Science:Math Exam Resources/Courses/MATH104/December 2011/Question 06 (b)
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q1 (l) • Q1 (m) • Q1 (n) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q2 (f) • Q2 (g) • Q3 • Q4 • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q6 (a) • Q6 (b) • Q6 (c) • Q6 (d) • Q6 (e) •
Question 06 (b) 

In this problem, you will approximate in two ways: one using linear approximation and the other using quadratic approximation. You may find it useful to draw the graph of for near 1 to help you answer some parts of this question. b) Use the formula where > 0, to estimate an error bound for your approximation of in part (a). It will be useful to remember how to find such an . 
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 

How does M relate to the second derivative of f(x)? 
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 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Here we are working with and . We find that the error takes the following value where is the maximum value of the absolute value of the second derivative on the interval [0.9, 1]. In other words, we would like to estimate how large the second derivative can be on this interval. Given that and since this is a decreasing function on the interval [0.9, 1] we can deduce that the maximum value it takes on the interval is at and hence This allows us to find the upper bound on the error: 