Chapter 3.4, Problem 59E

Social Security Assets The table gives actual and projected year end assets in Social Security trust funds, in trillions of current dollars, where Year represents the number of years since $2000$. Source: Social Security Administration. The polynomial function defined by $f(t)=0.0000329t^3-0.00450t^2+0.0613t+2.34$ models the data quite well.

a. To verify the fit of the model, find $f(10),f(20)$, and $f(30)$.

b. Use a graphing calculator with a command such as $\text{nDeriv}$ to find the slope of the tangent line to the graph of $f$ at the following $t$-values: $0,10,20,30$, and $35$.

c. Use your results in part b to describe the graph of $f$ and interpret the corresponding changes in Social Security assets.

$\begin{array}{cc}\text{Year}& \text{Trillions}\text{of}\text{Dollars}\\ 10& 2.45\\ 20& 2.34\\ 30& 1.03\\ 40& -0.57\\ 50& -1.86\end{array}$