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

Find the second order Taylor polynomial for at x=1. 
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 

Recall (or review) the definition of the 2nd order Taylor polynomial of a function at a point from your text. In this case, what is the function and what is ? 
Hint 2 

Recall that the second order Taylor polynomial of a function at a point is given by where, In our case, and . 
Hint 3 

Recall the derivative of arctan(x) 
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. For our problem, we have that and . Evaluating the derivatives, we get Thus, and so the 2nd order Taylor polynomial is 