Digging Deeper: Applications of Functions

After exploring the concept of a function, it is helpful to see how this concept can be usefully applied in the real world. Applications can enhance our understanding of functions and increase our interest in functions. On this page you will find some optional resources that illustrate a variety of applications of functions.

Here is a familiar function where the input, r, is the radius of a circle, and the output is the circumference of that circle:

$LaTeX: C\left(r\right)\:=\:2\cdot\pi\cdot r$

In the following function, r is again the radius of a circle, and the output is the area of that circle:

$LaTeX: A\left(r\right)\:=\:\pi\cdot r^2$

Next, we let r represent the radius of a sphere. The output of this function is the surface area of that sphere:

$LaTeX: S\left(r\right)\:=\:4\cdot\pi\cdot r^2$

Again, we let r represent the radius of a sphere. The output of this function is the volume enclosed by that sphere:

$LaTeX: V\left(r\right)=\frac{4}{3}\cdot\pi\cdot r^3$

In each of the above four functions, notice how the letter selected to name the function is the first letter of the description of the output of that function: C for circumference, A for area, S for surface area, and V for volume

Also, the fascinating constant $LaTeX: \pi$ appears in each of these four functions. We know that $LaTeX: \pi$ is an irrational number, which means that $LaTeX: \pi$ has an infinite, non-repeating decimal expansion. The number $LaTeX: \pi$ is approximately equal to 3.14159. The current world record for memorizing digits of $LaTeX: \pi$ is 70,030 digits.

Here is a YouTube video about Pi.

The YouTube video "Pi" by Numberphile is available here.

A TED-Ed video demonstrating a creative way to use Pies to approximate $LaTeX: \pi$ can be found here.

Applications Using Exponential Functions

In the series of three videos below you well see some applications of exponential functions. Exponential functions are very interesting, since they can be used to model exponential growth or exponential decay. When modeling exponential growth we will use increasing exponential functions, and when modeling exponential decay, we will use decreasing exponential functions. The first video in this series illustrates exponential growth with an example of the population growth of bacteria.

Additional viewing option: Turn on closed captions by selecting the "CC" button and then choosing "English."

.The YouTube video, "Applications of Exponential Functions (Part 1)" by davidtutorsmath can be found here.

In the second video in this series, an example illustrating exponential decay of Berrilium-11 is discussed. Also, an exponential decay model is used to illustrate how the value of a new car decreases over time.

Additional viewing option: Turn on closed captions by selecting the "CC" button and then choosing "English."

The YouTube video, "Applications of Exponential Functions (Part 2)" by davidtutorsmath can be found here.

In the third video in this series, an example illustrating the growth in the value of a classic car is discussed. Also, an example illustrating compound interest is presented.

Additional viewing option: Turn on closed captions by selecting the "CC" button and then choosing "English."

The YouTube video, "Applications of Exponential Functions (Part 3)" by davidtutorsmath can be found here.

Modeling with Functions

When we build models in mathematics, we often use one or more functions. Below is a video that focuses on modeling with functions. You will see an example where a function is constructed to give the total revenue received from selling an item as function of the price of the item. This function can be used to determine the price per item that will maximize revenue.

The YouTube video, "Modeling with Functions Part 1,"by ProfRobBob can be found here.

In the video below, "Modeling with Functions Part 2," you will see examples where a function is constructed based on geometrical ideas.

The YouTube video, "Modeling with Functions Part 2,"by ProfRobBob can be found here.

In "Modeling with Functions Part 3" below, you will see an example involving the surface area and the volume of a rectangular solid. A second example involving the volume of a cone is also presented.

The YouTube video, "Modeling with Functions Part 3," by ProfRobBob can be found here.