Science:Math Exam Resources/Courses/MATH110/April 2016/Question 08 (a)

MATH110 April 2016

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Question 08 (a)
(a) Use an appropriate linear approximation to estimate ${\frac {1}{4.16}}$. 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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Try using a linear approximation. You should set $f(x)={\frac {1}{x}}$ and center your approximation around $a=4$.

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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. We want to approximate the value ${\frac {1}{4.16}}.$ To this end, we define the function $f(x)={\frac {1}{x}}$. We will use a linear approximation: $f(x)\approx f(a)+f'(a)(x-a)$ We want to use $x=4.16$ and $a=4$. It is necessary to compute $f'(4)$. The derivative of $f(x)$ is $f'(x)=-{\frac {1}{x^{2}}}$ so $f'(4)=-{\frac {1}{4^{2}}}=-{\frac {1}{16}}$ so $f(4.16)\approx {\frac {1}{4}}-{\frac {1}{16}}(4.16-4)$ and therefore $f(4.16)\approx 0.25-{\frac {0.16}{16}}$ and we get the approximation $f(4.16)\approx 0.24.$ This is believable because we expect to get something just a little less than $0.25$, since we are dividing by a number that is just a little more than 4. Answer: ${\frac {1}{4.16}}\approx 0.24$