91Ó°ÊÓ

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\cos x+\sqrt{3} \sin x, \quad 0 \leq x \leq 2 \pi$$

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow \infty} \frac{\ln (x+1)}{\log _{2} x}$$

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$g(x)=x e^{-x}, \quad-1 \leq x \leq 1$$

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$v=\frac{2}{\pi} \cos \frac{2 t}{\pi}, \quad s\left(\pi^{2}\right)=1$$

It costs you \(c\) dollars each to manufacture and distribute backpacks. If the backpacks sell at \(x\) dollars each, the number sold is given by $$n=\frac{a}{x-c}+b(100-x)$$ where \(a\) and \(b\) are positive constants. What selling price will bring a maximum profit?

Temperature change It took 14 sec for a mercury thermometer to rise from \(-19^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at the rate of \(8.5^{\circ} \mathrm{C} / \mathrm{sec}\).

You operate a tour service that offers the following rates: \(\$ 200\) per person if 50 people (the minimum number to book the tour) go on the tour. For each additional person, up to a maximum of 80 people total, the rate per person is reduced by \(\$ 2\) It costs \(\$ 6000\) (a fixed cost) plus \(\$ 32\) per person to conduct the tour. How many people does it take to maximize your profit?

Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. $$\int\left(e^{-x}+4^{x}\right) d x$$

Production level Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost.

a. Identify the function's local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? c. Support your findings with a graphing calculator or computer grapher. $$f(x)=\sqrt{x^{2}-2 x-3}, \quad 3 \leq x<\infty$$