Question

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.)

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 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.)

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

Use Newton’s method to find solutions accurate to within 10−4
for x − 0.8 − 0.2 sin x = 0, x in[0, π/2]. (Choose ?0=π/4).

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...

: 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.

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.

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.

Use C++ in Solving Ordinary Differential Equations using
a
Fourth-Order Runge-Kutta of Your Own Creation
Assignment:
Design and construct a computer program in C++ that will
illustrate the use of a fourth-order
explicit Runge-Kutta method of your own design. In other words, you
will first have to solve the Runge-Kutta equations of condition for
the coefficients
of a fourth-order Runge-Kutta method. See the
Mathematica notebook on solving the equations for 4th order RK
method.
That notebook can be found at...

Use Newton's method to find the
number arcsin(1/3) rounded to 14 digits after the
decimal point by solving numerically the equation sin(x)=1/3 on the
interval [0,pi/6].
1) Determine f(a) and f(b).
2) Find analytically f', f'' and check if f '' is continuous on
the chosen interval [a,b].
3) Determine the sign of f' and f '' on [a,b] using their
plots.
4) Determine using the plot the upper bound C and the lower
bound c for |f'(x)|.
5) Calculate the...

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