Test: Numerical Methods - 2 - Software Development MCQ

Newton's method is an iterative numerical method used for finding roots of equations by approximating them. It uses the derivative of the function to iteratively update the approximation until a desired accuracy is achieved.

Simpson's formula is a numerical method used for approximating the definite integral of a function. It involves dividing the interval into smaller subintervals and approximating the integral using quadratic polynomials.

Simpson's formula is a numerical method that involves dividing the interval into smaller subintervals and approximating the integral using quadratic polynomials. It provides a more accurate approximation compared to the Trapezoidal rule.

Newton's method is a numerical method based on the idea of linear interpolation. It uses the derivative of the function to iteratively update the approximation until a root is found.

Newton's method can be used to find the maximum or minimum value of a function within a given interval. By finding the roots of the derivative of the function, the critical points can be determined, which correspond to the maximum or minimum values.

The given code snippet implements Newton's method to find the root of the equation f(x) = x^3 - x - 1. The initial approximation x0 is set to 1. The code iteratively updates the approximation until the absolute value of f(x) is smaller than the tolerance (1e-6). The computed root will be 0.6180339887498949.

The given code snippet implements Simpson's rule to approximate the definite integral of the function f(x) = sin(x) in the interval [0, pi/2]. The code uses n = 10 subintervals for the approximation. The computed result will be approximately 1.000785192546629.

To find the value of x0 that will converge to one of the roots of the equation f(x) = x^2 - 4x + 3 using Newton's method, we need to find the root. By initializing x0 = 2 and performing the iterations of Newton's method, the approximation will converge to the root x = 1.

To approximate the root within an error tolerance of 0.001 using the bisection method, we start with the initial interval [2, 4]. In each iteration, the interval is halved until the width of the interval becomes less than the tolerance. The number of iterations required can be determined by counting the halvings. In this case, the root can be approximated within the desired tolerance in 7 iterations.

To find the minimum value of f(x) = x^2 - 4x + 4 in the interval [0, 5], we can use the bisection method. By repeatedly dividing the interval and comparing the function values at the midpoints, the interval containing the minimum value can be narrowed down.

The given code snippet implements Newton's method to find the root of the equation f(x) = x^3 - 2x - 2. The initial approximation x0 is set to 2. The code iteratively updates the approximation until the absolute value of f(x) is smaller than the tolerance (1e-5). The computed root will be approximately 1.3518188902833395.

The given code snippet implements Simpson's rule to approximate the definite integral of the function f(x) = sin(x) in the interval [0, pi/2]. The code uses n = 4 subintervals for the approximation. The computed result will be approximately 0.8976839622050416.

To approximate the root within an error tolerance of 0.001 using the bisection method, we start with the initial interval [-2, 0]. In each iteration, the interval is halved until the width of the interval becomes less than the tolerance. The number of iterations required can be determined by counting the halvings. In this case, the root can be approximated within the desired tolerance in 11 iterations.

To find the value of x0 that will converge to one of the roots of the equation f(x) = x^2 - 2 using Newton's method, we need to find the root. By initializing x0 = 2 and performing the iterations of Newton's method, the approximation will converge to the root x = -1.414213562373095.

To find the maximum value of f(x) = x^3 - x^2 - 2x + 1 in the interval [0, 2], we can use the Trapezoidal rule. By approximating the integral of the function over the interval using trapezoids, we can determine the maximum value. The Trapezoidal rule can handle both continuous and discrete functions.