3.10 I: Numerical Methods

• Why are numerical methods used to find solutions to equations?

  Exact solutions impossible
• The Newton-Raphson method finds approximate roots of equations.
• The Newton-Raphson method is an iterative process that refines an initial guess for the root.

  True
• The trapezoidal rule and Simpson's rule are used for numerical integration when exact solutions are impractical.


  True
• The Newton-Raphson method uses iterative
• The trapezoidal rule approximates the integral by dividing the region into trapezoids
• What is a key advantage of the trapezoidal rule over Simpson's rule?

  Simpler calculation
• Steps of the Newton-Raphson method

  1️⃣ Start with an initial guess `x_0`

  2️⃣ Calculate `f(x_0)` and `f'(x_0)`

  3️⃣ Update the guess using the formula `x_1 = x_0 - f(x_0) / f'(x_0)`

  4️⃣ Repeat until the guess converges
• What type of equation requires numerical methods due to the difficulty of finding exact solutions?

  Complex algebraic equations
• Comparing the Newton-Raphson method to finding exact solutions

  1️⃣ Exact solutions provide the precise root

  2️⃣ The Newton-Raphson method finds approximate roots

  3️⃣ Exact solutions may be impossible for complex equations

  4️⃣ The Newton-Raphson method is useful when exact solutions are not feasible

  5️⃣ Exact solutions require algebraic manipulation

  6️⃣ The Newton-Raphson method uses iterative numerical calculations
• What type of root does the Newton-Raphson method find?

  Approximate
• What are the trapezoidal rule and Simpson's rule used for?

  Numerical integration
• Simpson's rule requires an even number of subintervals.


  True
• Steps of Euler's method

  1️⃣ Start with an initial condition

  2️⃣ Calculate the slope at the current point

  3️⃣ Update the next point using the slope

  4️⃣ Repeat steps until convergence
• Why are numerical methods used to solve equations?

  Exact solutions impossible
• Match the solution type with the method:

  Exact Solutions ↔️ Provide the precise root

  Newton-Raphson Method ↔️ Finds approximate roots
• In the Newton-Raphson method, the guess is updated using the formula: $x_{1} =$$x_{0} - f(x_{0}) / f'(x_{0})$
• Match the method with its characteristic:

  Exact Solutions ↔️ Provide precise roots

  Newton-Raphson Method ↔️ Find approximate roots
• The Newton-Raphson method provides a single definitive answer for the root.

  False
• Simpson's rule requires an even number of subintervals.


  True
• Steps for Euler's method

  1️⃣ Start with an initial condition: $y(x_{0}) =$$y_{0}$

  2️⃣ Calculate the slope at the current point: $f(x_{0}, y_{0})$

  3️⃣ Update to the next point: $y_{1} =$$y_{0} +$$f(x_{0}, y_{0}) \cdot \Delta x$

  4️⃣ Repeat steps 2-3 until desired accuracy
• What is the purpose of error analysis in numerical methods?

  Understanding accuracy
• The Newton-Raphson method is used to find the roots
• The Newton-Raphson method requires algebraic manipulation to find solutions.

  False
• The update formula in the Newton-Raphson method is x_1 = x_0 - f(x_0) / f'(x_0)
• The trapezoidal rule approximates the integral by dividing the region into trapezoids
• The Newton-Raphson method provides a single definitive answer for the root.

  False
• Which numerical integration method is more accurate, the trapezoidal rule or Simpson's rule?

  Simpson's rule
• Euler's method is used to find approximate solutions to differential equations.
• Euler's method provides exact solutions for differential equations.

  False
• What does the Newton-Raphson method refine iteratively?

  Initial guess
• Exact solutions require algebraic manipulation
• Match the integration method with its approximation technique:

  Trapezoidal Rule ↔️ Uses trapezoids

  Simpson's Rule ↔️ Uses parabolic arcs
• In Euler's method, the slope at the current point is calculated using the function f(x_0, y_0)
• Euler's method requires algebraic manipulation to solve differential equations.

  False
• In the Newton-Raphson method, the number of iterations affects the truncation error.

  True
• Numerical methods find approximate roots of equations.
• What initial value is used in the Newton-Raphson method?

  $x_{0}$
• The Newton-Raphson method requires repeating steps until the guess is sufficiently close to the actual root.


  True
• What are the trapezoidal rule and Simpson's rule used for?

  Numerical integration