How Do You Use the Substitution Method to Calculate Change?

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

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

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

$latex {P}'(t)=\left( 3t+3 \right){\left( {t}^{2}+2t+2 \right)}^{1/3};$

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 dollars. 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 dollars. 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.