91Ó°ÊÓ

Find the rate of change of the area \(A\) of a square with respect to a side \(s\) at a given value \(s_{1}\) of the side. Is the result intuitively reasonable? Ans. \(A^{\prime}=2 s_{1}\).

If an object is thrown downward with an initial speed of \(100 \mathrm{ft} / \mathrm{sec}\), the distance that it falls in \(t\) seconds is given by the formula \(s=100 t+\) \(16 t^{2}\). Determine the formula for the speed at any time \(t\). Calculate the speed of the object at the end of the fourth second of fall.

Write the formula for the radius of a circle in terms of the area. Is the function single-valued? If not, which of the single-valued functions do you think would be more useful and why?

The area of a rectangle is given by the formula \(A=l w\), where \(l\) and \(w\) are the length and width, respectively. Suppose that \(l\) is kept fixed. Find the rate of change of \(A\) with respect to \(w\) at a given value of \(w\). Interpret the result geometrically.

An automobile travels at 30 miles per hour. Write a formula which expresses the distance \(d\) traveled in feet as a function of the time \(t\) which represents the number of seconds of travel.

Calculate the instantaneous speed of the following relations between distance and time at the instant indicated: (a) \(s=4 t^{2}\) at \(t=3\). Ans. 24 . (b) \(s=\frac{1}{4} t^{2}\) at \(t=3\). (c) \(s=3 t^{2}\) at \(t=0\). Ans. 0 . (d) \(s=\frac{5}{2} t^{2}\) at \(t=2\).

A rectangle is required to have an area of 4 square feet but its dimensions may vary. If one side has length \(x\), express the perimeter \(p\) of the rectangle as a function of \(x\). We shall be working with functions constantly. Thus we may have to deal with (4) $$ s=t^{2} $$ (5) $$ y=x^{3}+3 x^{2}+5 $$

In using the method of increments we encounter the function \(\Delta y / \Delta x\). What are the independent and dependent variables in this function?