Question 1

Let f(x) = (x-c) e

^{-x+c}– 0.3 where c = 3.558.Using the simple iteration method, find the root of f(x) correct to TWO decimal places using x

_{0}= 2.0 + c. Hint: Check that the root satisfies x = ln(x-c) – ln 0.3 AND x = 0.3e^{x-c}+ c. You may need to iterate more than 5 times.Answer:

Question 2

Let f(x) = (x-c)

^{2}e^{-x+c}– 0.3 where c = 3.058.Using the Newton-Raphson iteration scheme, find the root (correct to 3 decimal places) of f(x) with x

_{0 }= 1.0 + c.Answer:

Question 3

Let f(x) = (x-c) sin(x-c) + (x-c) + 5 where c = -0.022. If we set a

_{0}= -6.7 + c, and b_{0}= -6.3+c and set (a_{0},b_{0}) as the starting interval for the r=0 iteration of the bisection method. Find a_{4}.Answer:

Question 4

Let f(x) = (x-c) e

^{-x+c}– 0.3 where c = 0.769. Using the simple iteration method, find the root of f(x) correct to TWO decimal places using x_{0}= 0.4 + c.Hint: Check that the root satisfies x = ln(x-c) – ln 0.3 AND x = 0.3e

^{x-c}+ c. You may need to iterate more than 5 times.Answer:

Question 5

Letf(x) = a/x where a = 6.207. If R={x: 0.5 <= x <= 1.0 }

Find the maximum value (correct to 3 decimal places) of| df/dx | in the interval of R.

Answer:

Question 6

Let f(x) = e

^{ax}+ bx where a = 4.463 and b = 1.746.The Newton-Raphson method is used to solve f(x) = 0 with x_{0}= 0.67. Find x_{2}correct to 3 decimal places.Answer:

Question 7

We shall let f(x) = ax

^{2}where a = 0.98. If f(x) fulfills the condition 1 (Refer to SU1-8)of the contraction mapping in the intervalR = {x : 0 <= x <= c}

Find the maximum value of c. Express your answer correct to 3 decimal places.

Answer:

Question 8

If f(x) = ax

^{2}where a = 1.477.If | df/dx | <= 1 in the interval R = { x: 0 <= x <= c }find maximum c (correct to 3 decimal places).

Answer:

Question 9

A bisection method is to be used to find the solution of f(x) = 0 In the 0th iteration (the zeroth iteration, i.e., r = 0), an interval of a <= x <= b is used,

where a = -0.401 and b = -0.337.

Find the length of the interval in the third iteration (i.e., r = 3).Give the answer correct to 3 decimal places.

Answer:

Question 10

Let f(x) = (x-c)

^{2}e^{-x+c}– 0.3 where c = 3.549.Using the Newton-Raphson iteration scheme, find the root (correct to 3 decimal places) of f(x) with x_{0 }= 4.0 + c.