The limited exponential growth function

Limited exponential growth: Limited exponential growth functions

The limited exponential growth function

Definition of the limited exponential growth function The limited exponential growth function is

$y(t)=a-c\cdot e^{-r\cdot t}$ with $r>0$ .

The graph of this function in the case $c>0$ is outlined below: growth continues to decrease and disappears over time so that the limit $a$ reached. It is true that $y(0)=a-c$ and that $y(t)$ approaches the value $a$ over time $t$ .

graphs of the limited exponential growth function

Think about the shape of the graph of $y(t)$ in case $c<0$

Aerobic power $P_{\mathrm{aer}}$ of a human leg muscle at the start of sprint is mathematically modelled as

$P_{\mathrm{aer}}(t)=P_{\mathrm{max}}\cdot\left(1-e^{-\lambda\cdot t}\right)$ with $\lambda\approx 0.035\;\mathrm{s}^{-1}$ .

The differential equation of the limited exponential growth function

$y(t)=a-c\cdot e^{-r\cdot t}$ with $r>0$ . The derivative of $y$ is, according to the calculation rules of differentiating,

$y'(t)=c\cdot r\cdot e^{-r\cdot t}$ Because $c\cdot e^{-r\cdot t}=a-y(t)$ (check!) we know that

$y'(t)=r\cdot\bigl(a-y(t)\bigr)$ This is the differential equation of limited exponential growth, which we will examine in the next theory page.