Science:Math Exam Resources/Courses/MATH102/December 2015/Question 11

MATH102 December 2015

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[hide] Question 11
Consider each of the labeled points (solid dot) on the graph of $f(x)$ as a starting point for Newton’s method. To which zero of the function $f(x)$ (empty dots $Z_{1}$ ,$Z_{2}$ , or neither) will Newton’s method converge for each one? You may assume that the graph of the function continues off the edges of the graph with no significant change in direction.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
Remember Newton's method is a iteration: $x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}$

[show] Hint 2
Graphically, you can find the point $x_{n+1}$ by drawing the tangent line to the graph at $x_{n}$ and looking at the intersection with the x-axis.

Checking a solution serves two purposes: helping you if, after having used all the hints, 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.

[show] Solution

A: use the equation $x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}$. After the first iteration, the next point will locate on the left side of $Z_{1}$, then a few times later, it will gradually converge to $Z_{1}$. B: the derivative at B is 0, thus Newton's method does not work in this case. C: after the first iteration, the next point will locate on the left side of $Z_{2}$, then a few times later, it will gradually converge to $Z_{2}$. D: after the first iteration, the next point will locate on the very far left side of $Z_{1}$(because the derivative is small at D), then if assume that the graph of the function continues off the edges of the graph with no significant change in direction, the point will approach $Z_{1}$ on the left side gradually with the iteration. answer: $\color {blue}A-Z_{1},B-neither,C-Z_{2},D-Z_{1}$