Numerical Methods

Subtopic: Trapezoidal & Simpson 1/3 Rules

$\lim_{x \to 0} \frac{1 - \cos(2x)}{x^2} = $ _____ (answer in integer).

Subtopic: Newton-Raphson Method

An approximate solution of the equation $x^3 - 17 = 0$ is to be obtained using the Newton-Raphson method. If the initial guess is $x_0 = 2$, the value at the end of the first iteration is $x_1 = $ _____ (rounded off to two decimal places).

Subtopic: Trapezoidal & Simpson 1/3 Rules

$\hat{i}$ and $\hat{j}$ denote unit vectors in the $x$ and $y$ directions, respectively. The outward flux of the two-dimensional vector field $\vec{v} = x\hat{i} + y\hat{j}$ over the unit circle centered at the origin is ______ (rounded off to two decimal places).

Subtopic: Newton-Raphson Method

For a single degree of freedom spring-mass-damper system subjected to harmonic forcing, the part of the motion (response) that decays due to damping is known as:

Subtopic: Trapezoidal & Simpson 1/3 Rules

Using Trapezoidal rule with one interval, the approximate value of the definite integral: ∫ dx 1 + x2 2 1 = ______ (rounded off to 2 decimal places).

Subtopic: Trapezoidal & Simpson 1/3 Rules

Given that $C$ is the unit circle in the counter-clockwise direction centered at the origin, the value of the contour integral $\oint_C \frac{\cos z}{z}\,dz$ is _________ (round off to 2 decimal places).

Subtopic: Trapezoidal & Simpson 1/3 Rules

In an aerospace engineering analysis of Numerical Methods (Runge-Kutta Methods for ODEs), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Subtopic: Trapezoidal & Simpson 1/3 Rules

In an aerospace engineering analysis of Numerical Methods (Runge-Kutta Methods for ODEs), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Subtopic: Newton-Raphson Method

Use Newton-Raphson method to solve the equation: 1 e xx  . Begin with the initial guess 5 . 0 x0  . The solution after one step is x _________.

Subtopic: Newton-Raphson Method

→ P/8 < P/8 - 3P/8 - 3P/8 - P/8 → P/8 → 3P/8 is

Subtopic: Trapezoidal & Simpson 1/3 Rules

The value of the integral f(4x" + 3x' + 2x + 1)dx evaluated numerically using Simpson's rule with one step is

Subtopic: Trapezoidal & Simpson 1/3 Rules

An aerospace system operating under standard sea level conditions is governed by Numerical Methods (Runge-Kutta Methods for ODEs). Compute the required parameter in standard SI units (rounded off to the acceptable range).

Subtopic: Newton-Raphson Method

A damped single degree-of-freedom system is vibrating under a harmonic excitation with an amplitude ratio of 2.5 at resonance. The damping ratio of the system is ______________ 2013 AEROSPACE ENGINEERING - AE AE 7/19

Subtopic: Trapezoidal & Simpson 1/3 Rules

The integration 1 3 0 x dx∫ computed using trapezoidal rule with 4n= intervals is ____.

Subtopic: Newton-Raphson Method

to Q.55 are multiple choice type. Consider the function . The Newton-Raphson iteration formula to find the root of the function starting from an initial guess at iteration k is ) sin( ) (x x x f  ) 0 (x (A) ) ( ) ( ) ( ) 1 ( cos 1 cossin k kk k k x x x xx   (B) ) ( ) ( ) ( ) 1 ( cos 1 cossin k kk k k x x x xx  

Subtopic: Newton-Raphson Method

Newton-Raphson method achieves an order of convergence equal to:

Subtopic: Trapezoidal & Simpson 1/3 Rules

2009 AE The value of the integral 1+ x + sin x - evaluated using the trapezoidal rule with two equal intervals is approximately

Subtopic: Trapezoidal & Simpson 1/3 Rules

The percentage error (with respect to the exact solution) in estimation of the integral ∫ x^3 dx from 0 to 1 using Simpson's rule is:

Subtopic: Newton-Raphson Method

The Newton-Raphson iteration formula to find the cube root of a positive number $c$ ($x = c^{1/3}$) is: