Chapter 6.2, Problem 40E

Paying Down a Mortgage You took a $\$200,000$ home mortgage at an annual interest rate of $3\%$. Suppose that the loan is amortized over a period of $30$ years, and let $P\left(t\right)$ denote the amount of money (in thousands of dollars) that you owe on the loan after $t$ years. A reasonable estimate of the rate of change of $P$ is given by $P\text{'}\left(t\right)=-4.1107e^{0.03t}$.

Approximate the net change in $P$ after $20$ years.

What is the amount of money owed on the loan after $20$ years?

Verify that the loan is paid off in 30 years by computing the net change in $P$ after $30$ years.