Science:Math Exam Resources/Courses/MATH104/December 2011/Question 06 (e)

MATH104 December 2011

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Question 06 (e)

In this problem, you will approximate $\ln(0.9)$ in two ways: one using linear approximation and the other using quadratic approximation. You may find it useful to draw the graph of $\ln(x)$ for $x$ near 1 to help you answer some parts of this question. e) Compare the error bound for the linear approximation to $\ln(0.9)$ you found in part (b) to the magnitude of the 2nd degree term of the quadratic approximation to $\ln(0.9)$. Which do you expect to be larger and why?

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
What is the 2nd degree term of the quadratic approximation? How does it compare to the error defined in part (b)? What is the key difference?

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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. We expect that the error term will be larger than or equal to the magnitude of the second-order term in the quadratic approximation. The reason for this is because the two terms have a similar form, but in the error, the $M$ term replaces the $\displaystyle {}f''(a)$ term in the quadratic correction, where $M$ is the maximum value of $\displaystyle {}|f''(x)|$ on the interval $[0.9,1]$. Indeed the error term was greater than the magnitude of the quadratic correction term in the quadratic approximation: (1/162 = 0.0061... > 0.005)

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