Preview Activity 11.3.1.
Suppose we take out an $18,000 car loan at interest rate \(r\) and we agree to pay off the loan in \(t\) years. The monthly payment, in dollars, is
\begin{equation*}
M(r,t) =
\frac{1500r}{1-\left(1+\frac{r}{12}\right)^{-12t}}.
\end{equation*}
(a)
If you take a loan with an interest rate of \(3\%\) (so \(r = 0.03\)) and will pay it off in \(t=4\) years, what will your monthly payment be?
(b)
Suppose the interest rate on loans is fixed at \(3\%\text{.}\) Express \(M\) as a function \(f\) of \(t\) alone using \(r=0.03\text{.}\) That is, let \(f(t) = M(0.03, t)\text{.}\) Sketch the graph of \(f\) on the axes below and write a couple of sentences to explain the meaning of the function \(f\text{.}\)
Axes for sketching the graph of \(f(t)\text{.}\) The horizontal axis shows \(t\)-values from 0 to 10 and the vertical axis shows \(f(t)\) values from 0 to 1000.
(c)
Find the instantaneous rate of change \(f'(4)\) and state the units on this quantity. What information does \(f'(4)\) tell us about our car loan? What information does \(f'(4)\) tell us about the graph you sketched in (b)?
(d)
Suppose that we want to examine how the monthly payment will change if we plan to take 4 years to pay off the loan. Express \(M\) as a function of the interest rate \(r\) with a fixed loan term of \(t=4\text{.}\) That is, let \(g(r) = M(r, 4)\text{.}\) Give the function \(g(r)\) and sketch a graph of \(g\) on the axes provided. Write a couple of sentences to explain the meaning of the function \(g\text{.}\)
Axes for sketching the graph of \(g(r)\text{.}\) The horizontal axis shows \(r\)-values from 0 to 0.10 and the vertical axis shows \(g(r)\) values from 0 to 1000.
(e)
Find the instantaneous rate of change \(g'(0.03)\) and state the units on this quantity. Write a couple of sentences to explain what \(g'(0.03)\) tell us about the car loan. What information does \(g'(0.03)\) tell us about the graph you sketched in (d)?
