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

Let Consider its 20th order Taylor polynomial at q=1; What is ? 
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 20th order Taylor polynomial of a function at a point from your text (don't let the change of notation scare you! In this case, f is p and x is q). 
Hint 2 

Recall that the second order Taylor polynomial of a function at a point is given by where, In our case, and . However even knowing this, it might not be clear how to proceed  Look at what is in the equation above and try to figure out a pattern to help you compute this value. 
Hint 3 

The previous hint tells us that
So we need to compute the 20th derivative of at the point . Try computing the first few derivatives and seeing if you can figure out the pattern. 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Proceeding as in the hints, we compute the 20th derivative to be
Now using the second hint, we see that
which completes the problem. 