Question

Use newtons method to find positive root of 3 sinx = x?

Answer 1

`2.27886266`

Explanation:

`3sinx = x` let's subtract our terms to find the difference equal to zero. `3sinx-x=0`

I want us to use the intermediate value theorem to find our region that the function crosses the x axis.

Recall if `f(a)<0,f(b)>0,f(c)=0`

plugging in `pi` our function is `-pi` and plugging in `pi/2` gives us.

`3-pi/2` which is of course a positive integer. Let's define the region where a zero is positive to `[pi/2, pi]`

I will use a calculator to solve the newton approximation, but remember our parent function for a tangent line `y-y1 = m(x-x1)`

we want our y to be equal to zero as we get closer and closer to the root, so `-y1 = m(x-x1)` also recall slope is `f'(x)` and `y1 = f(x)`. our starting variable is our upper limit `pi`.

`-f(x) = f'(x)(x-pi)` rearranging to `(-f(x))/(f'(x))+pi = x` use the new values as our `x1`

recall our derivative rules such that `d/dx 3sinx-x=0` is `3cos(x)-1 = 0`
Typing our values into the calculator seems to result in `2.27886266`

Let's graph it to be sure.

It appears to agree.