Linearization (College Board AP® Calculus BC): Exam Questions

The function $f\left(x\right)$ is approximated for values close to $x=-0.5$ using a linear function, $g\left(x\right)$. The graph of $g\left(x\right)$ is a tangent to the graph of $f\left(x\right)$ at $x=-0.5$.

It is known that $f^{\text{'}\text{'}}\left(x\right)>0$ for all values of $x$.

Which of the statements below are true?

I. $g\left(-0.6\right)$ will be an overestimate for $f\left(-0.6\right)$

II. $g\left(-0.6\right)$ will be an underestimate for $f\left(-0.6\right)$

III. $g\left(-0.6\right)=f\left(-0.6\right)$

A function $f$ is defined as $f\left(x\right)=3\sin \frac{x}{2}$.

Find the equation of a linear function which approximates $f$ for values close to $x=1.5$.

Find the equation of the function which is a linearization of $y=\sqrt{x}$ at an integer value of $x$, which could be used to approximate the value of $\sqrt{101}$.

Find the equation of the function which is a linearization of $y=\sqrt[3]{x}$ at an integer value of $x$, which could be used to approximate $\sqrt[3]{8.1}$.

When a tangent to the graph of $f\left(x\right)$ is drawn at $x=0$ and used to approximate $f\left(0.1\right)$, for which of the following functions will this be an overestimate?

A linear approximation for $y=\cos x$ is created using a tangent to $y=\cos x$ at the point $\left(\frac{\pi }{4},\frac{\sqrt{2}}{2}\right)$. Find the percentage error when this linear approximation is used to approximate $\cos \frac{7\pi }{32}$.

A tangent to the graph of $f\left(x\right)=\sin x$ at $x=0$ is used to estimate the value of $\sin x$ at values close to 0. Which of the following statements is true for the estimate of $f\left(0.2\right)$?

I. It will be an underestimate

II. It will be an overestimate

III. It will be the exact value

Consider the cubic function $f\left(x\right)=x^3-3x^2-9x+27$.

Several different tangents are created as linear approximations for $f\left(x\right)$, and then used to approximate the value of $f\left(x\right)$ nearby points.

Which one of the following statements is false?