91Ó°ÊÓ

Watching a Ferris wheel An observer stands \(20 \mathrm{m}\) from the bottom of a Ferris wheel on a line that is perpendicular to the face of the wheel, with her eyes at the level of the bottom of the wheel. The wheel revolves at a rate of \(\pi \mathrm{rad} / \mathrm{min}\), and the observer's line of sight with a specific seat on the Ferris wheel makes an angle \(\theta\) with the horizontal (see figure). At what time during a full revolution is \(\theta\) changing most rapidly?

Suppose you own a tour bus and you book groups of 20 to 70 people for a day tour. The cost per person is \$30 minus \(\$ 0.25\) for every ticket sold. If gas and other miscellaneous costs are \(\$ 200,\) how many tickets should you sell to maximize your profit? Treat the number of tickets as a nonnegative real number.

All rectangles with an area of 64 have a perimeter given by \(P(x)=2 x+128 / x,\) where \(x\) is the length of one side of the rectangle. Find the absolute minimum value of the perimeter function on the interval \((0, \infty) .\) What are the dimensions of the rectangle with minimum perimeter?

Determine whether the following statements are true and give an explanation or counterexample. a. The function \(f(x)=\sqrt{x}\) has a local maximum on the interval \([0, \infty)\). b. If a function has an absolute maximum on a closed interval, then the function must be continuous on that interval. c. A function \(f\) has the property that \(f^{\prime}(2)=0 .\) Therefore, \(f\) has a local extreme value at \(x=2\). d. Absolute extreme values of a function on a closed interval always occur at a critical point or an endpoint of the interval.

Sketch a graph of a function \(f\) that is continuous on \((-\infty, \infty)\) and has the following properties. $$\begin{aligned} &f^{\prime}(x)<0 \text { and } f^{\prime \prime}(x)<0 \text { on }(-\infty, 0) ; f^{\prime}(x)<0 \text { and }\\\ &f^{\prime \prime}(x)>0 \text { on }(0, \infty) \end{aligned}$$

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$f(x)=x^{4}-2 x^{3}+1$$

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$f(x)=-x^{4}-2 x^{3}+12 x^{2}$$

The pressure \(P\), temperature \(T\), and volume \(V\) of an ideal gas are related by \(P V=n R T\), where \(n\) is the number of moles of the gas and \(R\) is the universal gas constant. For the purposes of this exercise, let \(n R=1 ;\) therefore \(P=T / V\) a. Suppose that the volume is held constant and the temperature increases by \(\Delta T=0.05 .\) What is the approximate change in the pressure? Does the pressure increase or decrease? b. Suppose that the temperature is held constant and the volume increases by \(\Delta V=0.1 .\) What is the approximate change in the pressure? Does the pressure increase or decrease? c. Suppose that the pressure is held constant and the volume increases by \(\Delta V=0.1 .\) What is the approximate change in the temperature? Does the temperature increase or decrease?

a. Find the critical points of \(f\) on the given interval. b. Determine the absolute extreme values of \(f\) on the given interval. c. Use a graphing utility to confirm your conclusions. $$f(x)=x^{3} e^{-x} \text { on }[-1,5]$$

A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume that gasoline costs \(\$ p /\) gallon and the vehicle gets \(g\) miles per gallon. Also assume that the driver earns \(\$ w /\) hour. a. A plausible function to describe how gas mileage (in mi/gal) varies with speed \(v\) is \(g(v)=v(85-v) / 60 .\) Evaluate \(g(0)\) \(g(40),\) and \(g(60)\) and explain why these values are reasonable. b. At what speed does the gas mileage function have its maximum? c. Explain why the formula \(C(v)=L p / g(v)+L w / v\) gives the cost of the trip in dollars, where \(L\) is the length of the trip and \(v\) is the constant speed. Show that the dimensions are consistent. d. Let \(L=400 \mathrm{mi}, p=\$ 4 /\) gal, and \(w=\$ 20 / \mathrm{hr} .\) At what (constant) speed should the vehicle be driven to minimize the cost of the trip? e. Should the optimal speed be increased or decreased (compared with part (d)) if \(L\) is increased from \(400 \mathrm{mi}\) to \(500 \mathrm{mi}\)? Explain. f. Should the optimal speed be increased or decreased (compared with part (d)) if \(p\) is increased from \(\$ 4 /\) gal to \(\$ 4.20 /\) gal? Explain. g. Should the optimal speed be increased or decreased (compared with part (d)) if \(w\) is decreased from \(\$ 20 /\) hr to \(\$ 15 /\) hr? Explain.