The demand, $D$, for a new rollerball pen is given by $D=0.009 p^{3}-0.5 p^{2}+180 p$, where $p$ is the price in dollars.
a) Find the rate of change of quantity with respect to price, $\mathrm{dD} / \mathrm{dp}$.
b) How many units will consumers want to buy when the price is $\$ 25$ per unit?
c) Find the rate of change at $p=25$, and interpret this result.
d)Would you expect $\mathrm{dD} / \mathrm{dp}$ to be positive or negative?
Answer
Final Answer: The rate of change of quantity with respect to price, \(\frac{dD}{dp}\), is \(\boxed{0.027p^{2} - 1.0p + 180}\).
Step 1 ：We are given the demand function, \(D = 0.009p^{3} - 0.5p^{2} + 180p\), where \(p\) is the price in dollars.
Step 2 ：We are asked to find the rate of change of quantity with respect to price, \(\frac{dD}{dp}\). This is a calculus problem and we can find the derivative using the power rule for differentiation, which states that the derivative of \(x^n\) is \(n*x^{n-1}\).
Step 3 ：Applying the power rule to each term in the demand function, we get \(\frac{dD}{dp} = 0.027p^{2} - 1.0p + 180\).
Step 4 ：Final Answer: The rate of change of quantity with respect to price, \(\frac{dD}{dp}\), is \(\boxed{0.027p^{2} - 1.0p + 180}\).
