Question

Use Newton's method to approximate a root of the equation 5sin(x)=x as follows.
Let x1=2 be the initial approximation.
The second approximation x2 is:
and the third approximation x3 is:

Answer 1

5 sin x = x
let y = x - 5 sin x, search for y = 0
dy/dx = y' = 1 - 5 cos x
Xn+1 = Xn + y(Xn)/y'at Xn
X1 = 2
y = 2 - 5 sin 2 = 2 - 4.54 = -2.54
y'=1 - 5 cos 2 = 3.08
X2 = 2 -2.54/3.08 = 1.17

y = 1.17 - 5 sin 1.17 = -3.43
y' = 1 - 5 cos 1.17 = -.951
X3 = 1.17 -3.43/-.951 = 4.77

This is unlikely to work the way you want because you are jumping from cycle to cycle of the original sine wave

Answer 2

The answers are wrong for this one.

Answer 3

sorry, sign wrong. I drew my picture wrong
5 sin x = x
let y = x - 5 sin x, search for y = 0
dy/dx = y' = 1 - 5 cos x
Xn+1 = Xn - y(Xn)/y'at Xn
X1 = 2
y = 2 - 5 sin 2 = 2 - 4.54 = -2.54
y'=1 - 5 cos 2 = 3.08
X2 = 2 + 2.54/3.08 = 2.82

y = 2.82 - 5 sin 2.82 = 1.24
y' = 1 - 5 cos 2.82 = 5.74
X3 = 2.82 -1.24/5.74 = 2.60

Answer 4

The answer is still wrong :(

Answer 5

check my arithmetic carefully

Answer 6

the correct answers are
x2=2.82658
x3=2.60457

Answer 7

How did you solve this is this the same process?