1.9 Numerical Methods

• Exact solutions are used for complex polynomials and transcendental equations.

  False
• What is the primary goal of using iteration in mathematics?

  Find approximate solutions
• Fixed point iteration involves applying a function $g(x)$ to the previous approximation
• Match the iterative method with its convergence condition:

  Fixed Point Iteration ↔️ $|g'(x)| < 1$

  Newton-Raphson ↔️ $f'(x) ≠ 0$

  Bisection ↔️ $f(a)f(b) < 0$
• The Newton-Raphson method converges quickly if the initial guess is far from the root.

  False
• The Newton-Raphson method uses the formula $x_{n + 1} =$$x_{n} - \frac{f(x_{n})}{f'(x_{n})}$ and requires $f'(x) \neq 0$ for convergence.
• What are the key applications of iteration in numerical methods?

  Solving equations, finding roots, optimization
• What is the convergence condition for the Newton-Raphson method?

  $f'(x) \neq 0$
• The convergence condition for the Newton-Raphson method is $f'(x) ≠ 0$


  True
• Simpson's rule uses parabolic arcs to approximate the integral
• To approximate \int_{0}^{1} x^{2} dx</latex> using Simpson's rule with n=4, the result is 7/108
• In Euler's method, the next value of x is calculated using $x_{n + 1} =$$x_{n} +$$h$


  True
• Numerical methods are used when finding an exact algebraic solution is too difficult
• Steps involved in the iterative process to find an approximate solution

  1️⃣ Apply a function or algorithm repeatedly

  2️⃣ Refine the solution until desired accuracy
• Iterative methods can refine solutions to the desired level of accuracy.

  True
• Fixed point iteration converges if |g'(x)| < 1 in the region of interest.

  True
• Steps of the Newton-Raphson method

  1️⃣ Start with an initial guess $x_{0}$

  2️⃣ Apply the iteration formula: $x_{n + 1} =$$x_{n} - \frac{f(x_{n})}{f'(x_{n})}$

  3️⃣ Repeat until $|x_{n + 1} - x_{n}| < \text{tolerance}$
• The Newton-Raphson method finds the root of $f(x) =$$x^{3} - 2x - 5$ using an initial guess of $x_{0} =$$2$.


  True
• The equation x^{3} - 2x - 5 = 0</latex> has a simple algebraic solution.

  False
• Steps to find a root using the Newton-Raphson method

  1️⃣ Start with an initial guess x_0

  2️⃣ Apply the iteration formula $x_{n + 1} =$$x_{n} - \frac{f(x_{n})}{f'(x_{n})}$

  3️⃣ Repeat until $|x_{n + 1} - x_{n}| < \text{tolerance}$
• For fixed-point iteration to converge, $|g'(x)| < 1$ in the region of interest.


  True
• After a few iterations, the root of $f(x) =$$x^{3} - 2x - 5$ using Newton-Raphson method with $x_{0} =$$2$ is approximately 2.09455
• Numerical integration approximates the value of an integral using numerical methods
• Numerical integration is used when analytical techniques are insufficient

  True
• Euler's method iteratively calculates the next value of y based on the current value and the slope
• The local truncation error of Euler's method is $O(h^{2})$


  True
• Why is the Newton-Raphson method used for the equation $x^{3} - 2x - 5 =$$0$?


  No simple algebraic solution
• Iteration can be used to determine the values of x where a function $f(x)$ equals zero
• What is the formula for fixed point iteration?

  $x_{n + 1} =$$g(x_{n})$
• What does the Newton-Raphson method use to approach the root of a function?

  Derivative of the function
• What is a condition for the Newton-Raphson method to converge quickly?

  $f'(x) \neq 0$
• Why are numerical methods used in mathematics?

  Exact solutions are difficult
• Iteration is used when exact algebraic solutions are impossible.
• What is the formula for fixed-point iteration?

  $x_{n + 1} =$$g(x_{n})$
• The Trapezoidal Rule is a numerical method for approximating the value of an integral.


  True
• The convergence condition for fixed-point iteration is |g'(x)| < 1</latex>


  True
• The trapezoidal rule has an accuracy of $O(h^{2})$


  True
• Euler's method solves first-order ordinary differential equations


  True
• The convergence rate of Euler's method is first-order, meaning it converges as h approaches 0