91Ó°ÊÓ

$$\begin{gathered} \mathrm{Given}\mathrm{that}u=u\left(t\right),v=v\left(t\right),\mathrm{and}w=w\left(t\right)\mathrm{are}\mathrm{functions}\mathrm{of}t\mathrm{and}\mathrm{that}k\mathrm{is}\mathrm{a}\mathrm{constant}, \\ \mathrm{calculate}\mathrm{the}\mathrm{derivative}\mathrm{of}\mathrm{each}\mathrm{function}f\left(t\right)\mathrm{in}\mathrm{Exercises}27–36.\mathrm{Your}\mathrm{answers}\mathrm{may} \\ \mathrm{involve}u,v,w,\frac{du}{dt},\frac{dv}{dt},\frac{dw}{dt},k,and/ort.f\left(t\right)=u+v+w \end{gathered}$$

Use optimization techniques to answer the questions in Exercises 25–30.
Find the area of the largest rectangle that fits inside a circle of radius $10$.

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$to determine the intervals on which each function $\mathit{f}$is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{x}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}}$

Find the critical points of each function f .Then use a graphing utility to determine whether f has a local minimum, a local maximum, or neither at each of these critical points.

$f\left(x\right)=3x^4+8x^3-18x^2$

Calculate each of the limits in Exercises 21-48. Some of these limits are made easier by L'Hôpital's rule, and some are not.

$\underset{x→∞}{\lim }\left(e^x-\frac{e^x}{x+1}\right)$

Use the second-derivative test to determine the local extrema of each function $f$in Exercises $29-40$. If the second-derivative test fails, you may use the first-derivative test. Then verify your algebraic answers with graphs from a calculator or graphing utility. (Note: These are the same functions that you examined with the first-derivative test in Exercises $39-50$of Section $3.2$.)

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

$$\begin{gathered} \mathrm{Given}\mathrm{that}u=u\left(t\right),v=v\left(t\right),\mathrm{and}w=w\left(t\right)\mathrm{are}\mathrm{functions}\mathrm{of}t\mathrm{and}\mathrm{that}k\mathrm{is}\mathrm{a}\mathrm{constant}, \\ \mathrm{calculate}\mathrm{the}\mathrm{derivative}\mathrm{of}\mathrm{each}\mathrm{function}f\left(t\right)\mathrm{in}\mathrm{Exercises}27–36.\mathrm{Your}\mathrm{answers}\mathrm{may} \\ \mathrm{involve}u,v,w,\frac{du}{dt},\frac{dv}{dt},\frac{dw}{dt},k,and/ort.f\left(t\right)=tv+kv \end{gathered}$$

L’Hˆopital’s Rule limit calculations: Calculate each of the

limits that follow. Some of these limits are easier to calculate

by using L’Hopital’s rule, and some are not.

$\underset{x→∞}{\lim }\frac{\ln x}{\ln (x+1)}$

Evaluation in Leibniz notation: Given that $r=r\left(t\right),s=s\left(t\right)$and $u=u\left(t\right)$are functions of $t$, answer each of the following.

$\left(i\right)$If $\frac{ds}{dt}=3s^2-4,$find ${\overline{)\frac{ds}{dt}}}_{s=2}$

$\left(ii\right)$If $r^2\frac{dr}{dt}-2r=0,$find ${\overline{)\frac{dr}{dt}}}_{r=3}$

$\left(iii\right)$If$4=2u\frac{du}{dt}$andlocalid="1649780328523" $u=2+3t,$find${\overline{)\frac{du}{dt}}}_{t=4}$

Parametric curves: Imagine the curve traced in the xy-plane by

the coordinates (x, y) = (3z + 1, $z^2$− 4) as z varies, where the

parameter z is a function of time t.

If the parameter z moves at 3 units per second and

z = 0 when t = 0, plot the points (x, y) in the plane

that correspond to t = 0, 1, 2, 3, and 4.