AP Calculus BC / Unit 7: Differential Equations
7.8 Exponential Models with Differential Equations
Notes
$\displaystyle \frac{dy}{dt}=ky$ has solution $y=y_0e^{kt}$.
- $k>0$: growth. $k<0$: decay.
- Half-life $\displaystyle =\frac{\ln2}{|k|}$; doubling time $\displaystyle =\frac{\ln 2}{k}$.
Practice
$\displaystyle \frac{dy}{dt}=0.05y$, $y(0)=200$. Then $y(t)=$
$y_0e^{kt}$.
A substance has $k=-0.1$ per year. Its half-life is about:
$\displaystyle \frac{\ln2}{0.1}\approx6.93$.
In $\displaystyle \frac{dy}{dt}=ky$, the rate of change is proportional to:
That's what makes it exponential.
Ready for more? Take the Unit 7 test.