Question

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 Newton’s method: 4x^(3) + y = 6 and x^(2)y = 1 using x0 = y0 = 1 and stop when accuracy 10^−2.

Answer #1

Apply Newton’s method to?(?)=?−2sin(?)=0. Compare the number of
iterations required for convergence to that of the fixed-point
iteration method with formulation ?=?(?)=2sin(?). Solve for x0 =
pi/4, piand -piwith14-digit accuracy [i.e., tol = 10-(14+1)=
10-15]. Compare your solutions to those obtained using Matlab’s
fsolve and fzero.

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).

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.)

: 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.

3) If A = 3
1 and B
= 1 7
0
-2
5 -1
Find
a) BA
b) determinant
B
c) Adjoint A
d)
A-1
4) Using matrix method solve the following simultaneous
equations
5x – 3y = 1
2x – 2y = -2
5) Given that f(x) = 6x - 5 g(x) = 3x +
4 and h(x) = 4x – 6
2
Find:-
i)...

1. Use Euler’s method with step size ∆x = 1 to approximate y(4),
where y(x) is the solution of the initial value problem:
y' = x2+ xy
y(0) = 1
2. The coroner arrives at a murder scene at 9:00 pm. He
immediately determines that the temperature of the body is 83◦F. He
waits one hour and takes the temperature of the body again; it is
81◦F. The room temperature is 68◦F. When was the murder
committed? Assume the man...

Using Euler's method Calculate the exact solution and
investigate the accuracy of your approximations. dy/dx=x-xy
y(1)=0
dx=0.2

Use the method for solving homogeneous equations to solve the
following differential equation.
(9x^2-y^2)dx+(xy-x^3y^-1)dy=0
solution is F(x,y)=C, Where C= ?

Let f(t) be the solution of y' = y(4t-1), y(0) = 4. Use Euler's
method with n = 3 to estimate f(1). (Round your
answer to three decimal places.)

dy/dx = x^4/y^2
initial condition y(1)= 1
a) use eulers method to approximate the solution at x=1.6 and
a step size od delta x = 0.2
b) solve the differential equation exactly using seperation
variabled and the intial condtion y(1)=1.
c) what is the exact value of y(1.6) for your solution from
part b.

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