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

Suppose you approximate by , where is the linear approximation of at . Provide an estimate for the absolute value of the error in your approximation. 
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 

Recall that an upper bound on the error in a linear approximation is given by
where is the error in a linear approximation at the point x and is an upper bound on the second derivative, that is for all values of x inside our interval . 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Following the hint, we first find a suitable value for M. To do this, we need the second derivative. Computing gives
and thus
where the above bound holds for any value of x. Plugging into the upper bound on the error in a linear approximation as given by the hint, we have
completing the question. 