Derive the formula for integration using Simpson's 3/8 rule. Use Secant Method to estimate the root of equation \(x^2 - 4x - 10 = 0\), with initial estimate \(x_1 = 4\) and \(x_2 = 2\).
What do you mean by boundary value problem? Use shooting method, solve the equation:
\(y'' = 6x^2\), with \(y(0) = 1\) and \(y(1) = 2\) in the interval \((0, 1)\) for \(y(0.5)\) taking \(h = 0.5\).
Write an algorithm and program to compute the interpolation using Lagrange Interpolation.
Attempt any Eight questions
[8×5=40]Show that the rate of convergence of Newton's Raphson method is quadratic.
The temperature of a metal strip was measured at various time intervals during heating and the values are given in the table below.
| Time('t' min) | 1 | 2 | 3 | 4 |
| Temp('T' °C) | 70 | 83 | 100 | 124 |
Given the following set of data points. Obtain the table of divided difference and use that table to estimate the value of \(f(1.5)\).
| x | 1 | 2 | 3 | 4 | 5 |
| \(f(x) = x^3 - 1\) | 0 | 7 | 26 | 63 | 124 |
Solve the following system of linear equation by Gauss Elimination with Pivoting.
\(2x + 2y + z = 6\)
\(4x + 2y + 3z = 4\)
\(x - y + 1 = 0\)
Determine the Eigen Values and corresponding Eigen Vectors for the matrix.
\[A = \begin{bmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}\]
The table below gives the values of distance travelled by a car at various time intervals during the initial running.
| Time('t' sec) | 5 | 6 | 7 | 8 | 9 |
| Temp('T' °C) | 10.0 | 14.5 | 19.5 | 25.5 | 32.0 |
Solve the following integral using trapezoidal rule for \(n = 8\).
\[I = \int_{2}^{4} (x^4 + 1)\, dx\]
Given the equation \(y' = 3x^2 + 1\) with \(y(1) = 2\), estimate \(y(2)\) by Euler's Method using \(h = 0.2\).
Solve the Poisson's Equation \(\nabla^2 f = 2x^2y^2\) over the square domain \(0 \leq x \leq 3\) and \(0 \leq y \leq 3\) with \(f = 0\) on the boundary and \(h = 1\).