Question

Please solve the following problem by using newton raphson method?

Where do the curves of `y = cosx` and `y = x3 −1` intersect?

Answer 1

`x approx 1.12656`

Explanation:

Consider the graphs of `y=cosx and y=x^3-1` below:
graph{(y-cosx)(y-x^3+1)=0 [-10, 10, -5, 5]}

We can see that the graphs intersect at some point greater than `x=1`

The intersection point occurs where: `cosx = x^3-1`

That is where: `cosx-x^3+1=0`

The Newton/Raphson iteration method says that for `f(x)=0`

`x_(i+1) = x_i - (f(x_i))/(f'(x_i))`

Where, `x_(i+1)` is a better estimate of `x` than `x_i`

In this case: `f(x) = cosx-x^3+1`
and `f'(x)= -sinx-3x^2`

We will take `x_0 =1` (from the graphs above).

We can then iterate as follows:

Hence, `x=approx 1.12656`