5.1.4. Comparison of Methods

Try these first

1. 1.

   What is required for the Bisection Method to work?

   Hint

   Think about the conditions of continuity and sign change.

2. 2.

   In which method do you not need the derivative of the function?

   Hint

   Recall which methods require or do not require derivatives.

3. 3.

   Which method requires two initial guesses?

   • Bisection Method
   • Newton-Raphson
   • Fixed Point Iteration

   Hint

   Think about methods needing two values.

4. 4.

   True or False: The Secant Method requires knowledge of the derivative of the function.

   • True
   • False

   Hint

   Recall the distinctive nature of the Secant Method.

5. 5.

   Given the function f(x) = x^2 - 3, apply the Bisection Method to find the root between x = 1 and x = 2. Document each step.

   Hint

   Ensure you check the sign changes at each bounding step.

6. 6.

   Derive an efficient way to estimate a root of f(x) = e^x - x using the Newton-Raphson method. Use an initial guess of 0.5.

   Hint

   Always evaluate f and f' accurately before each guess.