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148Lab3Wi09

# 148Lab3Wi09 - Math 148 Winter 2009 Lab Assignment...

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Math 148 - Winter 2009 Lab Assignment 3 (v0.11038) Exponential Models Due: February 16, 2009 Background Directions: Work out each problem and write the required answers and explana- tions clearly on the answer sheet on the last page of this lab. Submit the answer sheet to your recitation instructor. Material from this lab will appear on the next exam. Purpose: In Lab 2 you found the equation of a quadratic function by using given points on the graph, employing one of two equivalent forms: (intercept form) y = C ( x - z 1 )( x - z 2 ) ; or (vertex form) y = A ( x - h ) 2 + k . In Lab 3 you will determine an exponential function from given points on its graph, employing one of two equivalent forms for such functions: (base a) f ( x ) = Pa x and (base e) f ( t ) = Pe rt . These are the basic forms used in Chapter 5 to model exponential growth and decay. 1

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2 Part I. Two Forms for Exponentials: f ( x ) = Pa x and f ( x ) = Pe rx . (1) Assume that f ( x ) is an exponential function with f ( 0 ) = 5 and f ( 6 ) = 23 . (a) Write down the co¨ordinates of two points that you know are on the graph of f ( x ) . (b) Let’s use the form f ( x ) = Pa x to find an equation for this exponential function. (i) Find the value of P . (ii) Find the exact value of a . Round this value to the nearest hun- dredth. (iii) Using the value of P and the rounded value of a , write the equa- tion in the form f ( x ) = Pa x . (c) Now let’s use the form f ( x ) = Pe rx to find another equation for this exponential function. (i) Find the value of P . (ii) Find the exact value of r . Round this value to the nearest hun- dredth. (iii) Using the value of P and the rounded value of r , write the equa- tion in the form f ( x ) = Pe rx . (d) (i) Use the equation in (b)(iii) to find f ( 16 ) . Round your answer to the nearest hundredth. (ii) Use the equation in (c)(iii) to find f ( 16 ) . Round your answer to the nearest hundredth. (iii) Because Pa x and Pe rx are two forms of the same exponential func- tion, one would expect to obtain the same answer for f ( 16 ) in parts (i) and (ii). Briefly explain why in practice this might not happen. 1 (e) The two formulas f ( x ) = Pa x and f ( x ) = Pe rx provide two models for the same exponential growth or decay function. How are the exact values of a and r related? 2 1 For a hint , see the next section. 2 Hint : find a formula for a as a function of r .
3 Part II. Rounding Issues. Inappropriate rounding leads to inaccurate answers. Here’s the sample problem we will analyze: Suppose f ( x ) is an exponential function with f ( 0 ) = 3 and f ( 4 ) = 109 . Find the value f ( 15 ) .

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148Lab3Wi09 - Math 148 Winter 2009 Lab Assignment...

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