Marginal Analysis of a Demand Function

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Lesson Parent: 
Marginal Analysis as an Instantaneous Rate of Change
Problem: 

A product's demand function is found to be $n(p)=0.006p^4-0.15p^3+1.4p^2-5.0p+11$ where $n$ is the number of units bought and $p$ is the price per unit. How fast is the demand increasing when the price is $\$3$ per unit and $\$7$ per unit?

Answer: 

$n'(3)=-3.002$ and $n'(7)=-6.218$

SAMPLE SOLUTION

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Solution: 

Recall the definition of a demand function.

To find how fast the demand is increasing we need to determine $n'(p)$. We know that $n(p)$ is a polynomial and that the derivative of a polynomial is easily determined. Therefore, <Sign in to see all the formulas>.

At $p=\$3<Sign in to see all the formulas>n'(3)=0.024(3^3)-0.45(3^2)+1.8(3)-5.0=-3.002<Sign in to see all the formulas>p=\$7$ we have <Sign in to see all the formulas>. Note that the marginal demand is negative. This means that demand is decreasing and not increasing. Therefore, when the price increases from $\$3$ to $\$4$ then the number of units sold will decrease by approximately $3$ units. When the price increases from $\$7$ to $\$8$ then the number of units sold will decrease by approximately $6$ units. This makes sense because as the price increases we would expect less units to be sold.