# Cost Function in Calculus: Formula & Examples

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• 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

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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

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