Newton's Method Assignment.sagews

Newton's Method Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Question 1

Estimate the roots of the function $f$ using the algorithm developed in class (copy Example 5):

• Graph the function.

• Use the graph to choose a guess close to one root.

• Use your guess to perform Newton's Method (use 10 iterations).

• Repeat the process for any additional roots.

Part a

$f(x)=x^3-3x^2+2x-1$

[Hint: there is one answer.]

Part b

$f(x)=3\ln(x)-x$

[Hint: there are two answers.]

Question 2

Find the points of intersection of two functions $f_1$ and $f_2$ (copy example 6):

• Define a new function $f(x)=f_1(x)-f_2(x)$.

• Follow the steps above to find the roots of $f$. These are the x-coordinates of the points of intersection.

• Find the y-coordinates of the points of intersection by plugging the roots into both $f_1$ and $f_2$ (you should get the same answer).

Part a

$f_1(x)=e^x$, $f_2(x)=2-x$

[Hint: there is one answer.]

Don't forget to find the y-coordinate at the end.

Part b

$f_1(x)=\ln(x)$, $f_2(x)=x^2-10$

[Hint: there are two answers, and one is really close to 0.]

Don't forget to find the y-coordinate at the end.

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.