Only one problem on the Section 14.1 Homework was missed by very many people. In that problem you were given the rate of change of profit, *P*‘(*t*), and asked to calculate how much the profit changed. Since this is a question about *P*(*t*), you need to undo the derivative with an antiderivative in the form of the Fundamental Theorem of Calculus. With this function, we would write it as

$latex displaystyle intlimits_{a}^{b}{P'(t),dx=P(b)-P(a)}$

**Problem 1** The rate of change of profit (in thousands of dollars per year) after t years of operation is

$latex displaystyle {P}'(t)=left( 3t+3 right){{left( {{t}^{2}}+2t+2 right)}^{{scriptstyle{}^{1}!!diagup!!{}_{3};}}}$

Find the total profit in the first three years.

Since the rate was in thousands of dollars per year, the profit function must be in thousands of dollars or $46,340. I would probably round this answer to more decimal places so that I could read the profit from the definite integral to the nearest penny. In this case it would be 46.34094 thousand dollars or $46,340.93. By the way, the second number shown (37.48 thousand dollars) is the definite integral from 3 to 4 which represents the profit in the fourth year of operation.