Science:Math Exam Resources/Courses/MATH103/April 2013/Question 10
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Question 10 

Taylor Series: Find the first five terms of the Taylor series centred at 0 for i.e. the polynomial with the highest power of x. 
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 

First notice that the question is asking for the first 5 terms NOT the first 5 nonzero terms. Thus, it's possible to grind this out simply by taking derivatives. 
Hint 2 

The Taylor Series Formula about 0 is given by

Hint 3 

When taking derivatives, don't forget that your first function is the function defined by the integral and that you'll have to use the chain rule. 
Hint 4 

(Alternate Solution) If you want, you can start with the Taylor expansion of and go from there. 
Hint 5 

The Taylor expansion is given by . 
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 1 

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Please rate my easiness! It's quick and helps everyone guide their studies. We mechanically plug in the derivatives to the Taylor series formula given by
where . Now, we have . Using the FTC, we get . The next derivative is (via the chain rule) . The next derivative is (via the quotient rule) . The last derivative is . Thus, the fourth Taylor polynomial is
as required. 
Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Starting with
We see that . Integrating yields
Since the question is seeking , we have that
as required. 