Chapter 2 Functions and Graphs Section 5 Exponential Functions Exponential Function The equation x f (x) b , 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. 2
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? 3 Solution Complete the table: Day No. pennies 1
1 2 2 2^1 3 4 2^2 4 8 2^3
5 16 ... 6 32 7 64 4 Solution (continued) Now, if this pattern continued, how many pennies would you have on Dec. 31? Your answer should be 230 (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 230? 1,073,741,824 pennies!!! Move the decimal point two places to the left to find the amount in dollars. You should get: $10,737,418.24 5 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. 6 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: 7 Table of values x
y f (x) 2 x y 4 24 = 1/24 = 1/16 3 23 = 1/8 2 22 = 1/4 1 21 = 1/2
0 20 = 1 1 21 = 2 2 22 = 4 8 Basic Properties of the Graph of x f (x) b , b 0, b 1
All graphs will pass through (0,1) (y intercept) All graphs are continuous curves, with no holes of jumps. The x axis is a horizontal asymptote. If b > 1, then bx increases as x increases. If 0 < b < 1, then bx decreases as x increases. 9 1 f ( x) 2 x Graph of 2 x Using a table of values, you will obtain the following graph.
x f ( x ) b The graphs of and f ( x) b x will be reflections of each other about the y-axis, in general. 12 10 8 graph of y = 2^(-x) 6 approaches the positive x-axis as x gets large 4 2 passes through (0,1)
0 -4 -2 0 2 4 10 Graphing Other Exponential Functions Now, lets 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. 11 Preliminary Graph of f ( x) 3 x 12 Complete Graph 30 y = 3^x
25 20 15 Series1 10 5 0 -4 -2 0 2 4 13
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). 14 Properties of Exponential Functions For a and b positive, a 1, b 1, and x and y real, 1. Exponent laws: x x
y a a a y ax x y a xy a x y a y a x ab a x bx
x a ax b bx 2. ax = ay if and only if x = y 3. For x 0, ax = bx if and only if a = b. 15 Base e Exponential Functions Of all the possible bases b we can use for the exponential function y = bx, probably the most useful one is the exponential function with base e. The base e is an irrational number, and, like , 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 16 Exponential Function With Base e 1 1 x x The table to the left
illustrates what happens to the expression x 1 2 10 2.59374246 100 2.704813829 1000 10000 1000000
1 1 x x as x gets increasingly larger. 2.716923932 As we can see from the table, the values approach a 2.718145927 number whose 2.718280469 approximation is 2.718 17 Exponential Function With Base e y = ex and y = ex
Domain: (, ) Range: (0, ) 18 Relative Growth Rates Functions of the form y = cekt, 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 = cekt. 19
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) 760e Find the pressure at sea level (h = 0) Find the pressure at a height of 7 kilometers. 0.145 h
20 Solution Find the pressure at sea level (h = 0) P (0) 760e 0 760 Find the pressure at a height of 7 kilometers P (7) 760e 0.145(7) 275.43 21 Depreciation of a Machine A machine is initially worth V0 dollars but loses 10% of its value each year. Its value after t years is given by the formula t
V (t ) V0 (0.9 ) Find the value after 8 years of a machine whose initial value is $30,000. 22 Depreciation of a Machine A machine is initially worth V0 dollars but loses 10% of its value each year. Its value after t years is given by the formula Solution: V (t ) V0 (0.9t ) V (8) 30000(0.98 ) $12,914
t V (t ) V0 (0.9 ) Find the value after 8 years of a machine whose initial value is $30,000. 23 Compound Interest The compound interest formula is r A P 1 n nt 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, n represents the number of compounding periods per year, and t is the number of years 24 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. 25 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: r A P 1
n nt Substitute P = 1500, r = 0.0575, n = 4 and t = 5 to obtain 0.0575 A 1500 1 4 (4)(5) =$1995.55 26
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