Need calculus help with Newton's Method Problem

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Let f(x)=x^4-x^3. Show that the equation f(x)=75 has a solution in the interval [3,4] and use Newton’s method to find it. Please show work.

Dec 7th, 2015

Thank you for the opportunity to help you with your question!

We have the function as

f(X)=x^4-x^3

We have x^4-x^3=75

x^4-x^3-75=0

Let g(X)=x^4-x^3-75

g(3)=81-27-75=-21

g(4)=4^4-4^3-75=117

As the g(3) value is negative and the g(4) value is positive it implies that the function has a solution in the interval in the [3,4] as the graph crosses the x-axis in this interval.

Newtons method to find the solution;

x(n+1)=xn- f(xn)/f'(xn)

Here n=3

x(n+1)=xn -(xn^4-x^n3-75)/(4xn^3-3xn^2)

Let x_0 = 3.5...recursive relation is ( 3 x^4 - 2 x³ + 75 ) / ( 4 x³ - 3 x²)

Using a calculator 5 times the 10 place accuracy...3.2285

I hope this would help you,but if you have any doubt regarding this,you can surely ask. :)
Dec 7th, 2015

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Dec 7th, 2015
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