Question

Consider the demand function D = 10(20 - 0.5p)⁴, where p is the unit price of a commodity and D is the demand for that commodity. Compute the instantaneous rate of change of the demand with respect to unit price when the commodity is priced at $6 per unit.

Answer 1

To find the instantaneous rate of change of demand at a unit price of $6, calculate the derivative of the demand function with respect to price and substitute $6 into it. The rate of change is -98260, meaning demand decreases rapidly as price increases at this point.

Step-by-step explanation:

To compute the instantaneous rate of change of the demand with respect to unit price when the commodity is priced at $6 per unit, we need to find the derivative of the demand function with respect to p and then substitute p = 6 into the derivative. The demand function is D = 10(20 - 0.5p)^4.

First, we find the derivative of D with respect to p using the chain rule:

1. Differentiate D with respect to p: D' = 10*4*(20 - 0.5p)^3*(-0.5).
2. Simplify the derivative: D' = -20*(20 - 0.5p)^3.
3. Substitute p = 6 into the derivative: D'(6) = -20*(20 - 0.5*6)^3 = -20*(17)^3.
4. Calculate the value: D'(6) = -20*4913 = -98260.

The instantaneous rate of change of the demand when the unit price is $6 is -98260 units of commodity per dollar.