56. 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 $f^{\prime}(8) ?$ What are its units? (b) Is $f^{\prime}(8)$ positive or negative? Explain.

56. 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 $f^{\prime}(8) ?$ What are its units?

(b) Is $f^{\prime}(8)$ positive or negative? Explain.

The meaning of f′(8) and its units.

Given:

Consider the function f(p) is a quantity (in pounds) of coffee is sold by a coffee company at a price of p dollars per pounds. That is Q=f(p).

Note that, the derivative f′(x) is the instantaneous rate of change of the function f(x) with respect to x

The derivative f′(p) is the rate of change of quantity (in pounds) of coffee is sold with respect to price of p dollars per pounds.

The derivative f′(8) is the rate of change of quantity (in pounds) of coffee is sold with respect to price of 8 dollars per pounds.

The instantaneous rate of change is equal to f′(t)=limΔx→0ΔyΔt here y=f(t).

Here Δy is measured in pounds and Δt is measured in dollars.

Thus, the units are pounds per dollars.

To explain: Whether f′(8) is negative or positive.

The derivative f′(8) is the rate of change of quantity (in pounds) of coffee is sold with respect to price of 8 dollars per pounds.

Since the quantity of coffee is sold is decrease when the price is increasing.

If price is 8 dollars, then the quantity of coffee sold will decreasing.

Thus, the value f′(8) is negative.