Hifumi's Study NotesnotesCalculusCegep2Exponential Growth

Exponential Growth

Tags

Calculus

Cegep/2

Word count

215 words

Reading time

2 minutes

Growth rate of a function $y$ such that the derivative of $y$ is proportional to $y$

\begin{aligned} y^{'} & \propto y \\ y^{'} & = k y \\ y & = y_{0} e^{k x} \end{aligned}

Examples

A bacteria culture starts with 500 bacteria and after 3 hours there are 8000.
Find the function $P (t)$ that models the population after $t$ hours.
When will the population reach 30000 bacteria?
What is the doubling time?
When will the population reach 60000?

\begin{aligned} \frac{d P}{d t} & = k P (t) \\ \int \frac{1}{P} d P & = \int k d t \\ \ln | P | & = k t + C \\ | P | & = e^{k t + C} \\ P & = \pm e^{k t + C} \\ P > 0 ⟹ P & = e^{k t + C} \\ P & = e^{c} e^{k t} \\ 500 & = e^{c} e^{0} \\ e^{c} & = 500 \\ ∴ P & = 500 e^{k t} \\ 8000 & = 500 e^{k \cdot 3} \\ 16 & = e^{3 k} \\ k & \approx 0.924196 \\ ∴ P (t) & = 500 e^{0.924196 t} \end{aligned}

\begin{aligned} 30000 & = 500 e^{0.924196 t} \\ t & \approx 4.43 h \end{aligned}

\begin{aligned} 1000 & = 500 e^{0.924196 t} \\ t & \approx 0.75 h \end{aligned}

\begin{aligned} P (4.43 + 0.75) & = 60000 \\ t & \approx 5.18 h \end{aligned}

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