Q:

1a. One root x ₁ of a quadratic equation

is .

Calculate x ₁ if a=1.11, b= 111 and c=0.111.

Is the answer reliable? Note that x ₁ is given by the difference of two almost equal numbers and this could result in loss of significant figures.

By expanding , show that x ₁ can be approximated by

By multiplying the numerator and denominator by , show that x ₁ can be written as

For the given values of a, b, c, calculate x _1a and x _1b. Evaluate ax ²+bx+c for x=x ₁, x=x _1a, and x=x _1b. Comment on the results.

Q:

2a. The equation e ^x ?4x ²=0 has a root in the interval [0, 1]. Obtain the root (i) by the method of bisection and (ii) by Newton s method.

Q:

3a. The cubic x ³ 2x ?1 has a root in the interval [1, 2]. Use (i) the secant method and (ii) Newton s method to find the root.

Q:

4a Deduce that an iterative formula for finding the cube root of a real number A is

Use the formula to calculate the cube root of 30. Answer: 3.107

Q:

5a. Obtain the three roots of x ³ ?4x+1 by the fixed point iteration method. Test the convergence in each case.

Q:

6b. The Redlich-Kwong equation [Equation (1.6 2)] can be written as

Calculate b if P=87, T=486, V=12, A=0.08, and...