Problem

The demand, $D$, for a new rollerball pen is given by $D=0.007 p^{3}-0.5 p^{2}+130 p$, where $p$ is the price in dollars.
a) Find the rate of change of quantity with respect to price, $d D / d p$.
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?
a) $\frac{d D}{d p}=$

Answer

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Answer

Final Answer: The rate of change of quantity with respect to price, $\frac{d D}{d p}$, is $\boxed{0.021p^{2} - 1.0p + 130}$.

Steps

Step 1 :Given the demand function $D=0.007 p^{3}-0.5 p^{2}+130 p$, where $p$ is the price in dollars.

Step 2 :We are asked to find the rate of change of quantity with respect to price, $\frac{d D}{d p}$.

Step 3 :We can solve this by applying the power rule for differentiation, which states that the derivative of $x^n$ is $n*x^{n-1}$.

Step 4 :Applying the power rule to each term in the function, we get $\frac{d D}{d p} = 0.021*p^{2} - 1.0*p + 130$.

Step 5 :Final Answer: The rate of change of quantity with respect to price, $\frac{d D}{d p}$, is $\boxed{0.021p^{2} - 1.0p + 130}$.

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