- #1

crybllrd

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## Homework Statement

The statement [itex]\sqrt[4]{a}=x[/itex] means that [itex]x^{n}=a[/itex].

Using this, we can approximate the radical [itex]\sqrt[n]{a}[/itex] by approximating the

solution to the equation [itex]x^{n}-a=0[/itex].

Consider the function [itex]f(x)=x^{n}-a[/itex].

We can use Newton's Method to approximate where f(x)=0 and thus approximate the radical

[itex]\sqrt[n]{a}[/itex].

a) Use Newton's Method with the function [itex]f(x)=x^{n}-a[/itex]

to obtain a general formula approximating [itex]\sqrt[4]{a}[/itex].

b) Enter [itex]\sqrt[4]{100}[/itex] using your calculator and give the approximation to the

accuracy found by your calculator.

c) Use the formula found in (a) and make a table of values to approximate

[itex]\sqrt[4]{100}[/itex] to the same accuracy as your calculator. Use 3

as your initial guess.

d) How many iterations are required to obtain this same accuracy?

e)The fundamental theorem of algebra guarantees 4 solutions to x^4-100=0.

you just found one. Are there more real solutions? Use your tools of calculus to sustain

your answer.

## Homework Equations

[itex]x_{2}=x_{1}-\frac{f(x_{1}}{f'(x_{1}}[/itex]

## The Attempt at a Solution

a)[itex]f(x)=x^{n}-a[/itex]

Am I supposed to assume a is constant here? If so, then:

[itex]f '(x)=nx^{n-1}[/itex]

[itex]x_{2}=x_{1}-\frac{x^{n}-a}{nx^{n-1}}[/itex]

b) Easy enough, plugged it into the calc to get 3.16227766.

c) I want to make sure I have part a) right before making a chart.

d) This will be simple after part c)

e) I can tell I will be stuck on this final part. Any tips to help me get started?