Question

f(x)=10-x^3-3cosx=0 use newton iteration to estimate the root,

Answer #1

Find the root of the function: f(x)=2x+sin(x)-e^x,
using Newton Method and initial value of 0. Calculate the
approximate error in each step. Use maximum 4 steps (in case you do
not observe a convergence).

Use the fixed point iteration method to find a root for the
function g(x)=2^(-x) in the interval [0,1] with an error of
0.03%

For the following function, determine the highest real root
of
f(x) = 2x3 – 11.7x2 + 17.7x - 5
by using (a) graphical methods, (b) fixed point iteration (three
iterations, x0 = 3) (Hint: Be certain that you develop a solution
that converges on the root), and (c) Newton-Raphson method (three
iterations, x0 = 3).
Perform an error check on each of your final root approximations
(e.g. for the last of the three iterations).

Use intermediate theorem to show that theer is a root of
f(x)=-e^x+3-2x in the interval (0, 1)

Use the secant method to estimate the root of
f(x) = -56x + (612/11)*10-4 x2 -
(86/45)*10-7x3 + (3113861/55)
Start x-1= 500 and x0=900.
Perform iterations until the approximate relative error falls below
1% (Do not use any interfaces such as excel etc.)

Utilize Newton's Method to estimate the root of 3 sin x - x = 0
for x > 0 correct to the sixth decimal places. Show all work
below.
(Hint: start with x1 = 2)

Use
the root-solving method to obtain the root of x^4-10x^3-50x+24=0
within the error range of 10^-4. (Put in the initial x^(0)=3.5 /
Please make the significant figure 10^-5.)

Newton's method: For a function ?(?)=ln?+?2−3f(x)=lnx+x2−3
a. Find the root of function ?(?)f(x) starting with
?0=1.0x0=1.0.
b. Compute the ratio |??−?|/|??−1−?|2|xn−r|/|xn−1−r|2, for
iterations 2, 3, 4 given ?=1.592142937058094r=1.592142937058094.
Show that this ratio's value approaches
|?″(?)/2?′(?)||f″(x)/2f′(x)| (i.e., the iteration converges
quadratically). In error computation, keep as many digits as you
can.

Suppose f′(x) = √(x)*sin(x^2) and f(0) = 5.
Estimate f(b) for:
b = 0 : f(b) ≈ 5
b = 1: f(b) ≈
b = 2 : f(b) ≈
b = 3 : f(b) ≈
(Only the square root of x is being taken.)
(This is all the information the question provides. If it helps,
the chapter is on "Theorems about definite integrals.")

Use Newton-Raphson to find a solution to the polynomial equation
f(x) = y where y = 0 and
f(x) = x^3 + 8x^2 + 2x
- 40. Start with x(0) = 1 and continue until (6.2.2) is
satisfied with e= 0.0000005.

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