Science:Math Exam Resources/Courses/MATH104/December 2012/Question 06 (c)/Solution 1

The linear approximation formula is

${\displaystyle }L(x)=f(a)+f^{\prime }(a)(x-a).$

The number a is the value we are approximating around and thus in our case a=1. Since we are trying to approximate f(1.02) we take x=1.02. Using the information from part (a) we have that

${\displaystyle }f(1.02)\approx L(1.02)=f(1)+f^{\prime }(1)(1.02-1)=1+0(0.02)=1.$

This gives the estimate that $f(1.02)=1$. From part (b), the second derivative at x=1 is negative and therefore we see that the function is concave down at this point. This tells us that points near x=1 are smaller than what the tangent line predicts and therefore we expect that our estimate is an overestimate.