Science:Math Exam Resources/Courses/MATH104/December 2011/Question 01 (n)

MATH104 December 2011

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Question 01 (n)
Find an approximation to $(7.5)^{1/3}$ by using the linear approximation to $f(x)=x^{1/3}$ at $a=8$. Give your answer as a fraction, $p/q$, in lowest terms.

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
What is the definition of the linear approximation to a function $f(x)$ at the point $a$?

Hint 2
The linear approximation, $L(x)$, to a function, $f(x)$, at the point $a$, is given by $\displaystyle L(x)=f(a)+f'(a)(x-a).$

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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. The linear approximation, $L(x)$, to a function, $f(x)$, at the point $a$, is given by $\displaystyle L(x)=f(a)+f'(a)(x-a).$ From the information given we can write the linear approximation to the function $f(x)=x^{1/3}$ at $a=8$ as follows: ${\begin{aligned}L(x)&=8^{1/3}+{\frac {1}{3}}8^{-2/3}(x-8)\\&=2+{\frac {1}{12}}(x-8).\end{aligned}}$ Thus, the linear approximation gives us the approximate value for $(7.5)^{1/3}$: ${\begin{aligned}(7.5)^{1/3}\approx L(7.5)&=2+{\frac {1}{12}}(7.5-8)\\&=2-{\frac {1}{24}}\\&={\frac {47}{24}}.\end{aligned}}$ (This answer is already expressed in lowest terms.)

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