Higher Engineering Mathematics, Sixth Edition

86 Higher Engineering Mathematics

f( 1 )= 53 −e^1.^92 +5cos

1

3

− 9 ≈ 114

f( 2 )= 63 −e^3.^84 +5cos

2

3

− 9 ≈ 164

f( 3 )= 73 −e^5.^76 +5cos1− 9 ≈ 19

f( 4 )= 83 −e^7.^68 +5cos

4

3

− 9 ≈− 1660

From these results, let a first approximation to the root

ber 1 =3.

Newton’s formula states that a better approximation to

the root,

r 2 =r 1 −

f(r 1 )

f′(r 1 )

f(r 1 )=f( 3 )= 73 −e^5.^76 +5cos1− 9

= 19. 35

f′(x)= 3 (x+ 4 )^2 − 1 .92e^1.^92 x−

5

3

sin

x

3

f′(r 1 )=f′( 3 )= 3 ( 7 )^2 − 1 .92e^5.^76 −

5

3

sin1

=− 463. 7

Thus, r 2 = 3 −

19. 35

− 463. 7

= 3 + 0. 042

= 3. 042 = 3. 04 ,

correct to 3 significant figures.

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 figures.

Sincer 2 andr 3 are the same when expressed to the

required degree of accuracy, then the required root is

3.04, correct to 3 significant figures.

Now try the following exercise

Exercise 37 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. 3lnx+ 4 x=5, correct to 3 decimal places.
   [1.147]


6. x^3 =5cos2x, correct to 3 significant figures.
   [−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 35, page 81 and Problems 1 to
   4, Exercise 36, page 84 using Newton’s
   method.


9. A Fourier analysis ofthe instantaneousvalue
   of a waveform can be represented by:
   y=

(

t+

π

4

)

+sint+

1

8

sin3t

Use Newton’smethod todetermine the value

oftnear to 0.04, correct to 4 decimal places,

when the amplitude,y, is 0.880.

[0.0399]

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

y=− 7 .4e^0.^5 tsin3t.

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]