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Assignment 7, Math for Economics I (28 points)Due Friday, October 31st, in recitationWrite neat solutions for the problems below. Show all your work. If you only write the answer with no work, you willnot 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 off(x) =1(1 +x)4ata= 0, and use it to estimate1(0.97)4.Solution:f0(x) =-4(1 +x)5, sof0(0) =-4, andL(x) =f(0) +f0(0)(x-0) = 1-4x.Our approximation is1(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) ata= 1.(b) (2 pt) Use this linearization to approximate ln(1.03) and ln(1.8). Using a calculator, compute ln(1.03) andln(1.8).(c) (3 pts) Which approximation of ln(x) in part (b) was closest to the actual value of ln(x)? Justify why weshould have suspected this to be the case even if you hadn’t been able to compare your approximations inpart (b) with their actual values.Solution: (a) The linearization off(x) = ln(x) ata= 1. Sincef0(x) =1x, we getL(x) =f(1) +f0(1)(x-1),L(x) = ln(1) +11(x-1) = 0 + (x-1),L(x) =x-1.(b) Using the linearization we getln(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:thetangent line to a function whena= 1 usually approximates the function forx