Find the change in profit

Question

A company has a weekly marginal profit function given by P'(x)=30x-0.33x^2-235 dollars per item when x items are made and sold. Find the change in profit when the number of items made and sold each week increases from 132 to 153.

Answer 1

You are only given the weekly marginal profit (P'(x) = 30x - 0.33x² - 235) and not the actual profit (P(x)). To find the amount the profit changes, you need to know the profit function. That means you need to take the antiderivative of P'(x) = 30x - 0.33x² - 235 to get the answer.

So what function is it that when you take the derivative, you get P'(x) = 30x - 0.33x² - 235?

Wouldn't it be 15x² - 0.11x³ - 235x + (some constant)? Check this by taking the derivative of it to make sure it results in the marginal profit function.

So to find the change in profit, evaluate this antiderivative at 153 then at 132 and subtract them. Please note that you do NOT need to include the "some constant" because you add it in then subtract it off so it provides no change in the result.

So the change in profit is [15(153)² - 0.11(153)³ - 235(153)] - [15(132)² - 0.11(132)³ - 235(132)] = [351135 - 393973.47 - 35955] - [261360 - 252996.48 - 31020] = -78793.47 - -22656.48 = -56136.99 so the profit goes down by $56,136.99

Another way to do this, if you have a calculator like the TI-84 Plus CE is to use the integration button.

The change in profit = ₁₃₂∫¹⁵³ 30x - 0.33x² - 235 dx and using that, you get the same value.