Unit 5

Applications of Derivatives

A. Maximum and Minimum Problems

Lesson 2: Extreme Values of Distance and Time and Economics

Extreme Values in Economics

When selling a product, total revenue or income depends on the selling price of the product and the number of units sold. The selling price of a product is often determined by the number of units sold. If a product is selling well, the price per unit may decrease.

If it costs a company «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/math» dollars to produce «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» units of a product, the function defined by «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»C«/mi»«/math» is called the cost function. The total cost of producing «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» units of a product includes fixed costs (rent, utilities, depreciation on machinery, etc), which continue even if nothing is produced, and variable costs (labour, materials, etc), which depend on the number of items produced.


In the cost function, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» is the number of units produced and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»p«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/math» is the variable cost of production per unit.

Economists call the derivative of the cost function the marginal cost. This derivative is used to calculate minimum costs.

The average cost function, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»c«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»C«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«mi»x«/mi»«/mfrac»«/math», gives the cost per unit when «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»x«/mi»«/math» units are produced. When the marginal cost is equal to the average cost, the average cost is at a minimum.

Although a formal justification for this relationship is beyond the scope of this course, it can be thought of  intuitively. If the cost of production per unit (the marginal cost) is below the average cost, the average cost will decrease as more units are added. (For example, if the average cost is«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»$«/mo»«mn»10«/mn»«/math» and an item valued at «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»$«/mo»«mn»7«/mn»«/math» is added, the average cost will go down.) If the cost per unit is above the average cost, the average cost will increase as more units are added. It is only when the two costs are equal that adding more units will not change the average cost (the rate of change is zero); this point is the minimum of the average cost function.