Higher Engineering Mathematics

SOLVING EQUATIONS BY ITERATIVE METHODS 85

A

Thus, r 2 = 3 −

19. 35

− 463. 7

= 3 + 0. 042

= 3. 042 = 3 .04,

correct to 3 significant figure

Similarly, r 3 = 3. 042 −

f(3.042)

f′(3.042)

= 3. 042 −

(− 1 .146)

(− 513 .1)

= 3. 042 − 0. 0022 = 3. 0398 = 3 .04,

correct to 3 significant figure.

Sincer 2 andr 3 are the same when expressed to the
required degree of accuracy, then the required root
is3.04, correct to 3 significant figures.

Now try the following exercise.

Exercise 41 Further problems on Newton’s

method

In Problems 1 to 7, useNewton’s methodto

solve the equations given to the accuracy stated.

1.x^2 − 2 x− 13 =0, correct to 3 decimal

places. [−2.742, 4.742]

2. 3x^3 − 10 x=14, correct to 4 significant
   figures. [2.313]

3.x^4 − 3 x^3 + 7 x=12, correct to 3 decimal

places. [−1.721, 2.648]

4. 3x^4 − 4 x^3 + 7 x− 12 =0, correct to 3 deci-
   mal places. [−1.386, 1.491]
   5. 3 lnx+ 4 x=5, correct to 3 decimal places.
   [1.147]

6.x^3 =5 cos 2x, correct to 3 significant fig-

ures. [−1.693,−0.846, 0.744]

7. 300e−^2 θ+

θ

2

=6, correct to 3 significant

figures. [2.05]

8. Solve the equations in Problems 1 to 5,
   Exercise 39, page 80 and Problems 1 to
   4, Exercise 40, page 83 using Newton’s
   method.


9. A Fourier analysis of the instantaneous
   value of a waveform can be represented by:

y=

(

t+

π

4

)

+sint+

1

8

sin 3t

Use Newton’s method to determine the

value oftnear to 0.04, correct to 4 decimal

places, when the amplitude,y, is 0.880.

[0.0399]

10. A damped oscillation of a system is given
    by the equation:

y=− 7 .4e^0.^5 tsin 3t.

Determine the value oftnear to 4.2, correct

to 3 significant figures, when the magnitude

yof the oscillation is zero. [4.19]

11. The critical speeds of oscillation,λ,ofa
    loaded beam are given by the equation:

λ^3 − 3. 250 λ^2 +λ− 0. 063 = 0

Determine the value ofλwhich is approx-

imately equal to 3.0 by Newton’s method,

correct to 4 decimal places. [2.9143]