Science:Math Exam Resources/Courses/MATH110/April 2019/Question 07 (c)
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Question 07 (c) 

Find the Taylor polynomial of degree 3 of centered 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 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 

Given a threetimes differentiable function and a point , the Taylor polynomial of degree three is a polynomial such that have the same value, first, second and third derivatives at . Compare this to Questions 7(a) and 7(b). What similarities do you see in these formulas. Can you think of how noticing these similarities would allow you to answer this question more efficiently in an exam scenario? 
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. We are asked to find the Taylor polynomial of degree 3 for centred at . The formula for this quadratic approximation, which we denote by , is given by . Remember that is the third derivative of . We already know that , so we need only find . To do so, we apply the power rule three times, and find that , , and finally . Substituting this information into our formula for , we have . 