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)=0.006p4−0.15p3+1.4p2−5.0p+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.