Exponential Models (College Board AP® Calculus AB)

Study Guide

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Written by: Roger B

Reviewed by: Dan Finlay

Differential equations for exponential models

What type of differential equation corresponds to an exponential model?

There are many situations where assuming that the rate of change of a quantity is proportional to the size of the quantity provides a good model
- For example a population of bacteria
  - The more bacteria there are, the more new bacteria will be being produced
- Or a radioactive sample
  - The more radioactive atoms there are, the greater the number of atoms that will be undergoing decay
This is known as the exponential growth and decay model
The mathematical way of expressing this for a quantity $y$ is
- $\frac{d y}{d t} = k y$
  - $\frac{d y}{d t}$ is the rate of change of $y$
  - $k$ is the constant of proportionality
- If the quantity $y$ is increasing
  - then $k$ is positive
- If the quantity $y$ is decreasing
  - then $k$ is negative
  - Or alternatively, assume that $k$ is positive and write $\frac{d y}{d t} = - k y$

Solutions to exponential growth & decay models

How do I find the solutions for exponential growth and decay models?

The solution to the exponential growth and decay model $\frac{d y}{d t} = k y$ , with the initial condition $y = y_{0}$ when $t = 0$ , is
- $y = y_{0} e^{k t}$
It is a good idea to remember this result
- but it can also be derived using separation of variables

Solving the exponential growth and decay model using separation of variables

Start with $\frac{d y}{d t} = k y$
- Separate the variables
  - $\frac{1}{y} \frac{d y}{d t} = k$
- Integrate both sides with respect to $t$
  - $\int \frac{1}{y} d y = \int k d t$
- Integrate, including a constant of integration
  - $\ln |y| = k t + C$
- If $y$ represents a population then $y$ can never be negative, so we can ignore the modulus sign
  - $\ln y = k t + C$
- $y = y_{0}$ when $t = 0$ , so
  - $\ln (y_{0}) = k (0) + C \Rightarrow C = \ln (y_{0})$
- Rearrange
  - $\begin{array}{rcl} \ln y & = & k t + \ln (y_{0}) \\ e^{\ln y} & = & e^{k t + \ln (y_{0})} \\ y & = & e^{k t} \cdot e^{\ln (y_{0})} \\ y & = & y_{0} e^{k t} \end{array}$

Examiner Tips and Tricks

If an exam question is based on a real world example, be sure that your answers are given in the context of the question.

Worked Example

At any point in time, the rate of growth of a colony of bacteria is proportional to the current population size, $P$ .

(a) Write a differential equation to model the size of the population of bacteria.

Answer:

This is a description of an exponential growth model.

$\frac{d P}{d t} = k t$

At time $t = 0$ hours, the population size is 5000.

(b) Write down the particular solution of the differential equation from part (a).

Answer:

For $\frac{d y}{d t} = k y$ , with the initial condition $y = y_{0}$ when $t = 0$ , the particular solution is $y = y_{0} e^{k t}$

$P = 5000 e^{k t}$

After 1 hour, the population has grown to 7000.

(c) Determine how long it will take from time $t = 0$ , according to the model, for the population of bacteria to grow to 100 000.

Answer:

The question doesn't say what units of time to use, but looking at the information given it will be easiest to use hours

Substitute the values for $t = 1$ hour into the particular solution, and solve for $k$

$\begin{array}{rcl} 7000 & = & 5000 e^{k (1)} \\ e^{k} & = & \frac{7000}{5000} = \frac{7}{5} = 1.4 \\ k & = & \ln (1.4) \end{array}$

Using that value of $k$ , substitute $P = 100000$ into the particular solution and solve for $t$

$\begin{array}{rcl} 100000 & = & 5000 e^{\ln (1.4) t} \\ e^{\ln (1.4) t} & = & \frac{100000}{5000} = 20 \\ \ln (1.4) t & = & \ln (20) \\ t & = & \frac{\ln (20)}{\ln (1.4)} = 8.903356 . . . \end{array}$

8.903 hours (to 3 decimal places)

Last updated: 15 August 2024

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Exponential Models (College Board AP® Calculus AB)

Study Guide

Differential equations for exponential models

What type of differential equation corresponds to an exponential model?

Solutions to exponential growth & decay models

How do I find the solutions for exponential growth and decay models?

Solving the exponential growth and decay model using separation of variables

Examiner Tips and Tricks

Worked Example

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Author: Roger B

Author: Dan Finlay