01 Linear approximation
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School
CUNY College of Staten Island**We aren't endorsed by this school
Course
MTH 232
Subject
Mathematics
Date
Dec 17, 2024
Pages
7
Uploaded by bingmo92
Topic: Linear approximationQuestion: Find the linear approximation of the function at a= 0.f(x) = cosxAnswer choices:A L(x) = 1B L(x) =xC L(x) =x-1D L(x) =-x381
Solution: ATake the derivative.f(x) = cosxf(x) =-sinxEvaluate the original function at a= 0.f(0) = cos 0f(0) = 1Evaluate the derivative at a= 0.f(0) =-sin 0f(0) = 0Substitute all of these pieces into the linear approximation formula.L(x) =f(a) +f(a)(x-a)L(x) = 1 + 0(x-0)L(x) = 1382
Topic: Linear approximationQuestion: Use linear approximation to estimate e-0.1.Answer choices:A 0.1B 0C 0.9D 1.1383
Solution: CGiven the value of the function we’re asked to estimate, it’s clear that the function should be ex. Instead of trying to find f(-0.1), let’s use a linear approximation equation and a= 0to get an approximation for f(-0.1). Take the derivative.f(x) =exf(x) =exEvaluate the original function at a= 0.f(0) =e0f(0) = 1Evaluate the derivative at a= 0.f(0) =e0f(0) = 1Substitute all of these pieces into the linear approximation formula.L(x) =f(a) +f(a)(x-a)L(x) = 1 + 1(x-0)L(x) = 1 +xNow that we’ve built the linear approximation equation, we can substitute x=-0.1.384
L(-0.1) = 1-0.1L(-0.1) = 0.9385
Topic: Linear approximationQuestion: Find the linear approximation of the function at a= 2.f(x) = (x+ 4)2Answer choices:A L(x) = 1 +xB L(x) = 12 + 12xC L(x) =-12-12xD L(x) = 1-x386
Solution: BTake the derivative.f(x) = (x+ 4)2f(x) = 2(x+ 4)(1)f(x) = 2x+ 8Evaluate the original function at a= 2.f(2) = (2 + 4)2f(2) = 36Evaluate the derivative at a= 2.f(2) = 2(2) + 8f(2) = 12Substitute all of these pieces into the linear approximation formula.L(x) =f(a) +f(a)(x-a)L(x) = 36 + 12(x-2)L(x) = 36 + 12x-24L(x) = 12 + 12x387