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 $S(n)=0.1n^4-2.8n^3-19.1n^2+918n-575$. 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 $\$3,000<Sign in to see all the formulas>738.6<Sign in to see all the formulas>\$3,001$.