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# marh

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 x0= 2.0 + c.

Hint: Check that the root satisfies x = ln(x-c) – ln 0.3  AND x = 0.3ex-c + c. You may need to iterate more than 5 times.

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 x0 = 1.0 + c.

Question 3

Let

f(x) = (x-c) sin(x-c) + (x-c) + 5
where c = -0.022.If we set
a0 = -6.7 + c, and b0 = -6.3+c and set (a0,b0) as the
starting interval for the r=0 iteration of the bisection method.

Find a4.

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 x0= 0.4 + c.

Hint: Check that the root satisfies x = ln(x-c) – ln 0.3  AND x = 0.3ex-c + c. You may need to iterate more than 5 times.

Question 5

Let
f(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.

Question 6

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Let
f(x) = eax + bx
where a = 4.463 and b = 1.746.

The Newton-Raphson method is used to solve f(x) = 0 with x0 = 0.67.

Find x2 correct to 3 decimal places.

Question 7

We shall  let
f(x) = ax2 where a = 0.98. If f(x) fulfills the condition 1 (Refer to SU1-8)
of the contraction mapping in the interval

R = {x : 0 <= x <= c}

Find the maximum value of c. Express

Question 8

If
f(x) = ax2

where a = 1.477.

If
| df/dx |  <=  1

in the interval
R = { x:  0 <=  x <=  c }

find maximum c (correct to 3 decimal places).

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.

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 x0 = 4.0 + c.