Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (m)
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Question 01 (m) 

Estimate by considering the linear approximation of at . 
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 , what is the linearization of ? What is the error term associated to this approximation? 
Hint 2 

The linearization (or linear approximation) of about a point is given by and it holds that Do you remember how to obtain the constant ? 
Hint 3 

The constant in the error term for the linear approximation , is the maximum of the second derivative of in the interval between and . What is , what is in our case? 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We are looking at the linear approximation of about the point . Hence and the linear approximation is given by We now want to estimate the error of this approximation at the point . From the formula for the error of a linear approximation we find that Finally it remains to calculate , which is the largest absolute value of the second derivative of in the interval . Doing the math we obtain (the number 1 could also be used as an upper bound though it is not as tight of an upper bound), therefore the final answer is 