Science:Math Exam Resources/Courses/MATH110/April 2018/Question 08 (c)

MATH110 April 2018

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Question 08 (c)
In this question you will approximate the value of ${\textstyle e^{0.5}}$ in two different ways and answer a few questions about your calculations. Make sure you justify your answers. Answers without justification will not receive any points. You may assume that ${\textstyle e=2.718281...}$. (c) Which of the two estimates you computed above do you expect to be more accurate? 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
Think about which values in the linear approximation formula would affect the accuracy of the 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 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Answer for MATH 110: The accuracy of the linear approximation depends on the distance between the point ${\textstyle x=a}$ and the ${\textstyle x}$-value of the point we are trying to estimate: in general a function only behaves approximately like a linear function between two points that are very close together. This would suggest that the first approximation should be more accurate, since the ${\textstyle x}$-values only differ by ${\textstyle 0.5}$ whereas in the second approximation they differ by ${\textstyle e-1}$ which is more than three times as big. Answer: One would expect the linear approximation from ${\textstyle {\color {blue}(a)}}$ to be more accurate.

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. This answer is outside the curriculum of MATH110. But this can be a valid answer for other 100 level courses of which curriculum contains the error formula. In addition to the explanation from part (a), the accuracy of the linear approximation will also depend on how far the function is from being linear: if a function is truly linear then the linear approximation would give an accurate estimate even if the two ${\textstyle x}$-values used are very far apart, whereas an exponential function diverges very quickly from it’s tangent line. This is measured by the second derivative of the function. The second derivative of the function used in (b) at ${\textstyle x=1}$ is significantly smaller in absolute value than the second derivative of the function used in (a) at ${\textstyle x=0}$, so this would suggest that (b) should give a more accurate approximation. Answer: One would expect the linear approximation from ${\textstyle {\color {blue}(b)}}$ to be more accurate.