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Problem 56 Medium Difficulty

The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of $ p $ dollars per pound is $ Q = f(p) $.

(a) What is the meaning of the derivative $ f'(8) $? What are its units?

(b) Is $ f'(8) $ positive or negative? Explain.

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Video Transcript

Yeah in this problem we have the function Q. Equals a function of P. Where Q. Is the number of pounds to quantity, the number of pounds of coffee sold. And it is a function of P where P. Is the price uh in dollars per pound, the price of the coffee. F. Prime of a. In this particular case the eight is in the location of the P. So a. Is the price The current price of coffee. So currently the coffee is $8 a pound that prime of a. Uh lets us know the rate at which the number of pounds of coffee sold will now change with respect to a change in price. I'll say that again. Um Think of F. Prime of eight. Think of the derivative as to change in F over the change in the variable pick. So currently Because of this eight, that means the current price of coffee is $8 a pound As the price increases slightly above the $8. That prime of eight tells us how uh the quantity or the number of pounds of coffee sold will change with respect to change in the price as the price increases above E. $8 per pound. Now, what do we expect this to be a positive or a negative amount? Uh F prime Uh the derivative of F. evaluated at eight. Well as the price of coffee increases. Uh the number of pounds sold is going to decrease if it's more expensive people are not going to want to buy as much. And so f prime of eight, we will expect to be negative because uh negative, divided by a positive is a negative. So F prime of eight will be negative, it will be less than zero.