The differential equation that describes the growth of a continuously compounded investment is $$\frac{dy}{dt} = ry$$, where $y$ is the amount of the investment, $r$ is the annual interest rate, and $t$ is the time in years.

The solution to the differential equation $$\frac{dy}{dt} = ry$$ is $$y = Ce^{rt}$$, where $C$ is a constant.

The present value of an annuity that pays $1000 per year for 10 years at an annual interest rate of 5% (assuming continuous compounding) is $$\frac{1000}{0.05} \left( 1 - e^{-0.05 \cdot 10} \right)$$.

The monthly payment on a loan of $100,000 that is to be repaid over 30 years at an annual interest rate of 4% (assuming continuous compounding) is $$\frac{100,000}{30 \cdot 12} \left( 1 - e^{-0.04 \cdot 30} \right)$$.

The effective annual interest rate on a loan that has a nominal annual interest rate of 12% and is compounded monthly is $$13.38\%$$.

The doubling time of an investment that is continuously compounded at an annual interest rate of 7% is $$10\text{ years}$$. This means that it will take 10 years for the investment to double in value.

The future value of an investment of $1000 that is continuously compounded at an annual interest rate of 5% for 10 years is $$\$1643.85$$. This means that the investment will be worth $1643.85 at the end of 10 years.

The present value of an investment that will be worth $1000 in 10 years if the annual interest rate is 5% and the interest is compounded continuously is $$\$783.53$$. This means that you would need to invest $783.53 today in order to have $1000 in 10 years.

The annual interest rate on a loan that has a monthly payment of $1000, a loan term of 30 years, and a total amount borrowed of $100,000 is $$4.5\%$$. This means that the borrower will pay a total of $135,000 in interest over the life of the loan.

The total amount of interest paid on a loan of $100,000 that is repaid over 30 years at an annual interest rate of 5% (assuming continuous compounding) is $$\$135,000$$. This means that the borrower will pay a total of $235,000 over the life of the loan.

The present value of an annuity that pays $1000 per year for 10 years at an annual interest rate of 4% (assuming continuous compounding) is $$\$8203.46$$. This means that the present value of the annuity is $8203.46.

The future value of an annuity that pays $1000 per year for 10 years at an annual interest rate of 6% (assuming continuous compounding) is $$\$12387.65$$. This means that the future value of the annuity is $12387.65.

The present value of a perpetuity that pays $1000 per year at an annual interest rate of 5% (assuming continuous compounding) is $$\$20,000$$. This means that the present value of the perpetuity is $20,000.

The future value of a perpetuity that pays $1000 per year at an annual interest rate of 4% (assuming continuous compounding) is $$\$25,000$$. This means that the future value of the perpetuity is $25,000.