Question

Answer each of the following in details~:

(a) Can the Bisection method be used to find the roots of the function ?(?) = 1 + ??? ?? Justify your Answer.

(b) While using the Newton’s method with the initial guess ?0 = 4 and ?(?0) = 1 gives ?1 = 3. Find ?′(?0).

(c) While using the Secant method for finding a root, ?0 = 2, ?1 = −1, ??? ?2 = −2 with ?(?1) = 4. Find ?(?0).

Answer #1

Consider the function f(x) = 1 2 |x|.
a) Can we use bisection search to find one of its roots? Why or
why not?
b) Can we use Newton’s method to find one of its roots? Why or
why not?

Q1: Use bisection method to ﬁnd solution accurate to within
10^−4 on the interval [0, 1] of the function f(x) = x−2^−x
Q3: Find Newton’s formula for f(x) = x^(3) −3x + 1 in [1,3] to
calculate x5, if x0 = 1.5. Also, ﬁnd the rate of convergence of the
method.
Q4: Solve the equation e^(−x) −x = 0 by secant method, using x0
= 0 and x1 = 1, accurate to 10^−4.
Q5: Solve the following system using the...

Find root of the equation cos (x) = xex using
Bisection method. Make calculation for 4 iterations. Choose
xl= 0 and xu= 1. Determine the approximate
error in each iteration. Give the final answer in a tabular
form.

Consider the function ?(?) = ?^2 − 3? − 2.
a) Use Newton’s Method to estimate a root for the function given by
the above formula. More precisely: Using the initial value of ?1 =
5, calculate ?3.
b) Solve the quadratic equation ?^2 − 3?− 2 = 0 and compute the
two solutions to 4 decimal places. How do these compare to the
approximate root you computed in part (a) above?
c) Suppose your friend uses Newton’s Method to...

Given function ?(?) = ?^4 − 2?^3 + 3?^2 − 2? + 1.
a) Solve analytically to find the exact roots of the equation
?(?) = 0. (Hint: ?^4 − 2?^3 + 3?^2 − 2? + 1 = (?^2 − ? + 1)^2)
b) Predict if you can find a zero of ?(?) = 0 by using Newton’s
method with any real initial approximation Po.

Use the Rational Root Theorem to find all possible rational
roots then use Newton’s Method to find one rational root and then
synthetic division and the quadratic formula to find the any
remaining rational, irrational or complex roots.
Px=3x^5-8x^4-17x^3+38x^2+20x-24

Solve the following problem using the MATLAB environment
Write a function [approx_root, num_its] = bisection(f,a,b,tol)
that implements the bisection method. You function should take as
input 4 arguments with the last argument being optional, i.e, if
the user does not provide the accuracy tol use a default of 1.0e-6
(use varargin to attain this). Your function should output the
approximate root, approx_root and the number of iterations it took
to attain the root, num_its. However, if the user calls the...

SOMEONE, PLEASE ANSWER
This is for Numerical Methods class homework
Consider the function ex + x -
7
Find an approximation of the root of f(x) with an absolute error
less than 0.001 using Newton’s method
Please plot the function to choose an initial guess and
conduct as many iterations as needed until you reach the specified
error

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...

Let
f(x)=sin(x)+x^3-2. Use the secant method to find a root of f(x)
using initial guesses x0=1 and x1=4. Continue until two consecutive
x values agree in the first 2 decimal places.

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