Question

Consider a function *f*(*x*) =
2*x*^{3} − 11.7*x*^{2} +
17.7*x* − 5.

Identify the root of the given function after the third
iteration using the secant method. Use initial guesses
*x*_{–1} = 3 and *x*_{0} = 4.

CAN YOU PLZ SHOW ALL THE WORK. THANK YOU

Answer #1

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).

Consider the function, f(x) = - x4 - 2x3 -
8x2 - 5x
Use parabolic interpolation (x0 = -2, x1 =
-1, x2= 1, iterations = 4). Select new points
sequentially as in the secant method.

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.

Consider the function
g (x) = 12x + 4 - cos x. Given
g (x) = 0 has a unique solution
x = b in the interval (−1/2, 0), and you can use this
without justification.
(a) Show that Newton's method of starting point
x0
= 0 gives a number sequence with
b <··· <xn+1
<xn
<··· <x1
<x0
= 0
(The word "curvature" should be included in the argument!)
(b) Calculate
x1
and x2.
Use theorem 2 in section...

Consider the function g (x) = 12x + 4 - cos x.
Given g (x) = 0 has a unique solution x =
b in the interval (−1/2, 0), and you can use this without
justification.
(a) Show that Newton's method of starting point
x0 = 0 gives a number sequence with
b <··· <xn+1 <xn <···
<x1 <x0 = 0
(The word "curvature" should be included in the argument!)
(b) Calculate x1 and x2. Use
theorem 2 in section...

Consider the function g (x) = 12x + 4 - cos x.
Given g (x) = 0 has a unique solution x =
b in the interval (−1/2, 0), and you can use this without
justification.
(a) Show that Newton's method of starting point
x0 = 0 gives a number sequence with
b <··· <xn+1 <xn <···
<x1 <x0 = 0
(The word "curvature" should be included in the argument!)
(b) Calculate x1 and x2. Use
theorem 2 in section...

If we want to minimize a function f(x) = e^(x^2)
over R, then it is equivalent to finding the root of f '(x).
Starting with x0 = 1, can you perform 4 iterations of Newton's
method to estimate the minimizer
of f(x)? (Correct to four decimal places at each iteration).

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.

Find the root of the function given below that is greater than zero with the Newton-Raphson method. First guess value You can get x0 = 0
f (x) = x2 + x - 2

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.)

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