Precalculus / Unit 3: Exponential and Logarithmic Functions
3.6 Growth, Decay and the Number e
Notes
The number $e \approx 2.718$ is the base of natural growth.
- Continuous growth or decay: $A = Pe^{rt}$.
- Compound interest: $\displaystyle A = P\left(1 + \frac{r}{n}\right)^{nt}$.
- Doubling: after each doubling time the amount multiplies by 2. Half-life: after each half-life the amount is halved.
- Example: $1,000 at 5% compounded continuously for 10 years is $1000e^{0.5}$.
Practice
Which expression gives the value of $1,000 invested at 5% compounded continuously for 10 years?
Use $A = Pe^{rt}$ with $r = 0.05$ and $t = 10$.
A population of 500 bacteria doubles every 3 hours. How many are there after 9 hours?
Nine hours is 3 doublings: $500 \cdot 2^3 = 4000$.
A sample of 80 g has a half-life of 5 days. How much remains after 15 days?
Fifteen days is 3 half-lives: $\displaystyle 80 \cdot \frac{1}{8} = 10$.
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