Chapter 5.5, Problem 58E

(a)

To determine

To graph: The functions for amount paid toward the interest and amount paid toward the reduction of the principal given as $u=M-\left(M-\frac{P\left(r\right)}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$ and $v=\left(M-\frac{P\left(r\right)}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$, respectively in the same viewing window for a $\$120,000$ home mortgage for $30$ years at $7\frac{1}{2}\%$ that has a monthly payment of $\$839.06$, such that part of the monthly payment covers the interest charge on the unpaid balance, and the remainder of the payment reduces the principal, where $P$ 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.

(b)

To determine

Whether the greater part of the monthly payment is paid toward the interest or the principal in early years of mortgage, and also, approximate the time when the monthly payment is evenly divided between interest and principal reduction.

(c)

To determine

The graph of functions $u=M-\left(M-\frac{P\left(r\right)}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$ and $v=\left(M-\frac{P\left(r\right)}{12}\right){\left(1+\frac{r}{12}\right)}^{12t}$ for a period of $20\text{ years}$ with $M=\$966.71$, and check if greater part of the monthly payment is paid toward the interest or the principal in early years of mortgage, and approximate the time when the monthly payment is evenly divided between interest and principal reduction.