Problem 73 Medium Difficulty

Linear approximation and the second derivative Draw the graph of a function $f$ such that $f(1)=f^{\prime}(1)=f^{\prime \prime}(1)=1$ Draw the linear approximation to the function at the point
(1,1). Now draw the graph of another function $g$ such that $g(1)=g^{\prime}(1)=1$ and $g^{\prime \prime}(1)=10 .$ (It is not possible to represent the second derivative exactly, but your graphs should reflect the fact that $f^{\prime \prime}(1)$ is relatively small compared to $g^{\prime \prime}(1) .$ ) Now suppose linear approximations are used to approximate $f(1.1)$ and $g(1.1)$
a. Which function has the more accurate linear approximation near $x=1$ and why?
b. Explain why the error in the linear approximation to $f$ near a point $a$ is proportional to the magnitude of $f^{\prime \prime}(a)$

Answer

a) From the figure, $f(x)$ has the more accurate approximation at $x=1$
b) The error in the linear approximation to $f$ near a point a is proportional to the magnitude of $f^{\prime \prime}(a)$ because the curvature decreses and become similar to the tangent at points close to $a$ If the value of $f^{\prime \prime}(x)$ decreases.

Related Courses

Calculus 1 / AB

Calculus Early Transcendentals

Chapter 4

Applications of the Derivative

Section 6

Linear Approximation and Differentials

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Video Transcript

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