How Do You Undo A Rate with the Substitution Method?

Suppose that the profit for a company is increasing at a rate of


where the company has been in operation for t years. What is the total change in profit over the first three years?

In this problem, we are given the rate at which profit is changing over time. This is confirmed by the fact that the function is defined as P′(t), the derivative of profit. However, the question is about the corresponding profit function P(t). So we need to find this profit function by taking the antiderivative of P′(t),


Remember, the antiderivative undoes the derivative so the antiderivative of P′(t) is P(t). To do this antiderivative, we need to use the Substitution Method.


This means that


You might think that the total change in profit over the first three years is P(3), but this is the profit at the end of the third year. To find the total change in profit we need to calculate P(3) – P(0),