Cost Function in Calculus: Formula & Examples

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Graphing Sine and Cosine Transformations

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:00 The Cost Function: An…
  • 0:27 What Is the Cost Function?
  • 0:50 Additional Uses of the…
  • 1:30 Example Problems
  • 7:56 Lesson Summary
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Login or Sign up

Timeline
Autoplay
Autoplay
Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Joshua White

Josh has worked as a high school math teacher for seven years and has undergraduate degrees in Applied Mathematics (BS) & Economics/Physics (BA).

This lesson will explore the total cost function and the concepts of average and marginal cost. It will also cover how to find the minimum and maximum cost.

The Cost Function: An Introduction

How do companies determine the price that they charge to sell certain goods? Although it might seem random, companies frequently use a cost function to determine how many units of an item they should produce and what price they should sell it for. The cost function is just a mathematical formula that gives the total cost to produce a certain number of units. Let's take a more in depth look at the cost function and see how it works.

What Is the Cost Function?

The cost function, usually denoted C(x) where x represents a positive number and is generally an integer. If you want to know the cost of producing 50 units of an item, you would plug in 50 for every x in the cost function, and then, using order of operations, simplify the expression to a number, or dollar, figure.

Additional Uses of the Cost Functions

Besides the total cost, you can use the cost function to find the average cost and marginal cost of production. To find the average cost, you will simply divide the total cost by the total number of units produced. The marginal, or additional, cost represents the cost of producing one additional unit of the good. If you produce 100 battery chargers, the marginal cost will tell you how much extra it costs to produce that 100th charger. To find the marginal cost, you will find the total cost for the unit and subtract from it the total cost for producing one fewer units.

Now let's see how you would actually use the function.

Example Problems

1. The cost function to produce x tires is given as C(x)=.012x + 5,000.

First, let's find the cost to produce 1500 tires. To find this, you can simply plug in 1500 for x and then evaluate the cost function:

C(1500) = .012*1,500 + 5,000 = $5,018

Thus, it costs $5,018 to produce 1,500 tires.

Now, let's find the average cost of producing those 1500 tires. To find this, simply divide the total cost, $5,018, by the number of tires, 1500. You should get approximately $3.35. Note that although it costs on average $3.35 to produce each tire, the individual cost of producing each tire, or the marginal cost, is not $3.35. Let's see why.

To find the marginal cost of producing the 1500th tire, we can take the total cost of producing 1500 tires and subtract from that the total cost of producing 1499 tires.

C(1499) = (.012*1499) + 5000 = $5017.988

If you plug in 1499 for x in our original equation, you should get $5,017.99.

$5,018 - $5017.99 = $.01

Subtracting these two values gives $.01 or 1 cent. Thus, the marginal cost of producing the 1500th tire is approximately one cent.

2. The cost function for a property management company is given as C(x) = 50x + 100,000/x + 20,000 where x represents the number of properties being managed.

First, let's find the cost of managing 500 properties. Just substitute 500 in for x into the formula to find the answer:

C(500) = (50*500) + (100,000/500) + 20,000 = 25,000 + 200 + 20,000 = $45,200

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create An Account
Support