William W. answered • 11d

Math and science made easy - learn from a retired engineer

Newtons Method finds zeros. To find the intersection of f(x) and g(x), we can just equate the two functions. So letting f(x) = g(x), we can say 2x + 2 = √x + 6.

To find the zeros of a function, I set the function equal to zero. So let's take 2x + 2 = √x + 6 and subtract √x from both sides and subtract 6 from both sides to get 2x + 2 - √x - 6 = 0 or 2x - √x - 4 = 0. We can now say:

h(x) = 2x - √x - 4

Taking the derivative, we can say

h'(x) = 2 - 1/2x^{-1/2}

Let's let x_{1} = 1 (just arbitrarily)

Then x_{2} = x_{1} - h(x_{1})/h'(x_{1}) = 1 - -3/1.5 = 1 - -2 = 3

So x_{2} = 3

Then x_{3} = x_{2} - h(x_{2})/h'(x_{2}) = 3 - 0.2679491924/1.711324865 = 3 - .1565741245 = 2.843425876

So x_{3} = 2.843425876

Then x_{4} = x_{3} - h(x_{3})/h'(x_{3}) = 2.843425876 - 6.056620119x10^{-4}/1.703483375 = 2.843425876 - 3.555432479x10^{-4} = 2.843070333

So x_{4} = 2.843070333

This differs from x_{3} by < 0.001 so we have a solution per the instructions. The intersection point occurs at x = 2.843

To find the y value, we can just find f(2.843070333) which is 7.686 so the intersection point is (2.843, 7.686)