Chapter 4

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. $$f(t)=12 t-t^{3}, \quad-3 \leq t<\infty$$

Use I'Hópital's rule to find the limits. $$\lim _{\theta \rightarrow 0} \frac{\theta-\sin \theta \cos \theta}{\tan \theta-\theta}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{e^{x}}{x}$$

Determine all critical points for each function. $$f(x)=\frac{x^{2}}{x-2}$$

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. $$\int \frac{2}{5} \sec \theta \tan \theta d \theta$$

Give the acceleration \(a=d^{2} s / d t^{2},\) initial velocity. and initial position of an object moving on a coordinate line. Find the object's position at time \(t\). $$a=\frac{9}{\pi^{2}} \cos \frac{3 t}{\pi}, \quad v(0)=0, \quad s(0)=-1$$

Airplane landing path An airplane is flying at altitude \(H\) when it begins its descent to an airport runway that is at horizontal ground distance \(L\) from the airplane, as shown in the figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function \(y=a x^{3}+b x^{2}+c x+d,\) where \(y(-L)=H\) and \(y(0)=0\) a. What is \(d y / d x\) at \(x=0 ?\) b. What is \(d y / d x\) at \(x=-L ?\) c. Use the values for \(d y / d x\) at \(x=0\) and \(x=-L\) together with \(y(0)=0\) and \(y(-L)=H\) to show that $$y(x)=H\left[2\left(\frac{x}{L}\right)^{3}+3\left(\frac{x}{L}\right)^{2}\right]$$ GRAPH CANT COPY

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(t)=t^{3}-3 t^{2}, \quad-\infty

Use I'Hópital's rule to find the limits. $$\lim _{x \rightarrow 0} \frac{\sin 3 x-3 x+x^{2}}{\sin x \sin 2 x}$$

Determine all critical points for each function. $$y=x^{2}-32 \sqrt{x}$$

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. $$\int\left(e^{3 x}+5 e^{-x}\right) d x$$

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 \(8.5^{\circ} \mathrm{C} / \mathrm{sec}\).

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?

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. $$h(x)=\frac{x^{3}}{3}-2 x^{2}+4 x, \quad 0 \leq x<\infty$$

Find the limits. $$\lim _{x \rightarrow 1^{-}} x^{1 /(1-x)}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x(\ln x)^{2}$$

Determine all critical points for each function. $$g(x)=\sqrt{2 x-x^{2}}$$

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. $$\int\left(2 e^{x}-3 e^{-2 x}\right) d x$$

A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why?

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?

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. $$k(x)=x^{3}+3 x^{2}+3 x+1, \quad-\infty

Find the limits. $$\lim _{x \rightarrow 1^{+}} x^{1 /(x-1)}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=e^{x}-2 e^{-x}-3 x$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=2 x^{2}-8 x+9$$

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. $$\int\left(e^{-x}+4^{x}\right) d x$$

Classical accounts tell us that a 170 -oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme's speed exceeded 7.5 knots (sea or nautical miles per hour).

Wilson lot size formula One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is $$A(q)=\frac{k m}{q}+c m+\frac{h q}{2}$$ where \(q\) is the quantity you order when things run low (shoes, radios, brooms, or whatever the item might be), \(k\) is the cost of placing an order (the same, no matter how often you order), \(c\) is the cost of one item (a constant), \(m\) is the number of items sold each week (a constant), and \(h\) is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security). a. Your job, as the inventory manager for your store, is to find the quantity that will minimize \(A(q) .\) What is it? (The formula you get for the answer is called the Wilson lot size formula.) b. Shipping costs sometimes depend on order size. When they do, it is more realistic to replace \(k\) by \(k+b q\), the sum of \(k\) and a constant multiple of \(q .\) What is the most economical quantity to order now?

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. $$f(x)=\sqrt{25-x^{2}}, \quad-5 \leq x \leq 5$$

Find the limits. $$\lim _{x \rightarrow \infty}(\ln x)^{1 / x}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=x e^{-x}$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=x^{3}-2 x+4$$

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. $$\int(1.3)^{x} d x$$

A marathoner ran the 26.2 -mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero.

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. $$f(x)=\sqrt{x^{2}-2 x-3}, \quad 3 \leq x<\infty$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=x^{3}+x^{2}-8 x+5$$

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. $$\int\left(4 \sec x \tan x-2 \sec ^{2} x\right) d x$$

Show that at some instant during a 2 -hour automobile trip the car's speedometer reading will equal the average speed for the trip.

Show that if \(r(x)=6 x\) and \(c(x)=x^{3}-6 x^{2}+15 x\) are your revenue and cost functions, then the best you can do is break even (have revenue equal 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. $$g(x)=\frac{x-2}{x^{2}-1}, \quad 0 \leq x<1$$

Find the limits. $$\lim _{x \rightarrow 0^{+}} x^{-1 / \ln x}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{\ln x}{\sqrt{x}}$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=x^{3}(x-5)^{2}$$

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. $$\int \frac{1}{2}\left(\csc ^{2} x-\csc x \cot x\right) d x$$

Free fall on the moon On our moon, the acceleration of gravity is \(1.6 \mathrm{m} / \mathrm{sec}^{2} .\) If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later?

Production level Suppose that \(c(x)=x^{3}-20 x^{2}+20,000 x\) is the cost of manufacturing \(x\) items. Find a production level that will minimize the average cost of making \(x\) items.

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. $$g(x)=\frac{x^{2}}{4-x^{2}}, \quad-2

Find the limits. $$\lim _{x \rightarrow \infty} x^{1 / 12 x}$$

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. $$y=\frac{1}{1+e^{-x}}$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=\sqrt{x^{2}-1}$$