Exponential Models (College Board AP® Calculus AB)

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 is

  • 

    • is the rate of change of

    • is the constant of proportionality

  • If the quantity is increasing

    • then is positive

  • If the quantity is decreasing

    • then is negative

    • Or alternatively, assume that is positive and write

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 , with the initial condition when , is

  • 

• 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

  • Separate the variables

    • 

  • Integrate both sides with respect to

    • 

  • Integrate, including a constant of integration

    • 

  • If represents a population then can never be negative, so we can ignore the modulus sign

    • 

  • when , so

    • 

  • Rearrange

    •