6.4 - Exponential Growth and Decay

#calc/concept
Important to know how to do Separable Differential Equations.

Law of Exponential Change

If $y$ changes at a rate proportional to the amount present ($\frac{dy}{dt}=ky$), then $y=y_0{e}^{kt}$ where $y_0$ is the initial amount present and $k$ is the growth/decay constant.

• Essentially, what this means is that in an exponential growth problem, if you see $\pm e^c$, replace it with $A$.
• If you see the statement “the rate of change is proportional to the amount present”, then know that $\frac{dy}{dt}=ky$.

Exponential Growth/Decay Formulas

Investment Formula

$A=p{(1+\frac{r}{n})}^{nt}$
p is amount you invest, r is annual interest rate, n is amount of times money is compounded a year, and t is time. You can do half-life with this as well.

Continuous Compounding

$A=p{e}^{rt}$