Chapter 5, Problem 16PS

Mortgage A $120,000 home mortgage for 35 years at $9\frac{1}{2}\%$ has a monthly payment of $985.93. Part of the monthly payment goes for the interest charge on the unpaid balance, and the remainder of the payment is used to reduce the principal. The amount that goes for interest is

$u=M-\left(M-\frac{\mathrm{Pr}}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$

and the amount that goes toward reduction of the principal is

$v=\left(M-\frac{\mathrm{Pr}}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$

In these formulas, P is the amount of the mortgage, r is the interest rate (in decimal form), M is the monthly payment, and t is the time in years.

(a) Use a graphing utility to graph each function in the same viewing window. (The viewing window should show all 35 years of mortgage payments.)

(b) In the early years of the mortgage, the larger part of the monthly payment goes for what purpose? Approximate the time when the monthly payment is evenly divided between interest and principal reduction.

(c) Use the graphs in part (a) to make a conjecture about the relationship between the slopes of the tangent lines to the two curves for a specified value of t. Give an analytical argument to verify your conjecture. Find $u^{\text{'}}(15)$ and $v^{\text{'}}(15)$.

(d) Repeat parts (a) and (b) for a repayment period of 20 years $(M=\$1118.56)$. What can you conclude?