Example 6

Suppose a bacterial culture grows in such a way that at time t there are t³ bacteria. Find the rate of growth at time t = 1000 sec.

y = t³, y' = 3t²

by Example 1. At t = 1000,

y' = 3,000,000 bacteria/sec.

Suppose the cost of making x needles is √x dollars. What is the marginal cost after 10,000 needles have been made?

by Example 2. At x = 10,000,

dollars per needle.

Thus the marginal cost is one half of a cent per needle.