The cost ,in dollars, of producing x belts is given by C(x)= 683+16x-0

Before we can find how the average cost is changing, we first need to define the average cost function. This can be found by dividing the given cost function by x.

[Math Processing Error]C¯(x)=C(x)x=683+16x−0.077x2x=683x+16−0.077x

The rate of change of a function is defined as the derivative. Thus, we need to differentiate this function in order to find how the average cost is changing. Since the function we have contains only terms with x to a numerical power -- remember that [Math Processing Error]1x is the same as [Math Processing Error]x−1 -- we can use the power rule to do this.

[Math Processing Error]C¯′(x)=−683x2−0.077

Evaluating this at a point will tell us how the average cost is changing at that level of production.

[Math Processing Error]C¯′(256)=−683(256)2−0.077≈−0.08742

Thus, the average cost per belt is decreasing at a rate of 9 cents per belt when 256 have been produced.