Science:Math Exam Resources/Courses/MATH110/April 2019/Question 07 (b)

MATH110 April 2019

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Question 07 (b)
Using an appropriate quadratic approximation of $f(x)=x^{1/3}$, find another estimate of $(7)^{1/3}$. This time you do not need to simplify your answer.

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
Given a twice-differentiable function $f(x)$, its quadratic approximation at a point $x=a$ is a function $Q(x)=kx^{2}+lx+m$ such that $Q{\text{ and }}f$ have the same value, slope and second derivative at a. In this case $f(x)=x^{1/3}$ and the point $a=8$.

Hint 2
Once you have the formula for the quadratic approximation of $f(x)=x^{1/3}$, compare it to the linear approximation you calculated in part (a). Do you notice any similarities? Can you think of how this helps you calculate the quadratic approximation more efficiently in an exam scenario?

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. We are asked to use the quadratic approximation of $f(x)=x^{1/3}$ at the point $a=8$ to approximate $(7)^{1/3}$. The formula for this quadratic approximation, which we denote by $Q(x)$, is given by $Q(x)=f(a)+f'(a)(x-a)+{\frac {f''(a)}{2}}(x-a)^{2}$. We already know that $a=8,f(x)=x^{1/3}$, so we need only find $f'(a),f''(a)$. To do so, we apply the power rule twice, and find that: $f'(x)={\frac {d}{dx}}x^{1/3}={\frac {1}{3}}x^{-2/3}$, and then $f''(x)={\frac {d}{dx}}{\frac {1}{3}}x^{-2/3}=-{\frac {2}{9}}x^{-5/3}$ Substituting this information into our formula for $Q(x)$, we have $Q(x)=(8)^{1/3}+{\frac {1}{3}}(8)^{-2/3}(x-8)-{\frac {2}{9}}\cdot {\frac {8^{-5/3}}{2}}(x-8)^{2}$. Then, to estimate the value of $7^{1/3}$, we plug-in $x=7$ into our quadratic approximation. This gives $(7)^{1/3}\approx Q(7)=(8)^{1/3}+{\frac {1}{3}}(8)^{-2/3}(7-8)-{\frac {2}{9}}\cdot {\frac {8^{-5/3}}{2}}(7-8)^{2}$.