9C Exponential Modeling

# 9C Exponential Modeling - 9C Exponential Modeling Wednesday...

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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 0 b rt A ( t ) = amount at time t A 0 = initial amount (i.e. amount present at time t = 0) b = base 0 <b , b ± =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 , (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

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2 so equation (1) becomes A ( t ) = \$4000 e 0 . 07 × t . (2) This is an exponential function with A 0 = \$4000, base b = e and rate r =0 . 07. To Fnd the balance after 5 years, simply plug t = 5 into equation (2) to get A (5) = \$4000 e 0 . 07 × 5 = \$5676 . 27 (b) Suppose interest is compounded monthly. ±ind 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 non-continuous compounding is A ( t )= P ± 1+ r n ² 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 . 07 n = 12, so equation (3) becomes A ( t ) = \$4000 ± 0 . 07 12 ² 12 × t . (4) This is an exponential function with A 0 = \$4000 ,b = 1 + 0 . 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) 0 . 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 ² - 1 =2 is greater than 1. (b) ( x y ) z = x yz . Using these properties, we can rewrite an exponential function with base 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 ² 3 t = ± 2 - 1 ² 3 t = (2) - 1 × 3 t - 3 t (b) Consider the function A ( t ) = 6(0 . 02) - t . Rewrite the function so that the base is greater than 1. Solution The given function has base b =0 . 02, which is less than 1. We rewrite using properties (a) and (b) form above as follows: A ( t ) = 6(0 . 02) - t =6 ± ( 1 0 . 02 ) - 1 ² - t ± 1 0 . 02 ² - 1 ×- t = 6(50) t Graphing Now that we can write exponential functions so that the base is greater than 1, we are ready to graph exponential functions. There are two basic shapes of exponential functions, the “growth” shape and the “decay” shape. Here is how they look:

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9C Exponential Modeling - 9C Exponential Modeling Wednesday...

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