Exam 12: Applications of the Derivative

Solve the problem. -A company wishes to manufacture a box with a volume of 16 cubic feet that is open on top and is twice as long as it is wide. Find the width to the nearest foot of the box that can be produced using the minimum amount of material.

$\mathrm{s}$ is the distance (in $\mathrm{ft}$ ) traveled in time $\mathrm{t}$ (in s) by a particle. Find the velocity and acceleration at the given time. - $s=-5 t^{3}+3 t^{2}-5 t-4, t=2$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f(x)=0.1 x^{5}+5 x^{4}-8 x^{3}-15 x^{2}-6 x-35$

$\mathrm{s}$ is the distance (in $\mathrm{ft}$ ) traveled in time $\mathrm{t}$ (in s) by a particle. Find the velocity and acceleration at the given time. - $s=\frac{1}{t+2}, t=5$

Solve the problem. -The energy cost of a speed burst as a function of the body weight of a dolphin is given by $\mathrm{E}=43.5 \mathrm{w}^{-0.61}$ , where $\mathrm{w}$ is the weight of the dolphin (in $\mathrm{kg}$ ) and $\mathrm{E}$ is the energy expenditure (in $\mathrm{kcal} / \mathrm{kg} / \mathrm{km}$ ). Suppose that the weight of a $400-\mathrm{kg}$ dolphin is increasing at a rate of $8 \mathrm{~kg} / \mathrm{day}$ . Find the rate at which the energy expenditure is changing with respect to time.

Determine the location of each local extremum of the function. - $f(x)=x^{3}+\frac{7}{2} x^{2}+12 x+4$

Use the maximum/minimum finder on a graphing calculator to determine the approximate location of all local extrema. - $f(x)=x^{4}-3 x^{3}-21 x^{2}+74 x-2$

Solve the problem. -From a thin piece of cardboard 40 in. by 40 in., square corners are cut out so that the sides can be folded up to make a box. What dimensions will yield a box of maximum volume? What is the maximum volume? Round to the nearest tenth, if necessary.

Solve the problem. -The cost of a computer system increases with increased processor speeds. The cost $C$ of a system as a function of processor speed is estimated as $C=14 S^{2}-5 S+1800$ , where $S$ is the processor speed in MHz. Find the processor speed for which cost is at a minimum. Round to the nearest tenth if necessary.

Use the first derivative test to determine the location of each local extremum and the value of the function at thatextremum. - $f(x)=(4-2 x)^{3 / 5}-4$

Solve each problem. -The price $P$ of a certain computer system decreases immediately after its introduction and then increases. If the price $\mathrm{P}$ is estimated by the formula $\mathrm{P}=130 \mathrm{t}^{2}-2500 t+6900$ , where $\mathrm{t}$ is the time in months from its introduction, find the time until the minimum price is reached.

Solve the problem. -It is estimated that the total value of a stamp collection is given by the formula $v=46,300+100 t^{2}$ , 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 an absolute maximum at $(0,46,300)$ , and a local maximum at one other point. What is the value of $t$ at the local maximum? What is the discounted value of the collection at that time?

The graph of the derivative function $f^{\prime}$ is given. Find the critical numbers of the function $f$ . -

Solve the problem. -The cost function for the manufacture of graphing calculators is given by $C(x)=100,000+20 x+\frac{x^{2}}{10,000}$ , where $x$ is the number of graphing calculators manufactured. Using the appropriate domain, sketch the graph of the average cost $\bar{C}$ to manufacture $x$ graphing calculators. What is the significance of $\lim _{x \rightarrow \infty} \bar{C}$ ?

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

Solve each problem. -If the price charged for a bolt is $p$ cents, then $x$ thousand bolts will be sold in a certain hardware store, where $p=125-\frac{x}{14}$ . How many bolts must be sold to maximize revenue?

Use the first derivative test to determine the location of each local extremum and the value of the function at thatextremum. - $f(x)=3 x e^{-x}$

Sketch the graph and show all local extrema and inflection points. - $f(x)=\frac{4 x}{x^{2}+1}$

Find the coordinates of the points of inflection for the function. - $f(x)=\frac{5 x}{x^{2}+49}$

Find dy/dx by implicit differentiation. - $x^{3}+3 x^{2} y+y^{3}=8$