91Ó°ÊÓ

The range \(R\) of a projectile fired with an initial velocity \(v_{0}\) at an angle \(\theta\) with the horizontal is \(R=\frac{v_{0}^{2} \sin 2 \theta}{g}\), where \(g\) is the acceleration due to gravity. Find the angle \(\theta\) such that the range is a maximum.

Consider a function \(f\) such that \(f^{\prime}\) is decreasing. Sketch graphs of \(f\) for (a) \(f^{\prime}<0\) and (b) \(f^{\prime}>0\).

Consider the functions \(f(x)=\frac{1}{2} x^{2}\) and \(g(x)=\frac{1}{16} x^{4}-\frac{1}{2} x^{2}\) on the domain \([0,4]\). (a) Use a graphing utility to graph the functions on the specified domain. (b) Write the vertical distance \(d\) between the functions as a function of \(x\) and use calculus to find the value of \(x\) for which \(d\) is maximum. (c) Find the equations of the tangent lines to the graphs of \(f\) and \(g\) at the critical number found in part (b). Graph the tangent lines. What is the relationship between the lines? (d) Make a conjecture about the relationship between tangent lines to the graphs of two functions at the value of \(x\) at which the vertical distance between the functions is greatest, and prove your conjecture.

Vertical Motion The height of an object \(t\) seconds after it is dropped from a height of 500 meters is \(s(t)=-4.9 t^{2}+500\). (a) Find the average velocity of the object during the first 3 seconds. (b) Use the Mean Value Theorem to verify that at some time during the first 3 seconds of fall the instantaneous velocity equals the average velocity. Find that time.

If \(f\) is a continuous function such that \(\lim _{x \rightarrow \infty} f(x)=5\), find, if possible, \(\lim _{x \rightarrow-\infty} f(x)\) for each specified condition. (a) The graph of \(f\) is symmetric to the \(y\) -axis. (b) The graph of \(f\) is symmetric to the origin.

Maximum Volume A sector with central angle \(\theta\) is cut from a circle of radius 12 inches (see figure), and the edges of the sector are brought together to form a cone. Find the magnitude of \(\theta\) such that the volume of the cone is a maximum.

Numerical, Graphical, and Analytic Analysis The cross sections of an irrigation canal are isosceles trapezoids of which three sides are 8 feet long (see figure). Determine the angle of elevation \(\theta\) of the sides such that the area of the cross section is a maximum by completing the following. (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) \begin{tabular}{|c|c|c|c|} \hline Base 1 & Base 2 & Altitude & Area \\ \hline 8 & \(8+16 \cos 10^{\circ}\) & \(8 \sin 10^{\circ}\) & \(\infty 22.1\) \\ \hline 8 & \(8+16 \cos 20^{\circ}\) & \(8 \sin 20^{\circ}\) & \(\approx 42.5\) \\ \hline \end{tabular} (b) Use a graphing utility to generate additional rows of the table and estimate the maximum cross-sectional area. (Hint: Use the table feature of the graphing utility.) (c) Write the cross-sectional area \(A\) as a function of \(\theta\). (d) Use calculus to find the critical number of the function in part (c) and find the angle that will yield the maximum cross-sectional area. (e) Use a graphing utility to graph the function in part (c) and verify the maximum cross-sectional area.

Use a graphing utility to graph the function. Use the graph to determine whether it is possible for the graph of a function to cross its horizontal asymptote. Do you think it is possible for the graph of a function to cross its vertical asymptote? Why or why not? $$ f(x)=\frac{\cos 3 x}{4 x} $$

Power The formula for the power output \(P\) of a battery is \(P=V I-R I^{2}\), where \(V\) is the electromotive force in volts, \(R\) is the resistance, and \(I\) is the current. Find the current (measured in amperes) that corresponds to a maximum value of \(P\) in a battery for which \(V=12\) volts and \(R=0.5\) ohm. Assume that a 15 -ampere fuse bounds the output in the interval \(0 \leq I \leq 15 .\) Could the power output be increased by replacing the 15 -ampere fuse with a 20-ampere fuse? Explain.

Minimize the sum of the absolute values of the lengths of vertical feeder lines given by \(S_{2}=|4 m-1|+|5 m-6|+|10 m-3| .\) Find the equation for the trunk line by this method and then determine the sum of the lengths of the feeder lines. (Hint: Use a graphing utility to graph the function \(S_{2}\) and approximate the required critical number.)