Science:Math Exam Resources/Courses/MATH102/December 2016/Question B 01

MATH102 December 2016

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Question B 01
The zeros of the quartic function $f(x)$ are labeled from $z_{1}$ to $z_{4}$, running left to right. Using Newton’s method, starting at the points $x_{0}=A,B,C,$ and $D$ (black dots), which zero will be found? Several relevant tangent lines have been provided (dashed lines).

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
Using Newton's method, find the tangent line at each of the given points $A,B,C,D$, then find the points where each tangent line crosses the x-axis, the closest root to this point is the root that is approximated.

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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. The tangent line at $(A,f(A))$ crosses the x-axis at a point close to $z_{1}$, so starting at $x_{0}=A$ will approximate $z_{1}$ . At $(B,f(B))$ the tangent line again crosses the x-axis at a point close to $z_{1}$, so $x_{0}=B$ is used to approximate $z_{1}$ too. At $(C,f(C))$ the tangent line crosses the x-axis at a point close to $z_{4}$, this means that $x_{0}=C$ is a starting point to approximate $z_{4}$ . The tangent line at $(D,f(D))$ crosses the x-axis at a point close to $z_{3}$, so starting at $x_{0}=D$ will approximate $z_{3}$ . Answer: $\color {blue}z_{1},z_{1},z_{4},z_{3}$.