Use Newton's method to approximate a root of the equation 3sin(x)=x as follows.
Let x1=1 be the initial approximation.
The second approximation x2 is
and the third approximation x3 is
I got x2=-1.454
but can't get x3 :(
You are on the right track!
x1=1,
x2=-1.455,
x3=-3.787,
x4=-2.140,
...
eventually it will settle on x=-2.28.
Remember that in the case of multiple roots, the one obtained by Newton's method is very dependent on the initial approximation. I assumed x1=1 is in radians.
I agree with drwls's setup in
Newton's method
starting with x1=1 I also got
x2= -41.454 but then I got the next values as
x3 = -3.787
x4 = -2.1404
x5 = -2.2878
x6 = -2.1813
x7 = -2.283
etc
appears to converge to around x = -2.28
check:
3sin(-2.28) = -2.2766
there was supposed to be a reference to
drwls setup in
http://www.jiskha.com/display.cgi?id=1283845535
As you can see in MathMate's reply both of our results coincide.
To use Newton's method to approximate a root of the equation 3sin(x) = x, you start with an initial approximation (x1) and then use the formula x2 = x1 - (f(x1) / f'(x1)) to find the second approximation (x2). Repeat this process to find the third approximation (x3).
Let's begin by finding x2:
Given x1 = 1 as the initial approximation.
To find x2, you need to evaluate the function f(x) = 3sin(x) - x and its derivative f'(x).
Step 1: Calculate f(x1):
f(x1) = 3sin(1) - 1 = 3(0.841) - 1 ≈ 2.523
Step 2: Calculate f'(x1):
To find f'(x), take the derivative of f(x) with respect to x:
f'(x) = 3cos(x) - 1
Evaluate f'(x1):
f'(x1) = 3cos(1) - 1 = 3(0.540) - 1 ≈ 0.620
Step 3: Calculate x2 using the formula:
x2 = x1 - (f(x1) / f'(x1))
= 1 - (2.523 / 0.620)
≈ 1 - 4.073
≈ -3.073
Therefore, the second approximation x2 is approximately -3.073.
To find x3, repeat the same process:
Step 1: Calculate f(x2):
f(x2) = 3sin(-3.073) + 3.073 ≈ 3sin(-3.073) + (-3.073)
≈ -0.506 + (-3.073)
≈ -0.506 - 3.073
≈ -3.579
Step 2: Calculate f'(x2):
f'(x2) = 3cos(-3.073) - 1 ≈ 0.084
Step 3: Calculate x3 using the formula:
x3 = x2 - (f(x2) / f'(x2))
= -3.073 - (-3.579 / 0.084)
≈ -3.073 + 42.607
≈ 39.534
Therefore, the third approximation x3 is approximately 39.534.