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

1 Answer
Apr 1, 2018

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.
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It appears to agree.