Homework 7 Solutions - Assignment 7 Math for Economics I(28 points Due Friday October 31st in recitation Write neat solutions for the problems below

# Homework 7 Solutions - Assignment 7 Math for Economics I(28...

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Assignment 7, Math for Economics I (28 points) Due Friday, October 31st, in recitation Write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you will not be given any credit. Please do not forget to: Write your name and recitation section number. Staple your homework if you have multiple pages! 1. (3 pts) Find the linearization of f ( x ) = 1 (1 + x ) 4 at a = 0, and use it to estimate 1 (0 . 97) 4 . Solution : f 0 ( x ) = - 4 (1 + x ) 5 , so f 0 (0) = - 4, and L ( x ) = f (0) + f 0 (0)( x - 0) = 1 - 4 x . Our approximation is 1 (0 . 97) 4 L ( - 0 . 03) = 1 + 4 · 0 . 03 = 1 . 12. 2. (7 pts total) (a) (2 pts) Find the linearization of ln( x ) at a = 1. (b) (2 pt) Use this linearization to approximate ln(1 . 03) and ln(1 . 8). Using a calculator, compute ln(1 . 03) and ln(1 . 8). (c) (3 pts) Which approximation of ln( x ) in part (b) was closest to the actual value of ln( x )? Justify why we should have suspected this to be the case even if you hadn’t been able to compare your approximations in part (b) with their actual values. Solution : (a) The linearization of f ( x ) = ln( x ) at a = 1. Since f 0 ( x ) = 1 x , we get L ( x ) = f (1) + f 0 (1)( x - 1) , L ( x ) = ln(1) + 1 1 ( x - 1) = 0 + ( x - 1) , L ( x ) = x - 1 . (b) Using the linearization we get ln(1 . 03) L (1 . 03) = 1 . 03 - 1 = 0 . 03 , ln(1 . 8) L (1 . 8) = 1 . 8 - 1 = 0 . 8 . A calculator gives ln(1 . 03) = 0 . 29558 ... , while ln(1 . 8) 0 . 587786 ... . (c) Our approximation of ln(1 . 03) was closer than our approximation of ln(1 . 8). This fits our intuition: the tangent line to a function when a = 1 usually approximates the function for x
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