91Ó°ÊÓ

Suppose the radius $r$, volume $V$, and surface area $S$of a sphere are functions of time $t$.

(a) How are $\frac{dV}{dt}$and $\frac{dr}{dt}$ related?

(b) How are $\frac{dS}{dt}$ and$\frac{dr}{dt}$ related?

Find the error in the following incorrect calculation, and then calculate the limit correctly:

$$\begin{gathered} \underset{x→0}{\lim }\frac{x^2-x}{2^x-1} \\ =\underset{x→0}{\lim }\frac{2x-1}{\left(\ln 2\right)2^x} \\ =\underset{x→0}{\lim }\frac{2}{{\left(\ln 2\right)}^2{2}^x} \\ =\frac{2}{{\left(\ln 2\right)}^2{2}^0} \\ =\frac{2}{{\left(\ln 2\right)}^2} \end{gathered}$$

If$\underset{x→c}{\lim }\ln \left(f\right(x\left)\right)=\_\_\_\_$, then$\underset{x→c}{\lim }f\left(x\right)=∞$

Find the critical points of the function

$f^{\text{'}}\left(x\right)=\sqrt{1+x^2}-4$

Describe in words, and then illustrate in pictures, the four types of arcs that are the building blocks of most continuous, differentiable graphs.

Find the error in the following incorrect calculation. Then

calculate the limit correctly.

$$\begin{gathered} \underset{x→∞}{\lim }\frac{e^{-x}}{x^2} \\ =\underset{x→∞}{\lim }\frac{-e^{-x}}{2x} \\ =\underset{x→∞}{\lim }\frac{e^{-x}}{2} \\ =\frac{0}{2} \\ =0 \end{gathered}$$

Find all roots, local maxima and minima, and inflection points of each function f. In addition, determine whether any local extrema are also global extrema on the domain of f.

$f\left(x\right)=x^{\frac{4}{3}}-x^{\frac{1}{3}}$

Find the critical points of the function

$f^{\text{'}}\left(x\right)=\frac{(x-1)(x-2)}{x-3}$

If $\underset{x→c}{\lim }\ln \left(f\right(x\left)\right)=\_\_\_\_$, then$\underset{x→c}{\lim }f\left(x\right)=0$

Find the locations and values of any global extrema of each function f in Exercises 11–20 on each of the four given intervals. Do all work by hand by considering local extrema and endpoint behavior. Afterwards, check your answers with graphs.

$f\left(x\right)=\frac{3x-2}{x-1}$on the intervals

$$\begin{gathered} \left(a\right)[0,2] \\ \left(b\right)[-2,0] \\ \left(c\right)[-1,1] \\ \left(d\right)(-1,1) \end{gathered}$$