I need help with this question Calculus

Question

Apply Newton's Method to approximate the x-value(s) of the indicated point(s) of intersection of the two graphs. Continue the iterations until two successive approximations differ by less than 0.001. [Hint: Let h(x) = f(x) − g(x).]

f(x) = 2x + 2

g(x) =

x + 6

 

Answer 1

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 x₁ = 1 (just arbitrarily)

Then x₂ = x₁ - h(x₁)/h'(x₁) = 1 - -3/1.5 = 1 - -2 = 3

So x₂ = 3

Then x₃ = x₂ - h(x₂)/h'(x₂) = 3 - 0.2679491924/1.711324865 = 3 - .1565741245 = 2.843425876

So x₃ = 2.843425876

Then x₄ = x₃ - h(x₃)/h'(x₃) = 2.843425876 - 6.056620119x10^-4/1.703483375 = 2.843425876 - 3.555432479x10^-4 = 2.843070333

So x₄ = 2.843070333

This differs from x₃ 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)