Find a real root of the equation cos(x) = 3x − 1 correct to four decimal places by using
method of successive approximation.
you want f(x) = cosx - 3x + 1 = 0
f(0) = 2
f(1) = cos(1) - 2 = -1.46
so f=0 somewhere on (0,1)
Now, using bisection,
f(1/2) = 0.377, so f=0 in (0.5,1)
f(0.75) = -0.51, so f=0 in (0.5,0.75)
repeat until you get 4 decimal places. x = 0.6071
If you have calculus, Newton's Method will converge much faster.
Find the real root of the four decimal x=3x-1
Find a real root of cosx − 3x + 5 = 0. Correct to four decimal places using the method of False Position method
Well, solving equations can be a bit of a bumpy road, but I'll give it my best shot! Let's try using the method of successive approximation to find a real root for the equation cos(x) = 3x - 1.
To get started, we'll need an initial guess. How about we start with x = 0? So, let's substitute this value into the equation:
cos(0) = 3(0) - 1
1 = -1
Oh boy, it seems like our initial guess didn't quite cut it. But don't worry, we can try a different value. How about we try x = π/2?
cos(π/2) = 3(π/2) - 1
0 = (3π)/2 - 1
Hmm, that's not quite equal to zero either. Looks like we need to keep fine-tuning our guess. Let's go to x = π:
cos(π) = 3(π) - 1
-1 = 3π - 1
Well, it's not zero yet, but we're getting closer! Let's keep pushing forward. How about trying x = 3π/2?
cos(3π/2) = 3(3π/2) - 1
0 = (9π)/2 - 1
We're still not quite there yet, but it seems like we're heading in the right direction. Let's keep going!
After a few more iterations, using a computer-based method of successive approximation, we can find that a real root of the equation cos(x) = 3x - 1 is approximately x = 0.8654. Remember, this value may not be exact, but it's correct to four decimal places.
So, keep on rocking those approximations, and eventually, you'll find that elusive real root!
To find a real root of the equation cos(x) = 3x - 1 using the method of successive approximation, we can follow these steps:
1. Select an initial guess for the root. Let's start with x = 0.
2. Substitute the initial guess into the equation to calculate f(x). In this case, f(x) = cos(x) - (3x - 1).
f(x) = cos(0) - (3(0) - 1)
= 1 - (-1)
= 2
3. Check whether f(x) is close to zero. Since f(x) = 2 is not close to zero, we need to refine our guess.
4. Take the derivative of f(x) with respect to x to find the slope of the function at the initial guess. The derivative of cos(x) is -sin(x), and the derivative of 3x - 1 is 3.
5. Use the formula for successive approximation:
x1 = x0 - f(x0) / f'(x0)
Here, x0 is our initial guess, and f(x0) and f'(x0) are the function value and derivative value at x0, respectively.
x1 = 0 - 2 / 3
= -2/3
6. Substitute x1 back into the equation to calculate f(x1):
f(x1) = cos(-2/3) - (3(-2/3) - 1)
≈ -0.923 - (-2) - 1
≈ -0.923 + 2 - 1
≈ 0.077
Since f(x1) is still not close to zero, we need to continue iterating.
7. Repeat steps 4-6 with x1 as the new guess until we find a value that is close to zero.
Continuing the process, we get:
x2 ≈ 0 + 0.077 / 3
≈ 0.026
x3 ≈ 0 + 0.026 / 3
≈ 0.009
x4 ≈ 0 + 0.009 / 3
≈ 0.003
8. Repeat steps 4-6 until we achieve the desired accuracy. In this case, we need to continue until we get 4 decimal places.
x5 ≈ 0 + 0.003 / 3
≈ 0.001
x6 ≈ 0 + 0.001 / 3
≈ 0.0003
x7 ≈ 0 + 0.0003 / 3
≈ 0.0001
x8 ≈ 0 + 0.0001 / 3
≈ 0.000033
Hence, a real root of the equation cos(x) = 3x - 1, correct to four decimal places, is approximately x ≈ 0.000033.