91Ó°ÊÓ

An object is projected upward from ground level with an initial velocity of 500 feet per second. In this exercise, the goal is to analyze the motion of the object during its upward flight. (a) If air resistance is neglected, find the velocity of the object as a function of time. Use a graphing utility to graph this function. (b) Use the result of part (a) to find the position function and determine the maximum height attained by the object. (c) If the air resistance is proportional to the square of the velocity, you obtain the equation $$\frac{d v}{d t}=-\left(32+k v^{2}\right)$$ (d) Use a graphing utility to graph the velocity function \(v(t)\) in part \((c)\) for \(k=0.001 .\) Use the graph to approximate the time \(t_{0}\) at which the object reaches its maximum height. (e) Use the integration capabilities of a graphing utility to approximate the integral $$\int_{0}^{t_{0}} v(t) d t$$ where \(v(t)\) and \(t_{0}\) are those found in part (d). This is the approximation of the maximum height of the object. (f) Explain the difference between the results in parts (b) and (e).

Think About It In Exercises \(79-82,\) L'Hopital's Rule is used incorrectly. Describe the error. $$\begin{aligned} \lim _{\rightarrow \infty} x \cos \frac{1}{x} &=\lim _{x \rightarrow \infty} \frac{\cos (1 / x)}{1 / x} \\ &=\lim _{x \rightarrow \infty} \frac{[-\sin (1 / x)]\left(1 / x^{2}\right)}{-1 / x^{2}} \\ &=\lim _{x \rightarrow \infty} \sin \frac{1}{x} \\ &=0 \end{aligned}$$

True or False? In Exercises \(83-86\) , determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false. The slope of the graph of the inverse tangent function is positive for all \(x.\)

Analyzing a Logarithmic Equation Consider the function \(f(x)=\log _{10} x\) (a) What is the domain of \(f ?\) (b) Find \(f^{-1}\) . (c) Let \(x\) be a real number between 1000 and \(10,000 .\) Determine the interval in which \(f(x)\) will be found. (d) Determine the interval in which \(x\) will be found if \(f(x)\) is negative. (e) When \(f(x)\) is increased by one unit, \(x\) must have been increased by what factor? (f) Find the ratio of \(x_{1}\) to \(x_{2}\) given that \(f\left(x_{1}\right)=3 n\) and \(f\left(x_{2}\right)=n .\)

Relative Extrema and Points of Inflection In Exercises \(87-92\) , locate any relative extrema and points of inflection. Use a graphing utility to confirm your results. $$y=\frac{x^{2}}{2}-\ln x$$

Inflation When the annual rate of inflation averages 5\(\%\) over the next 10 years, the approximate cost \(C\) of goods or services during any year in that decade is \(C(t)=P(1.05)^{t}\) where \(t\) is the time in years and \(P\) is the present cost. (a) The price of an oil change for your car is presently \(\$ 24.95 .\) Estimate the price 10 years from now. (b) Find the rates of change of \(C\) with respect to \(t\) when \(t=1\) and \(t=8\) . (c) Verify that the rate of change of \(C\) is proportional to \(C .\) What is the constant of proportionality?

Population Growth A population of bacteria \(P\) is changing at a rate of $$\frac{d P}{d t}=\frac{3000}{1+0.25 t}$$ where \(t\) is the time in days. The initial population (when \(t=0 )\) is \(1000 .\) \begin{equation}\begin{array}{l}{\text { (a) Write an equation that gives the population at any time } t .} \\ {\text { (b) Find the population when } t=3 \text { days. }}\end{array}\end{equation}

True or False? In Exercises \(95-100\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false An indeterminate form does not guarantee the existence of a limit.

In Exercises \(91-108,\) find the indefinite integral. $$\int \frac{e^{1 / x^{2}}}{x^{3}} d x$$

Population Growth A lake is stocked with 500 fish, and the population \(p\) is growing according to the logistic curve \(p(t)=\frac{10,000}{1+19 e^{-t / 5}}\) where \(t\) is measured in months. (a) Use a graphing utility to graph the function. (b) Find the fish populations after 6 months, 12 months, 24 months, 36 months, and 48 months. What is the limiting size of the fish population? (c) Find the rates at which the fish population is changing after 1 month and after 10 months. (d) After how many months is the population increasing most rapidly?