Lesson 2: Exponential Models

Property

To reveal growth rates for different time periods, rewrite $y = a \cdot b^x$ by multiplying the exponent by $\frac{n}{n}$:

This shows the growth factor for $\frac{1}{n}$ of the original time unit.

Property

Compound interest follows an exponential model where the amount $A$ after $t$ time periods is given by

where $P$ is the principal (initial amount), $r$ is the interest rate per period, and $t$ is the number of time periods. This is a specific case of the general exponential model $y = ab^x$ where $a = P$ and $b = (1 + r)$.

Examples

Property

When interest is compounded continuously, the amount $A(t)$ in an account after $t$ years is given by the function

Examples

• If you invest 2000 dollars at 4% interest compounded continuously, the value of your account after $t$ years is $A(t) = 2000e^{0.04t}$.
• To find how long it takes for a 1000 dollar investment to double to 2000 dollars at 6% interest compounded continuously, you solve the equation $2000 = 1000e^{0.06t}$.
• The value of a 5000 dollar investment after 8 years at 3.5% interest compounded continuously is $A(8) = 5000e^{0.035(8)} \approx 6615.65$ dollars.

Explanation

Continuous compounding is the theoretical limit of earning interest. It's as if interest is being calculated and added to your balance at every single moment. This formula gives the maximum amount of money an investment can earn at a given rate.