5.1.3.3. Newton-Raphson Method

Try these first

1. 1.

   What is the formula for the Newton-Raphson method?

   Hint

   Remember it's the current guess minus the function value divided by the derivative.

2. 2.

   What do you need to calculate to apply the Newton-Raphson method?

   Hint

   Think about evaluating the slope of the function.

3. 3.

   What is the primary advantage of the Newton-Raphson method?

   • It always finds a root.
   • It converges quickly when near the root.
   • It requires more initial guesses.

   Hint

   Evaluate the options based on the speed of convergence.

4. 4.

   True or False: The Newton-Raphson method can work without the derivative.

   • True
   • False

   Hint

   Think about how the method is defined and its necessity for derivatives.

5. 5.

   Apply the Newton-Raphson method to approximate the root of the equation x^3 - x - 2 = 0 starting from an initial guess of x0 = 1. Show detailed calculations for three iterations.

   Hint

   Make sure to calculate both f(x) and f'(x) correctly at each iteration.

6. 6.

   Using the equation sin(x) - x/2 = 0, find the root starting with x0 = 1. Illustrate the iterations clearly until convergence is evident.

   Hint

   Consider using a calculator for the sin function and be careful with radians.