5. Interpolation & Numerical Methods

Try these first

1. 1.

   Define an algebraic equation.

   Hint

   Think of equations with x raised to whole number powers.

2. 2.

   What is the primary condition for the Bisection Method to work?

   Hint

   Remember the sign change condition.

3. 3.

   What kind of equations can be solved using numerical methods?

   • Only algebraic
   • Only transcendental
   • Both algebraic and transcendental

   Hint

   Consider examples discussed in class.

4. 4.

   The Bisection Method can only be applied if:

   • True
   • False

   Hint

   Think about the conditions for continuity.

5. 5.

   Use the Newton-Raphson method to find the root of the equation f(x) = x^2 - 3 starting from an initial guess x₀ = 2. Show the calculations for three iterations.

   Hint

   Calculate f(x) and its derivative at each iteration.

6. 6.

   Explain how you would use the Fixed Point Iteration to solve for x in the equation x = 2x - 1. Iterates from x₀ = 1 until convergence.

   Hint

   Keep track of how the value changes each iteration.