Ex 1) Use Bisection method and write code for nding a root of ex
3x = 0 correct to four
decimal digits.
p
Ex 2)
Perform three iterations for computing
2, starting with x0 = 1 using Newton’s
p
metthod, and of the Bisection method for
2, starting with interval [1; 2]. How many
iterations are needed for each method in order to obtain 10
6 accuracy? Write code
1
N
for this problem. (Newton’s method for pN will be xn+1 =
xn +
:)
2
xn
Ex 3)
Write code for nding root of ex = sin x closest to 0 using Bisection method.
Ex 4)
Write a program to solve for a root of the equation e x2 = cos x + 1 on [0; 4]. What
happens in Newton’s method if we start with x0 = 0 or with x0 = 1?
Ex 5)
(Circuit Problem) A simple circuit with resistance R, capacitance C in series with
a battery of voltage V is given by
Q = CV 1 e T =(RC) ;
where Q is the charge of the capacitor and T is the time needed to obtain the charge. We wish to solve for the unknown C. For example, using Bisection method, solve this exercise
f(x) = 10x 1 e 0:004=(2000x) 0:00001:
Plot the curve.
Ex 6) In celestial mechanics, Kepler’s Equation is important. It reads
x = y sin y;
in which x is a planet’s mean anomaly, y its eccentric anomaly, and the eccentricity of its orbit. Taking = 0:9, construct a table of y for 30 equally spaced values of x in the interval 0 x . Use Newton’s Method to obtain each value of y. The y corresponding to an x can be used as the starting point for the iteration when x is changed slightly.
Ex 7) Using Newton’s Method, produce a table of x versus y, where y is de ned implicitly as a function of x. Use G(x; y) = 3x7 + 2y5 x3 +y3 3 and start at x = 0, proceeding in steps of 0.1 to x = 10.
Ex 8) Starting with (0; 1), perform two iterations of Newton’s method on the following system
(4x1x22
x1 = 1:
4x12
x22
= 0
