Skills Review for The Logistic Equation

Learning Outcome

  • Apply continuous growth/decay models

The Logistic Equation section will expose us to differential equations and population growth and carrying capacity. Here we will review exponential models.

Use Exponential Models

For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use e as the base are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.

The Continuous Growth/Decay Formula

For all real numbers r, t, and all positive numbers a, continuous growth or decay is represented by the formula

[latex]P\left(t\right)=P_0{e}^{rt}[/latex]

where

  • [latex]P_0[/latex] is the initial value,
  • r is the continuous growth or decay rate per unit time,
  • and t is the elapsed time.

If r >[latex]0[/latex], then the formula represents continuous growth. If r < [latex]0[/latex], then the formula represents continuous decay.

For business applications, the continuous growth formula is called the continuous compounding formula and takes the form

[latex]A\left(t\right)=P_0{e}^{rt}[/latex]

where

  • [latex]P_0[/latex] is the principal or the initial amount invested,
  • r is the growth or interest rate per unit time,
  • and t is the period or term of the investment.

In our next example, we will calculate continuous growth of an investment. It is important to note the language that is used in the instructions for interest rate problems.  You will know to use the continuous growth or decay formula when you are asked to find an amount based on continuous compounding.

Example: Using the Continuously Compounded Interest Formula

A person invested [latex]$1,000[/latex] in an account earning a nominal [latex]10\%[/latex] per year compounded continuously. How much was in the account at the end of one year?

How To: Given the initial value, rate of growth or decay, and time [latex]t[/latex], solve a continuous growth or decay function

  1. Use the information in the problem to determine [latex]P_0[/latex], the initial value of the function.
  2. Use the information in the problem to determine the growth rate r.
    1. If the problem refers to continuous growth, then r > [latex]0[/latex].
    2. If the problem refers to continuous decay, then r < [latex]0[/latex].
  3. Use the information in the problem to determine the time t.
  4. Substitute the given information into the continuous growth formula and solve for P(t).

In our next example, we will calculate continuous decay. Pay attention to the rate – it is negative which means we are considering a situation where an amount decreases or decays.

Example: Using the Continuous Decay Model

Radon-222 decays at a continuous rate of [latex]17.3\%[/latex] per day. How much will [latex]100[/latex] mg of Radon-[latex]222[/latex] decay to in [latex]3[/latex] days?

Try It

A certain bacteria has an initial population of [latex]1000[/latex] with a growth rate of 3%. What will the population of the bacteria be in 5 years?