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

MATH104 December 2011

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Question 06 (c)

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. c) Is your approximation for $\ln(0.9)$ too large or too small? Explain. Use this information together with the error bound found in part (b) to construct an interval you can guarantee contains the true value of $\ln(0.9)$.

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 concavity help with determining an over or under estimate by the linear approximation?

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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 find the second derivative of f(x)=ln(x), ${\begin{aligned}f'(x)&={\frac {1}{x}}\\f''(x)&=-{\frac {1}{x^{2}}}.\end{aligned}}$ By looking at the sign of the second derivative of the function $\ln(x)$, we see that the function is concave down for all $x$. Thus, the linear approximation is too large. (One can see this visually by drawing a tangent line to the $\ln(x)$ function at any point on the curve). Since, we know the linear approximation overestimates the value $\ln(0.9)$, we know that the true value of $\ln(0.9)$ must lie within the interval: ${\Big [}-0.1-{\frac {1}{162}},-0.1{\Big ]}$, where we have used our upper bound on the error from part (b) to help define the lower bound on our interval.

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