91Ó°ÊÓ

Find \(\left(f^{-1}\right)^{\prime}(a)\) for the function \(f\) and the given real number \(a\). \(f(x)=x^{3}-\frac{4}{x}, \quad a=6\)

Find the point on the graph of \(y=e^{-x}\) where the normal line to the curve passes through the origin. (Use Newton's Method or the zero or root feature of a graphing utility.)

Explain why the domains of the trigonometric functions are restricted when finding the inverse trigonometric functions.

Modeling Data A meteorologist measures the atmospheric pressure \(P\) (in kilograms per square meter) at altitude \(h\) (in kilometers). The data are shown below.$$ \begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{h} & 0 & 5 & 10 & 15 & 20 \\ \hline \boldsymbol{P} & 10,332 & 5583 & 2376 & 1240 & 517 \\ \hline \end{array} $$(a) Use a graphing utility to plot the points \((h, \ln P)\). Use the regression capabilities of the graphing utility to find a linear model for the revised data points. (b) The line in part (a) has the form \(\ln P=a h+b\) Write the equation in exponential form. (c) Use a graphing utility to plot the original data and graph the exponential model in part (b). (d) Find the rate of change of the pressure when \(h=5\) and \(h=18\)

Are the derivatives of the inverse trigonometric functions algebraic or transcendental functions? List the derivatives of the inverse trigonometric functions.

Complete the table to determine the amount of money \(P\) (present value) that should be invested at rate \(r\) to produce a balance of \(\$ 100,000\) in \(t\) years. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline t & 1 & 10 & 20 & 30 & 40 & 50 \\\ \hline \boldsymbol{P} & & & & & & \\ \hline \end{array} $$ $$ \begin{aligned} &r=5 \%\\\ &\text { Compounded continuously } \end{aligned} $$

The yield \(V\) (in millions of cubic feet per acre) for a stand of timber at age \(t\) is \(V=6.7 e^{(-48.1) / t}\) where \(t\) is measured in years. (a) Find the limiting volume of wood per acre as \(t\) approaches infinity. (b) Find the rates at which the yield is changing when \(t=20\) years and \(t=60\) years.

Vertical Motion An object is dropped from a height of 400 feet. (a) Find the velocity of the object as a function of time (neglect air resistance on the object). (b) Use the result in part (a) to find the position function. (c) If the air resistance is proportional to the square of the velocity, then \(d v / d t=-32+k v^{2}\), where \(-32\) feet per second per second is the acceleration due to gravity and \(k\) is a constant. Show that the velocity \(v\) as a function of time is \(v(t)=-\sqrt{\frac{32}{k}} \tanh (\sqrt{32 k} t)\) by performing the following integration and simplifying the result. \(\int \frac{d v}{32-k v^{2}}=-\int d t\) (d) Use the result in part (c) to find \(\lim _{t \rightarrow \infty} v(t)\) and give its interpretation. (e) Integrate the velocity function in part (c) and find the position \(s\) of the object as a function of \(t\). Use a graphing utility to graph the position function when \(k=0.01\) and the position function in part (b) in the same viewing window. Estimate the additional time required for the object to reach ground level when air resistance is not neglected. (f) Give a written description of what you believe would happen if \(k\) were increased. Then test your assertion with a particular value of \(k\).

Describe the relationship between the graph of a function and the graph of its inverse function.

To estimate the amount of defoliation caused by the gypsy moth during a year, a forester counts the number of egg masses on \(\frac{1}{40}\) of an acre the preceding fall. The percent of defoliation \(y\) is approximated by \(y=\frac{300}{3+17 e^{-0.0625 x}}\) where \(x\) is the number of egg masses in thousands. (Source: USDA Forest Service) (a) Use a graphing utility to graph the function. (b) Estimate the percent of defoliation if 2000 egg masses are counted. (c) Estimate the number of egg masses that existed if you observe that approximately \(\frac{2}{3}\) of a forest is defoliated. (d) Use calculus to estimate the value of \(x\) for which \(y\) is increasing most rapidly.