Help with this Calc question

Question

The demand, D, for a new rollerball pen is given by D=0.009p^3 - 0.3-^2 + 180p, where p is the price in dollars.

a) Find the rate of change of quantity with respect to price, dP/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; The number of consumers who want to buy will [increase or decrease] by [] when the price is $[] per unit.

d) Would you expect dD/dp to be positive or negative?

Answer 1

1) For this first part, we need to simply take the derivative of D with respect to price, so this gives us dD/dp = .027p² - 0.6p + 180.

2) To find the demand when the price is $25, we insert p=25 into the original function rule and evaluate:

D = .009(25)³ - 0.3(25)² + 180(25) which equals 4453.13, which we will round up to 4454 units.

3) To find the rate of change at p = 25, we insert 25 in for "p" in the derivative formula from part 1 :

0.027(25)² - 0.6(25) + 180 and this gives us 181.875, a positive rate of change when p = 25.

4) For this last part, looking at it from simply a consumer's point of view, as the price of the pens goes lower, one would expect that more of them would sell. From a mathematical perspective, the rate of change, dD/dp, is a quadratic function representing an upward-opening parabola that, for all domain values, sits above the x-axis and whose y-intercept is 180. So yes, one would expect the rate of change to be positive.