5.1. Solution of Algebraic and Transcendental Equations

Try these first

1. 1.

   Identify whether the equation x^2 - 5x + 6 = 0 is algebraic or transcendental.

   Hint

   Look for polynomial terms.

2. 2.

   What is the general principle of the Bisection Method?

   Hint

   Think about interval halving.

3. 3.

   What type of equations does the Bisection Method solve?

   • Linear equations
   • Algebraic equations
   • Transcendental equations

   Hint

   Think about the types of functions.

4. 4.

   True or False: The Secant Method requires derivatives.

   • True
   • False

   Hint

   Consider the definition of derivative-free methods.

5. 5.

   Prove the convergence of the Newton-Raphson Method when applied to f(x) = x^3 − 2x + 2.

   Hint

   Consider the derivative and the behavior near the root.

6. 6.

   Given the equation x^2 + 4x + 4 = 0, apply the Bisection Method between points -5 and -3. Show steps.

   Hint

   Keep checking the signs!