91Ó°ÊÓ

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=x^{2}(3-x) $$

Consider the graph of the function \(f(x)=-x^{2}-x+6 .\) (a) Find the equation of the secant line joining the points (-2,4) and (2,0) (b) Use the Mean Value Theorem to determine a point \(c\) in the interval (-2,2) such that the tangent line at \(c\) is parallel to the secant line. (c) Find the equation of the tangent line through \(c\). (d) Use a graphing utility to graph \(f,\) the secant line, and the tangent line.

A right circular cylinder is to be designed to hold 22 cubic inches of a soft drink (approximately 12 fluid ounces). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) $$ \begin{array}{|c|c|c|} \hline \text { Radius } r & \text { Height } & \text { Surface Area } \\ \hline 0.2 & \frac{22}{\pi(0.2)^{2}} & 2 \pi(0.2)\left[0.2+\frac{22}{\pi(0.2)^{2}}\right] \approx 220.3 \\ \hline 0.4 & \frac{22}{\pi(0.4)^{2}} & 2 \pi(0.4)\left[0.4+\frac{22}{\pi(0.4)^{2}}\right] \approx 111.0 \\ \hline \end{array} $$ (b) Use a graphing utility to generate additional rows of the table. Use the table to estimate the minimum surface area. (c) Write the surface area \(S\) as a function of \(r\). (d) Use a graphing utility to graph the function in part (c) and estimate the minimum surface area from the graph. (e) Use calculus to find the critical number of the function in part (c) and find the dimensions that will yield the minimum surface area.

The perimeter of a rectangle is 20 feet. Of all possible dimensions, the maximum area is 25 square feet when the rectangle's length and width are both 5 feet. Are there dimensions that yield a minimum area? Explain.

Use a graphing utility to (a) graph the function \(f\) on the given interval, (b) find and graph the secant line through points on the graph of \(f\) at the endpoints of the given interval, and (c) find and graph any tangent lines to the graph of \(f\) that are parallel to the secant line. $$ f(x)=x-2 \sin x,[-\pi, \pi] $$

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.

S represents weekly sales of a product. What can be said of \(S^{\prime}\) and \(S^{\prime \prime}\) for each of the following? (a) The rate of change of sales is increasing. (b) Sales are increasing at a slower rate. (c) The rate of change of sales is constant. (d) Sales are steady. (e) Sales are declining, but at a slower rate. (f) Sales have bottomed out and have started to rise.

Sketch the graph of \(f(x)=2-2 \sin x\) on the interval \([0, \pi / 2]\) (a) Find the distance from the origin to the \(y\) -intercept and the distance from the origin to the \(x\) -intercept. (b) Write the distance \(d\) from the origin to a point on the graph of \(f\) as a function of \(x\). Use a graphing utility to graph \(d\) and find the minimum distance. (c) Use calculus and the zero or root feature of a graphing utility to find the value of \(x\) that minimizes the function \(d\) on the interval \([0, \pi / 2]\). What is the minimum distance? (Submitted by Tim Chapell, Penn Valley Community College, Kansas City, MO.)

Graph a function on the interval [-2,5] having the given characteristics. Relative minimum at \(x=-1\) Critical number at \(x=0,\) but no extrema Absolute maximum at \(x=2\) Absolute minimum at \(x=5\)

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.) $$ \begin{array}{|c|c|c|c|} \hline \text { Base 1 } & \text { Base 2 } & \text { Altitude } & \text { Area } \\ \hline 8 & 8+16 \cos 10^{\circ} & 8 \sin 10^{\circ} & \approx 22.1 \\ \hline 8 & 8+16 \cos 20^{\circ} & 8 \sin 20^{\circ} & \approx 42.5 \\ \hline \end{array} $$ (b) Use a graphing utility to generate additional rows of the table and estimate the maximum cross-sectional area. (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.