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.] $$ \text { Cisco Systems Stock Price (\$) } $$$$ \text { Interval: }[1,5] $$

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 \(10.5 .]\) $$ f(x)=3 x+1 ; a=1 $$

(Compare Exercise 29 of Section 10.4.) The percentage of mortgages issued in the United States that are subprime (normally classified as risky) can be approximated by $$ A(t)=\frac{15}{1+8.6(1.8)^{-t}} \quad(0 \leq t \leq 9) $$ where \(t\) is the number of years since the start of 2000 . a. Estimate \(A(6)\) and \(A^{\prime}(6)\). (Round answers to two significant digits.) What do the answers tell you about subprime mortgages? b. \(\mathrm{T}\) Graph the extrapolated function and its derivative for $0 \leq t \leq 16\( and use your graphs to describe how the derivative behaves as \)t$ becomes large. (Express this behavior in terms of limits if you have studied the sections on limits.) What does this tell you about subprime mortgages? HINT [See Example 5.]

The median home price in the U.S. over the period 2004-2009 can be approximated by $$ P(t)=-5 t^{2}+75 t-30 \text { thousand dollars } \quad(4 \leq t \leq 9) $$ where \(t\) is time in years since the start of \(2000 .^{52}\) a. Compute the average rate of change of \(P(t)\) over the interval \([5,9]\), and interpret your answer. HINT [See Section \(10.4\) Example 3.] b. Estimate the instantaneous rate of change of \(P(t)\) at \(t=5\), and interpret your answer. HINT [See Example 2(a).] c. The answer to part (b) has larger absolute value than the answer to part (a). What does this indicate about the median home price?

(a) use any method to estimate the slope of the tangent to the graph of the given function at the point with the given \(x\) -coordinate and \((\boldsymbol{b})\) find an equation of the tangent line in part (a). In each case, sketch the curve together with the appropriate tangent line. HINT [See Example \(2(\mathrm{~b}) .]\) $$ f(x)=2 x+4 ; x=-1 $$

Use the method of Example 4 to list approximate values of \(f^{\prime}(x)\) for \(x\) in the given range. Graph \(f(x)\) together with \(f^{\prime}(x)\) for \(x\) in the given range. $$ f(x)=\frac{10 x}{x-2} ; \quad 2.5 \leq x \leq 3 $$