Science:Math Exam Resources/Courses/MATH100/December 2015/Question 11 (b)
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Question 11 (b) 

If , show that the absolute value of the error when we approximate using the second Maclaurin polynomial is less than . 
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 

Recall the Taylor's inequality: If for , then the remainder of the Taylor series of at satisfies the inequality for . 
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. Note that Maclaurin series is the special case of Talyor series at . Since is given and we know that and for any in the real line, we have for
Then, using the Taylor's inequality in the Hint, we have
for . Therefore, we obtain
