Newton's Method  Summary and Q&A
TL;DR
Newton's method is a technique for finding zeros of a function by iteratively improving an initial guess using the function's derivative.
Questions & Answers
Q: How does Newton's method help in finding zeros of a function?
Newton's method provides a technique to approximate zeros of a function by iteratively improving an initial guess using the function's derivative. It helps in narrowing down the value of x where the function equals zero.
Q: How is the first iteration performed in Newton's method?
In the first iteration, a value of x, usually close to the actual zero, is chosen as the initial guess. This value is then used in the formula x_n+1 = x_n  f(x_n)/f'(x_n) to calculate a more accurate zero.
Q: What is the significance of the derivative in Newton's method?
The derivative of the function is used in Newton's method to calculate subsequent values that asymptotically approach the zero of the function. It helps in determining the direction and magnitude of each iteration step.
Q: How many iterations are usually required in Newton's method?
The number of iterations required in Newton's method varies depending on the initial guess and the specific function. In many cases, two or three iterations are sufficient to obtain a solution that is very close to the actual zero.
Summary & Key Takeaways

Newton's method is used to approximate zeros of a function.

By picking a starting value and iteratively applying a formula, a more accurate zero can be obtained.

The method relies on the function's derivative to calculate subsequent values.