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Suppose that the total profit in hundreds of dollars from selling $x$ items is given by $P(x)=-x^2+10x-22$. Find the marginal profit at $x=3$.

A. $-\$600$

B. $\$400$

C. $-\$100$

D. $\$600$

I am having trouble figuring out how to start this problem. The other day when I worked on this homework I simply plugged $3$ in for $x$ and I ended up with $ -1$ which isn't any of the answer choices. As my teacher kinda skimmed over this in class, I was wondering if any of you could point me in the right direction of how to solve this problem. As this is calculus and we are working on rate of change and limits currently I am assuming I will have to use a formula but I am not entirely sure where to start.

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2 Answers

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The key word is marginal profit, which measures how much the profit is changing at a specific number of units.

Thus, marginal profit would be the derivative of the profit function as that is precisely what we want. Finally, the remaining step would be to find how much the profit is changing at $x=3$ units which is just a substitution into the derivative.

Note: the question says that profit is measured in hundreds of dollars.

Hint for the derivative below:

$$P'(x)=-2x+10$$

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The unit is in hundred dollars. So when you got $P(3)=-1$, you should actually multiply the answer by \$100, so it's -\$100.
Well in this question you are looking for Marginal profit, which is to take the derivative of the total profit. $$P'(x)=-2x+10$$$$P'(3)=4$$ Therefore the answer is \$400

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