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

1 Answer
Bryson
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.
Desmos.com

It appears to agree.

Related questions
Impact of this question
7009 views around the world
You can reuse this answer
Creative Commons License