5.1.3. Numerical Methods for Solving Equations

Try these first

1. 1.

   Define algebraic equations.

   Hint

   Think about simple equations with powers of x.

2. 2.

   What is the Bisection Method?

   Hint

   Consider what it means to bisect.

3. 3.

   What type of equations do numerical methods typically solve?

   • Only algebraic
   • Only transcendental
   • Both algebraic and transcendental

   Hint

   Recall what each type of equation involves.

4. 4.

   True or False: The Bisection Method always converges.

   • True
   • False

   Hint

   Think about the method's principle.

5. 5.

   Given the equation x³ - 5x + 3 = 0, apply the Bisection Method from x = 1 to x = 3 and find the root to three decimal places.

   Hint

   Calculate the midpoint iteratively and evaluate the function at each midpoint.

6. 6.

   Using the Newton-Raphson Method, find the root of the function f(x) = x² - 2 with an initial guess of x₀ = 1.5.

   Hint

   Remember the formula for the derivative and update accordingly.