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Unformatted text preview: 9C: Exponential Modeling Wednesday February 23, 2011 All functions in this section have domains and ranges consisting of numbers. Basics General Form of an Exponential Function A ( t ) = A b rt A ( t ) = amount at time t A = initial amount (i.e. amount present at time t = 0) b = base 0 < b , b 6 = 1 r = growth or decay rate t = time Example 1. Compound Interest Consider an account with initial principal $ 4000 and annual interest rate 7%. (a) Suppose interest is compounded continuously. Find the balance of the account as a function of time t where t = 0 corresponds to the initial deposit date. Find the balance of the account after 5 years. Solution The compound interest formula for continuously compounded interest is A ( t ) = Pe rt , (1) where A ( t ) is the account balance at time t , P is the initial deposit amount, e 2 . 1718, r is the yearly interest rate and t is time (measured in years). Using the given information, we have P = $4000 r = 0 . 07 , 1 2 so equation (1) becomes A ( t ) = $4000 e . 07 t . (2) This is an exponential function with A = $4000, base b = e and rate r = 0 . 07. To find the balance after 5 years, simply plug t = 5 into equation (2) to get A (5) = $4000 e . 07 5 = $5676 . 27 (b) Suppose interest is compounded monthly. Find the balance of the account as a func tion of time t where t = 0 corresponds to the initial deposit date. Solution The formula for noncontinuous compounding is A ( t ) = P 1 + r n nt , (3) where P is the initial deposit amount, r is the yearly interest rate, n is the number of compounding periods per year, and t is time (measured in years). Using the given information, we have P = $4000 r = 0 . 07 n = 12, so equation (3) becomes A ( t ) = $4000 1 + . 07 12 12 t . (4) This is an exponential function with A = $4000 , b = 1 + . 07 12 1 . 0058 r = 12 . To get the balance after 5 years, simply substitute t = 5 into (4) to get A (5) = $4000(1 . 0058) . 07 5 = $5670 . 50 Graphing Exponential Functions Before we can graph exponential functions, we need to learn how to rewrite a given expo nential function (if necessary) so that the base is greater than 1. Rewriting exponential functions Recall the following two properties 3 (a) If b < 1 is any given number then b 1 > 1. For example, Example 2. Suppose b = 1 2 . Since 1 2 is less than 1, 1 2 1 = 2 is greater than 1. (b) ( x y ) z = x yz . Using these properties, we can rewrite an exponential function with base b < 1 so that the base is greater than 1. As will be illustrated below, this changes the sign of the rate. Example 3. (a) Consider the function A ( t ) = (1 / 2) 3 t . Rewrite the function so that the base is greater than 1. Solution The given function has base b = 1 / 2, which is less than 1. We rewrite using properties (a) and (b) form above as follows: A ( t ) = 1 2 3 t = 2 1 3 t = (2) 1 3 t = 2 3 t (b) Consider the function A ( t ) = 6(0 . 02) t . Rewrite the function so that the base is greater than 1....
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 Spring '08
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