Summary of Exponential Functions

Basic Equation: MATH ($a$ and $k$ are always positive).

For exponential growth, $a>0$.

For exponential decay, $a<0$.

Note: If $a>0$ you can still get exponential decay if you replace the basic equation by MATH.

$P_{0}$ is called the initial value of $P$ since $P_{0}=P(0)=$ value of $P$ when $t=0$.




Discrete or Simple Growth Rate: Let us choose a unit of time called $U$ (for example, $U$ might be one second, or one week, or 20 years). Suppose that over some time interval I you observe a percentage growth $R$, expressed as a decimal (e.g. $R=5\%=0.05$). Suppose that $P_{0}$ is the size of $P$ at the beginning of this time interval I. Then the size $T$ units later is given by MATH, where $T$ is measured in units of size $U$.

For example, a "population" $P$ increases at the rate of $17\%$ every 10 years, over a period of 100 years (say I = the years 1700 to 1800). Let us take our unit $U$ to be a decade; that is $U=100~~years$. Suppose, finally, that $\ P$ is 12,000 at the beginning (the year 1700). Then the population $T$ decades later will be $12,000(1+0.17)^{T}$. On the other hand, if you want the population $t$ years later, then $T=t/10$ decades, so
MATH

This says that the annual percentage growth rate is about 1.6% (=0.016).

If you want to model a rate of decrease, or decay, of $R$ percent in $T$ units, then use the formula MATH. For example, a 3.8% decay rate will give MATH.




Continuous Growth: Here the model is $P(t)=P_{0}e^{rT}$, where $P_{0}$, $r$, (a percentage expressed as a decimal), and $T$ are as before. This is based on the "continuous compounding" model, which is why $e$ appears. As long as $r$ and $T$ are not too big, the functions $P_{0}(1+r)^{T}$ and $P_{0}e^{rT}$ are fairly close to each other. For large values of the percentage rate, the continuous model is noticeably bigger.




Effective Rates: We can compare the continuous and discrete growth models by asking: "What simple (discrete) growth rate $R$ produces the same growth model as the continuous growth rate $r$ ?" In other words, we set the models equal and solve for $R$:


MATH

$R=e^{r}-1$ is called the effective simple rate corresponding to the continuous rate $r$. In general, $R>r$ because $e^{r}\geq 1+r$. If you are given $R$, then you can find $r$ using MATH.

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