Shaun Ault

AP Calculus Review: Exponential Functions

Exponential functions play a key role in a wide array of applications including population growth. These important functions show up on both the AP Calculus AB and BC exams. So here’s what you should know about them for the test.

Exponential Functions — Definitions

An exponential function is one that involves a constant positive base to a variable exponent. The most basic exponential is: f(x) = ax, where a > 0 is a constant.

Other variations include coefficients that scale the graph horizontally or vertically. Adding or subtracting a constant shifts the graph up or down.

y = B a^(kx) + C, general exponential

Furthermore, there is a certain constant called e (Euler’s constant) that is so useful as a base that we call ex the natural exponential function.

Exponential functions

An exponential function. Here, the base is e, and the graph has been shifted down by 4 units.

Facts and Properties

The exponential function f(x) = ax satisfies the following properties.

  • The domain is all real numbers, (-∞, ∞).
  • The range is all positive real numbers, (0, ∞).
  • The y-intercept is (0, 1), but there is no x-intercept.
  • If a > 1, then f increases on its domain.
  • If 0 < a < 1, then f decreases on its domain.
  • The graph is concave up on its entire domain.
  • The line y = 0 is a horizontal asymptote.

Differentiating Exponential Functions

The natural exponential has the remarkable property that it is its own derivative. That property alone is what makes the function ex so essential to almost all branches of science.

When the base is something other than e, then the derivative formula involves a multiple of the natural logarithm of the base.

exponential derivatives

Check out the following article for more details about logarithms and exponent properties: AP Calculus Review: Properties of Exponents and Logarithms.

Example 1

Find the absolute minimum and maximum values of Example exponential function

Solution

The function g(x) is technically not an exponential function, however it does involve an exponential.

Let’s take the first derivative. Don’t forget product rule!

Derivative of example

Find the critical numbers by setting g ' equal to 0 and solving. Note that because the exponential function itself can never be 0, the only factor that matters here is 1 – 6x2.

Solving for x = 1/sqrt(6) and -1/sqrt(60

Finally, plug the two critical numbers into the original function to determine minimum and maximum. A sketch of the graph also helps to prove that the absolute min and max do occur at those points.

  • g(-0.4082) = -0.2476 (minimum value)
  • g(0.4082) = 0.2476 (maximum value)

Graph of example exponential.

Integrating Exponential Functions

Because the derivative of ex is equal to ex, you can expect its antiderivative to be the exact same thing. (But don’t forget the “+ C“!)

Exponential Antiderivatives

Example 2

Let e to the 2x. Find the value of c guaranteed by the Mean Value Theorem for Integrals so that f(c) is equal to the average value of f on the interval [1, 3].

Solution

We must first find the average value of the function on the given interval. This is a job for an integral. Note that there will be a substitution: u = 2x.

Computing the average value of an exponential function

Next, set f(c) = 99.01 and solve for c.

Solution to Mean Value Theorem for Integrals example

Related Topics

The exponential also shows up in a number of applications on the AP Calculus exams.

Exponential Growth Models

Any situation in which the rate of growth is proportional to the amount present lends itself directly to an exponential model.

The differential equation y ' = ky, where k is a constant, has the general solution, y = Aekx. Here, the value of the constant A is equal to the initial population, y(0).

Example 3

A certain bacteria culture has 1000 cells initially.

The amount X, in grams, of a radioactive isotope decays according to the equation dX/dt = -0.03X, where time t is measured in days. Determine the half-life of the isotope to the nearest day. (Note: the half-life of a substance is the amount of time it takes for exactly one-half of the substance to decay.)

Solution

First, we can build the exponential model (Aekx) based on the given information. Here, the initial amount is A = 1000, and the decay constant is k = -0.03. So, the appropriate exponential model would be:

X = 1000 e^(-0.03t)

Next, to find the half-life, we set X(t) = 500 (because that’s half of 1000), and solve for t.

Solution to half-life problem

Thus the half-life is about 21 days.

The Logistics Growth Model

the logistics growth model is an equation the models population growth up to a constant carrying capacity M.

Logistics curve

The logistics growth model shows how populations might grow under limited resources.

The natural exponential ex makes an appearance as part of the solution to the logistics differential equation.

Logistics function

For more information and examples, check out: AP Calculus BC Review: Logistics Growth Model.

The Maclaurin Series for ex

Taylor and Maclaurin series only show up on the AP Calculus BC exam. You should probably memorize the Maclaurin series for the natural exponential function.

Maclaurin series for e^x

Example 4

Find a power series expansion for x times e to the x-squared.

Solution

We just have to work with the Maclaurin series for the natural exponential. I’ve color coded certain parts to make it a little easier to follow.

solution to Maclaurin series example

Summary

  • Exponential functions have the basic form f(x) = ax, where a > 0 is a constant. But the graph can be scaled or shifted by including appropriate coefficients and constant terms.
  • The derivative of ex is equal to ex. The derivative of ax is ax ln a.
  • The antiderivative (integral) of ex is equal to ex + C. The antiderivative of ax is ax / (ln a) + C.
  • The general solution of the differential equation y ' = ky is the exponential function, y = Aekx.
  • The solution to the logistics growth model equation also involves exponentials.
  • The Maclaurin series (power series) of the natural exponential function is:
    Maclaurin series for e^x
About Shaun Ault

Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!


Leave a Reply

Magoosh blog comment policy: To create the best experience for our readers, we will approve and respond to comments that are relevant to the article, general enough to be helpful to other students, concise, and well-written! :) If your comment was not approved, it likely did not adhere to these guidelines. If you are a Premium Magoosh student and would like more personalized service, you can use the Help tab on the Magoosh dashboard. Thanks!


Share1
Tweet
Share
Pin