5.1.2.1.1. Algebraic Equations

Try these first

1. 1.

   Define an algebraic equation and provide an example.

   Hint

   Think of simple polynomial forms.

2. 2.

   What is the Bisection Method?

   Hint

   Consider the process of dividing an interval where a function changes sign.

3. 3.

   What is a characteristic of transcendental equations?

   • Involves polynomial expressions
   • Involves trigonometric functions
   • Is solvable using Bisection

   Hint

   Consider functions outside simple algebra.

4. 4.

   True or False: The Bisection Method always converges.

   • True
   • False

   Hint

   Think about what conditions are necessary for a method.

5. 5.

   Using the Newton-Raphson method, approximate the root of the equation x^3 - 2x - 5 = 0, starting with an initial guess of x0=2.

   Hint

   Calculate f(x) and f'(x) and apply the Newton-Raphson formula.

6. 6.

   Find the root of the equation sin(x) = 0.5 using the Secant Method with initial guesses x0 = 0.5 and x1 = 1.

   Hint

   Use initial guesses to establish a pattern of convergence.