Science:Math Exam Resources/Courses/MATH100/December 2016/Question 05 (a)

MATH100 December 2016

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Question 05 (a)
Which of the following graphs is a good approximation of $f(x)={\frac {x^{2}-2x}{x+7}}$ for $x$ on the small interval $\left[-{\frac {1}{10}},{\frac {1}{10}}\right]$? graphs

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
Consider the linear approximation of $f$ at $0$.

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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 function is continuous and differentiable near $0$. Then its derivative is $f'(x)={\frac {(x+7)(2x-2)-(x^{2}-2x)}{(x+7)^{2}}},$ and hence we get $f'(0)={\frac {-14}{7^{2}}}=-{\frac {2}{7}}<0.$ Therefore, its linear approximation at $x=0$ has negative slope and passes through $(0,f(0))=(0,0)$. Answer: the second picture in the top row.