Chapter 3.5, Problem 97E

Subprime Mortgages during the Housing Crisis (Com- pare Exercise 29 in Section 10.4.) The percentage of mort- gages issued in the United States during the period 2000–2009 that were subprime (normally classified as risky) can be approximated by

$A\left(t\right)=\frac{15}{1+8.6{\left(1.8\right)}^{-t}}\left(0\le t\le 9\right),$ where t is the number of years since the start of 2000.

a. Estimate $A\left(6\right)\text{ and }A\text{'}\left(6\right)$. (Round answers to two significant digits.) What do the answers tell you about subprime mortgages?

b. Graph the extrapolated function and its derivative for $0\le t\le 16$, 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 sub- prime mortgages? [HINT: See Example 5.]