# Exponential Growth Using Calculus

This section covers:

We can use Calculus to measure Exponential Growth and Decay by using Differential Equations and Separation of Variables. Note that we studied Exponential Functions here and Differential Equations here in earlier sections.

# Introduction to Exponential Growth and Decay

Remember that Exponential Growth or Decay means something is increasing or decreasing an exponential rate (faster than if it were linear). We usually see Exponential Growth and Decay problems relating to populations, bacteria, temperature, and so on, usually as a function of time.

# Solving Exponential Growth Problems using Differential Equations

It turns out that if a function is exponential, as many applications are, the rate of change of a variable is proportional to the value of that variable.

So, we have: $$\displaystyle \frac{{dy}}{{dt}}=ky$$ or $${y}’=ky$$. This is where the Calculus comes in: we can use a differential equation to get the following:

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Exponential Growth and Decay Formula For a function $$y>0$$ that is differentiable function of $$t$$, and $${y}’=ky$$:

$$y=C{{e}^{{kt}}}$$

$$C$$ is the initial value of $$y$$, $$k$$ is the proportionality constant.

For $$k>0$$, we have exponential growth, and for $$k<0$$, we have exponential decay.

Here’s how we got to this equation (using a Differential Equation), which is good to know for future problems. Note that since $${{e}^{C}}$$ is a constant, we can just turn this into another constant “$$C$$”. (Note that $$y>0$$.)

\displaystyle \begin{align}\frac{{dy}}{{dt}}&=ky\\dy&=ky\cdot dt\\\frac{{dy}}{y}&=k\,dt\\\int{{\frac{1}{y}\,dy}}&=\int{{kdt}}\\\ln \left( y \right)&=kt+{{C}_{1}}\\{{e}^{{\ln \left( y \right)}}}&={{e}^{{kt+{{C}_{1}}}}}\\y&={{e}^{{kt}}}\cdot {{e}^{{{{C}_{1}}}}}={{e}^{{kt}}}\cdot C=C{{e}^{{kt}}}\end{align}

Before we get into the Exponential Growth problems, let’s do a few more practice differential equation problems. Remember that we can cross-multiply to get started:

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# Exponential Growth Word Problems

Now let’s do some Exponential Growth and Decay Calculus problems:

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Here are a few more Exponential Growth problems:

Learn these rules and practice, practice, practice!

On to Derivatives and Integrals of Inverse Trig Functions — you are ready!

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