Calc Question- Newtons Method?

Question

Find the smallest positive value of x which satifies
x= 4.800cos(2.000x)

for some reason i cant get it

is it the same as saying F(x)=
and then just geting x1,x2, x3 etc.?

or do i have to do it a different way?

Answer 1

Newtons method is an iteration algorithm for finding the root of a function F - a root is a place where the value of the function is 0.

You start with a guess, x, and compute a new approximation, x' with the formula:

x' = x - F(x)/F'(x)

where F'(x) is the derivative of the function F at x. If the result is good enough (i.e. if F(x) is close enough to 0), you are done. Otherwise, you iterate again using x' as your guess, and again, etc. as needed.

Newton's method is only guaranteed to work if the function is smooth, there is a simple root, and the starting guess is close enough to the root. And when it works, it gives you better and better approximations to the "true" answer, but it almost never actually gives the exact true answer.

So what you need are:

a) a function F(x)
b) a good enough starting guess
c) a stopping tolerance (how accurate do you have to be?)

For the guess, we can try to solve a simpler function that doesn't need iteration. For example, we know that the series for cos x is:

cos x = 1 - (1/2!)x^2 + (1/4!)x^4 - ...

For small enough x we can ignore the x^4 and higher terms and get the equation:

x = (4.8)(1 - (1/2)x^2)

This is a quadratic and we have a formula for solving quadratics. So we can use the solution to this as our first guess.

You can also try 0 as the another obvious first guess.