x and y = e x Domain: (, ) Range: (0, ) 22

23 Growth & Decay Applications Relative Growth Rates Functions of the form y = ce kt, where c and k are constants and the independent variable t represents time, are often used to model population growth and radioactive decay. Note that if t = 0, then y = c. So, the constant c represents the initial population (or initial amount.) The constant k is called the relative growth rate. If the relative growth rate is k = 0.02, then at any time t, the population is growing at a rate of 0.02y persons (2% of the population) per year. We say that population is growing continuously at relative growth rate k to mean that the population y is given by the model y = ce kt. 23

24 Growth and Decay Applications: Cholera Bacteria The cholera bacterium multiplies exponentially. The number grows constinuously at a relative growth rate 1.386, that is: N = N 0 e 1.386t where N is the number of bacteria present after t hours and No is at the start (t = 0). How many bacteria will be present: A. In 0.6 hour? B. In 3.5 hours? If we start with 25 bateria, 24

25 Growth and Decay Applications: Cholera Bacteria The cholera bacterium multiplies exponentially. The number grows constinuously at a relative growth rate 1.386, that is: N = N 0 e 1.386t where N is the number of bacteria present after t hours and No is at the start (t = 0). If we start with 25 bateria, How many bacteria will be present: Solution: A. In 0.6 hour? N = 25e 1.386(0.6) N = 57 bacteria B. In 3.5 hours? N = 25e 1.386(3.5) N = 3,1977 bacteria 25

26 Growth and Decay Applications: Atmospheric Pressure The atmospheric pressure p decreases with increasing height. The pressure is related to the number of kilometers h above the sea level by the formula: P( h) h e Find the pressure at sea level (h = 0) Find the pressure at a height of 7 kilometers. 26

27 Growth and Decay Applications: Atmospheric Pressure The atmospheric pressure p decreases with increasing height. The pressure is related to the number of kilometers h above the sea level by the formula: P( h) h e P Solution: Find the pressure at sea level (h = 0) P(0) 760e Find the pressure at a height of 7 kilometers (7) (7) 760e

28 Depreciation of a Machine A machine is initially worth V 0 dollars but loses 10% of its value each year. Its value after t years is given by the formula V ( t) V (0.9 t ) Find the value after 8 years of a machine whose initial value is $30,

29 Depreciation of a Machine A machine is initially worth V 0 dollars but loses 10% of its value each year. Its value after t years is given by the formula V ( t) V (0.9 t ) 0 Solution: V ( t) V (0.9 t ) 0 8 V(8) 30000(0.9 ) $12,914 Find the value after 8 years of a machine whose initial value is $30,

30 Compound Interest The fee paid to use another s money is called interest. Interest paid on interest reinvested is called compound interest. The compound interest formula is A = P(1 + r m )mt Here, A is the future value of the investment, P is the initial amount (principal or future value), r is the annual interest rate as a decimal, m represents the number of compounding periods per year, and t is the number of years. We can say that the interest i = r/m 30

31 Compound Interest If a principal, P, is invested at an annual rate, r, and compounded continuously, then the amount in the account at the end of t years is given by A = Pe rt where the constant e is the base of the exponential function. This is called the Continuously Compound Interest Formula 31

32 Compound Interest Problem Find the amount to which $1500 will grow if deposited in a bank at 5.75% interest compounded quarterly for 5 years. 32

33 Compound Interest Problem Find the amount to which $1500 will grow if deposited in a bank at 5.75% interest compounded quarterly for 5 years. Solution: Use the compound interest formula: A = P (1 + r m ) mt Substitute P = 1500, r = , m = 4 and t = 5 to obtain A (4)(5) =$

34 Chapter 2 Functions and Graphs Section 5 Exponential Functions END Last Update: February 21/2013