As a manufacturer produces more units of some product, the cost to produce each unit should decrease as the worker skills and workflow improvements. This decline in cost-per-unit is called an experience curve or learning curve. One model of a learning curve is the unit learning curve model, which we will explore here.

The unit learning curve model assumes that each time the number of units to be produced doubles, say from n to 2n, the cost to produce the 2nth unit is some fraction of the cost to produce the nth unit. This fraction, r, is usually expressed as a percent.

For example, suppose there is a 90% learning curve and let C(x) be the cost to produce the xth unit.

Then: C(2n) = 0.90C(n).

It turns out that with a learning curve of r the cost to produce the xth unit is

C(x) = C(1)x ^ln(r)/ln(2) = C(1)x ^{log2 (r)}

5. It is much more important to find out what price would maximize the profit, rather than the revenue, of course. If C(x) is the cost to manufacture the xth switchboard, then the total cost to manufacture q switchboards would be

T(q) = C(1) + C(2) + C(3) + + C(q 1) + C(q).

In Part IV of this project, we will find an upper bound estimate on this total cost under the learning curve model we have been using: T(q) 319, 575q ^0.848 48, 575. Calculate the profit generated when revenue is maximized, assuming that T(q) 319, 575q ^0.848 48, 575.

Part IV: Finally, lets determine why 319, 575q ^0.848 48, 575 is a reasonable upper bound for the total cost to manufacture q switchboards,

T(q) = C(1) + C(2) + C(3) + + C(q 1) + C(q).

6. Notice that T(q) is an approximation of the area under the curve y = C(x) on the interval [0, q] with q rectangles and the right-hand rule. Use a definite integral to calculate the area under the curve y = C(x) exactly, in the case where C(x) = 271, 000x ^{log2 (0.9) .}

Connections:

7. Explain why the definite integral you calculated in Part IV is an upper bound on the total cost of manufacturing q switchboards. (Hint: start by explaining why T(q) is an underestimate for the integral.)

8. In Part I you calculated lim_x C(x) and in Part II you calculated lim_x C'(x). Explain why your answer in Part II makes sense in terms of your answer in Part I.