Science:Math Exam Resources/Courses/MATH104/December 2012/Question 01 (k)

MATH104 December 2012

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Question 01 (k)
The quadratic approximation of a demand function q(p) at p=10 is $\displaystyle {}T_{2}(p)=2-4(p-10)+3(p-10)^{2}.$ Calculuate $\epsilon (10)$ where $\epsilon (p)$ is the price-elasticity of demand.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Recall that the formula for the price-elasticity of demand is $\displaystyle \epsilon (p)={\frac {p\cdot q'(p)}{q(p)}}$.

Hint 2
The problem requests that we fine $\epsilon (10)$ which is given by $\displaystyle \epsilon (10)={\frac {10\cdot q'(10)}{q(10)}}$ so we need to find the values $q(10)$ and $q'(10)$. How can we do this with the given information?

Hint 3
Recall that the quadratic approximation is just the second degree Taylor polynomial given by $\displaystyle T_{2}(p)=q(10)+q'(10)(p-10)+{\frac {1}{2}}q''(10)(p-10)^{2}.$ Compare this with the given to find the unknown values as given in the previous hint.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Using the hints, we have ${\begin{aligned}\epsilon (p)&={\frac {p\cdot q'(p)}{q(p)}}\\\epsilon (10)&={\frac {10\cdot q'(10)}{q(10)}}={\frac {10(-4)}{2}}=-20\end{aligned}}$ where the values of $q(10)=2$ are read from the given degree 2 Taylor polynomial expansion and similarly for $q'(10)$.

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