Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 29(b)

MATH100 A December 2023

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Question 29(b)
Graph of $f$ (b) Label $x_{0},x_{1},x_{2},$ and $x_{3}$ on the graph below. Will the iterations of Newton’s method started in part (a) eventually converge toward the root? Justify your answer in 3-5 sentences.

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
The points $x_{0},\dots ,x_{3}$ are numbers on the $x$-axis.

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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. Example of Newton's method. In part (a) we computed $x_{2}=x_{0}=0$ and $x_{1}=x_{3}=-1$. These are labelled on the $x$-axis of the adjacent graph (the points $(x_{n},f(x_{n}))$ and the linear approximations are indicated by the red lines). Since Newton's method for a particular function only uses the number $x_{n}$ to produce $x_{n+1}$, it follows from part (a) that if $x_{n}=0$, then $x_{n+1}=-1$ and, if $x_{n}=-1$, then $x_{n+1}=0$. Therefore, the sequence of estimates is given by $x_{n}={\begin{cases}0&{\text{ if }}n{\text{ is even}}\\-1&{\text{ if }}n{\text{ is odd}}\end{cases}}$ and the sequence will not converge to a root.