Calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. HINT [See Example 1.] $$ \begin{array}{|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{f ( x )} & 3 & 5 & 2 & -1 \\ \hline \end{array} $$ Interval: \([1,3]\)

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=\frac{2}{x} $$

The processor speed, in megahertz (MHz), of Intel processors can be approximated by the following function of time \(t\) in years since the start of \(1995:^{65}\) $$ P(t)=\left\\{\begin{array}{ll} 180 t+200 & \text { if } 0 \leq t \leq 5 \\ 3,000 t-13,900 & \text { if } 5

Sketch the graph of a function whose derivative never exceeds 1.

Find the equation of the tangent to the graph at the indicated point. HINT [Compute the derivative algebraically; then see Example \(2(\mathrm{~b})\) in Section \(10.5 .]\) $$ f(x)=-2 x-4 ; a=3 $$

(Compare Exercise 30 of Section 10.4.) The value of subprime (normally classified as risky) mortgage debt outstanding in the U.S. can be approximated by $$ A(t)=\frac{1,350}{1+4.2(1.7)^{-t}} \text { billion dollars } \quad(0 \leq t \leq 9) $$ where \(t\) is the number of years since the start of 2000 . a. Estimate \(A(7)\) and \(A^{\prime}(7)\). (Round answers to three significant digits.) What do the answers tell you about subprime mortgages? b. I Graph the function and its derivative and use your graphs to estimate when, to the nearest year, \(A^{\prime}(t)\) is greatest. What does this tell you about subprime mortgages? HINT [See Example 5.]