Exam 12: Applications of the Derivative

Solve the problem. -A company is constructing an open-top, square-based, rectangular metal tank that will have a volume of $56.5 \mathrm{ft}^{3}$ . What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.

Solve the problem. -The position of a particle at time $t$ is given by $s$ , where $s^{3}+12 s t+4 t^{3}-5 t=0$ . Find the velocity $\mathrm{ds} / \mathrm{dt}$ .

If the price charged for a bolt is $p$ cents, then $x$ thousand bolts will be sold in a certain hardware store, where $\mathrm{p}=89-\frac{\mathrm{x}}{28}$ . How many bolts must be sold to maximize revenue?

Solve the problem. -It is estimated that the total value of a stamp collection is given by the formula $v=44,700+100 t^{2}$ , where $t$ is the number of years from now. If the inflation rate is running continuously at $4 \%$ per year so that the (discounted) present value of an item that will be worth $\$ v$ in $t$ years' time is given by $\mathrm{p}=\mathrm{ve}^{-.04 \mathrm{t}}$ . Sketch the graph of the discounted value as a function of time at which the stamp collection is sold. The graph has one local minimum. What is the value of $t$ at the local minimum? What is the discounted value of the collection at that time?

Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither. - $f(x)=\frac{x+5}{x-8}$

Identify the intervals where the function is changing as requested. -Decreasing

Solve the problem. -Find the acceleration function $a(t)$ if $s(t)=-9 t^{3}-2 t^{2}-6 t-7$ .

Find $\frac{d y}{d x}$ at the given point. - $6 x \ln y=11 ;(1, e)$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f(x)=x^{5}-15 x^{4}-3 x^{3}-172 x^{2}+135 x+0.002$

Find the largest open interval where the function is changing as requested. -Increasing $f(x)=\frac{1}{x^{2}+1}$

Find the location of the indicated absolute extrema for the function. -Maximum

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f(x)=0.1 x^{3}-15 x^{2}+46 x-86$

Find the absolute extremum within the specified domain. -Maximum of $f(x)=x^{2}-4 ;[-1,2]$

Evaluate $\mathrm{f}^{\prime \prime}(\mathrm{c})$ at the point. - $f(x)=e^{-x}, c=0$

Find dy/dx by implicit differentiation. - $x y+x=2$

Find the location of the indicated absolute extrema for the function. -Maximum

Find $\frac{d y}{d x}$ at the given point. - $6 x e^{5 y}=15 ;(1,0)$

Assume $x$ and $y$ are functions of $t$ . Evaluate $d y / d t$ . - $\mathrm{x}^{3}+\mathrm{y}^{3}=9 ; \frac{\mathrm{dx}}{\mathrm{dt}}=-3, \mathrm{x}=1, \mathrm{y}=2$

Determine the location of each local extremum of the function. - $f(x)=\frac{x^{4}}{4}-\frac{4}{3} x^{3}-\frac{9}{2} x^{2}+36 x+4$

Find $\frac{d y}{d x}$ at the given point. - $x^{2}+3 y^{2}=13 ;(1,2)$