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Help with this calculus problem please

Help with this calculus problem please-example-1

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You're using Newton's method to find the roots to

f(x) = g(x)x ⁶ = cos(x)

Recast this as finding the roots to

h(x) = f(x) - g(x) = x ⁶ - cos(x)

First, get the tangent line approximation L(x) to h(x) at an arbitrary starting point x = x₀ ; in a small neighborhood of x₀, we have

h(x)L(x) = h(x) + h'(x) (x - x₀)

Find the root to the tangent line function:

0 = h(x) + h'(x) (x - x₀) → x = x₁ = x₀ - h(x)/h'(x)

x₁ is the first approximation of a root to h(x). Repeat the process - that is, take the tangent line to h(x) at x₁ to find the next approximation x₂, and so on - until you meet the required condition.

For the smaller root, take x₀ = -π/2 :

x₀ ≈ -1.5708

x₁ = x₀ - h(x)/h'(x) ≈ -1.31348

x₂ = x₁ - h(x)/h'(x) ≈ -1.11366

x₃ = x₂ - h(x)/h'(x) ≈ -0.982446

x₄ = x₃ - h(x)/h'(x) ≈ -0.928013

x₅ = x₄ - h(x)/h'(x) ≈ -0.920039

x₆ = x₅ - h(x)/h'(x)-0.919887

Stop at x₆ because |x₆ - x₅| ≤ 0.001.

For the larger root, you can start the process over with a different seed for x₀, such as π/2. But since h(x) is an even function and symmetric about the y-axis, the larger root will just be the same root with positive sign, 0.919887.

User Elias Platek
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