Marginal Analysis of Advertising

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

The number of units sold after spending n thousand dollars on advertising is approximated by S(n)=0.1n4−2.8n3−19.1n2+918n−575. Determine the marginal sales when the amount spent on advertising is $3,000 and $10,000.

Answer: 

96 and 738.6 when spending $3,000 and $10,000, respectively.

SAMPLE SOLUTION

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Solution: 

The marginal sales function is given by S′(n). Note that S(n) is a polynomial function and we know the derivative of a polynomial. Therefore, <Sign in to see all the formulas>.

The problem states that n is in thousands of dollars. So if we want to determine the marginal sales then n=3 when spending $3,000 and $n=10<Sign in to see all the formulas>\<Sign in to see all the formulas>. At n=3 we have

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and at n=10 we have

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When spending $\$10,000<Sign in to see all the formulas>96<Sign in to see all the formulas>\<Sign in to see all the formulas>. However, when spending $\$3,000<Sign in to see all the formulas>738.6<Sign in to see all the formulas>$3,001$.