1irishman said:Homework Statement
Find the 5th root of 36 accurate to four decimal places
Homework Equations
xn+1 = xn - f(xn)/f'(xn)
The Attempt at a Solution
First I attempted to write the fifth root of 36 in exponential form as show below:
Let the 5th root of 36 = x
Let f(x) = x^1/5 - 36
So, f'(x) = 1/5x^-4/5 Is this right so far?
LCKurtz said:No. You need to solve x5 - 36 = 0.
LCKurtz said:No. You need to solve x5 - 36 = 0.
1irishman said:oh okay, but i am confused because i thought that the fifth root of a number in exponent
form was that number raised to the 1/5. I don't understand how it is x raised to the fifth.
1irishman said:oh okay. So then it is like below?
f(2) = 2^5 - 36
= - 4
and
f'(2) = 5(2^4)
= 80
Newton's Method is an algorithm used for finding the roots of a mathematical function. It is a numerical method that uses an initial guess to iteratively approach the root of a function until a desired level of accuracy is reached.
Newton's Method involves taking the derivative of a function at a given point and using that slope to find the next point on the function. This process is repeated until the function approaches the root, or the value at which the function equals zero.
The 5th root of a number is the value that, when multiplied by itself 5 times, equals the original number. It is useful in many mathematical applications, including finding the side lengths of a regular pentagon and solving certain types of equations.
The accuracy of Newton's Method in finding the 5th root depends on the initial guess and the complexity of the function. Generally, the more iterations that are performed, the more accurate the result will be.
Newton's Method may fail to find the 5th root if the function is not well-behaved or if the initial guess is not close enough to the root. It may also converge to a local minimum or maximum instead of the desired root. Additionally, it can be computationally intensive for more complex functions.