Tutorial-01
Bisection method
(1)
Define Algebraic and Transcendental equations.
(2) Use bisection method to find a root of the equation x3  4 x 10  0 in the interval
1, 2 .find the relative percentage error at each iteration. Use four iteration
(3) Solve x  cos x by Bisection method correct to two decimal places
(4) Find the negative root of x 3  7 x  3  0 by bisection method up to three decimal
Places.
(5) Find the five iteration of the bisection method to obtain a root of the equation
f ( x)  Cos x  x e x  0
3
(6) Find a root of the equation x  4 x  9  0 using the bisection method in four stages.
3
2
(7)Using bisection method find the real root of equation x  4 x  10  0 in [1,2] correct to three
decimal places.
Newton – Raphson Method
3
(1) Using Newton –Raphson method, find a root of the equation x  x  1  0 correct
to four decimal places.
(2) Find the zero of the function f ( x)  x3  cos x with starting point x  1 by using by
0
Newton Raphson method
(3) Find a root of x 4  x3  10 x  7  0 correct to three decimal places between a  2 and
b  1 by Newton – Raphson Method.
(4) Find the smallest root of the equation Sin x  e x correct upto to four decimal places
using the Newton-Raphson starting With x  0.6
0
(5) Derive an iterative formula to find N and hence find approximate value of
65 and
3 , correct up to three decimal places
(6) Use an iterative formula to find the value of
(7) Derive an iterative formula to find 1
5 and 27
1
N and hence find the value 53 correct up to three
decimal places
(8) Derive a Newton-Raphon iteration formula for finding the cube root of a positive number N.
Hence find cube root of 12.
(9) Find real root of equation x 3  cos x  0 with x0  1 correct up to four decimal places using
Newton-Raphson method.Could x0  0 be used for this problem ?
Secant method
(1) Derive Secant Method and solve x e x  1  0 correct to three decimal places between
0 and 1.
(2) Find the positive solution of f ( x)  x  2sin x by the secant method, starting from .
x  2 , x  1.9
0
1
(3) Find the smallest root of the equation Sin x  e x correct upto to four decimal places
using the Newton-Raphson starting With x  0.6
0
(4) Use the secant method to find the root of equation x2  4 x 10  0 .
3
(5) Use secant method to find root of x  5 x  7  0 correct to three decimal places.
(6) Find a real root of the following equations using secant method
x
(a) e  3x  sin x  0 , x0  0, x1  1 , correct up to four decimal places.
x
(b) xe  cos x , x0  0, x1  1 , up to x5 .