Question

17. I am using Newton’s method to ﬁnd the negative root of f(x)
= 3−x^{2}.

(a) What would be a good guess for x_{1}? Draw the line
tangent to f(x) at your x_{1} and explain why using
Newton’s method would lead to the negative root of the
function.

(b) What would be a bad guess for x_{1}? Draw the line
tangent to f(x) at your x_{1} and explain why using
Newton’s method would not lead to the negative root of the
function.

Answer #1

(a) We know, to find the roots of a solution using Newton Raphsons method

xn+1= xn- f(xn)/f'(xn)

Where xn= root after nth iteration

f'(xn)= value of the differentiation at xn.

Given f(x)=3-x2

f'(x)=-2x

So x1= x0+(3-x02)/2x0

A good guess value would be -1, since at x0=-1, x1=-2 which is negative. The tangent will lead to a negative root.

(b) 1 will be a bad root because at x0=1, x1=2 and hence it is not converging to a point, but diverging, since the actual root is plus and minus 1.732. But taking this tangent will lead to the positive solution.

Complete four iterations of Newton’s Method for the function
f(x)=x^3+2x+1 using initial guess x1= -.5

2. (a) For the equation e^x = 3 - 2 x , find a function, f(x),
whose x-intercept is the solution of the equation (i.e. a function
suitable to use in Newton’s Method), and use it to set up xn+1 for
Newton’s Method.
(b) Use Newton's method to find x3 , x4 and x5 using the initial
guess x1 = 0 . How many digits of accuracy are you certain of from
these results?
(c) Use x1+ ln 2 and show...

4. Let f(x) = 14xex+1 + 30ex+1 −7x3 −43x2 −95x−75. (a) Apply
Newton’s method to ﬁnd both roots of the function in the interval
[−5 2, 1 2], to as much precision as possible. For each root, print
out the sequence of iterates, the errors ei, and the error ratios
ei+1/e_i and ei+1/e2 i. (b) In each case, determine if the error
converges linearly or quadratically. Explain brieﬂy why and what
you conclude from it.

Approximate the zero for f(x) = (x^3)+(4x^2)-3x-8 using newton's
method
Use x1 = -6
A)Find x2,x3,x4,x5,x6
B)Based on the result, you estimate the zero for the function to
be......?
C)Explain why choosing x1 = -3 would have been a bad idea?
D) Are there any other bad ideas that someone could have chosen
for x1?

: Consider f(x) = 3 sin(x2) − x.
1. Use Newton’s Method and initial value x0 = −2 to approximate
a negative root of f(x) up to 4 decimal places.
2. Consider the region bounded by f(x) and the x-axis over the
the interval [r, 0] where r is the answer in the previous part.
Find the volume of the solid obtain by rotating the region about
the y-axis. Round to 4 decimal places.

Find the root of f(x) = exp(x)sin(x) - xcos(x) by the
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starting with an initial value of xo = 1.0.
Solve by using Newton’method until satisfying the tolerance
limits of the followings;
i. tolerance = 0.01
ii. tolerance = 0.001
iii. tolerance= 0.0001
Comment on the results!

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2)determine the equation of the line tangent to f(x) at
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b) Solve the quadratic equation ?^2 − 3?− 2 = 0 and compute the
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x values agree in the first 2 decimal places.

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