Solve your IGNOU Doubts

Question:

Derive a suitable numerical differentiation formula of 0(h²) to find $f{}''$ (2.4) with h= 0.1 given the table

x	0.1	1.2	2.4	3.9
$f(x)$	3.41	2.68	1.37	-1.48

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Question:

Using finite differences, show that the data

x	-3	-2	-1	0	1	2	3
$f(x)$	-13	7	3	1	1	3	7

represents a second degree polynomial. Obtain this polynomial using interpolation and find $f$ (2.5).

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Question:

Prove that the function f defined by

$f(x)=\left\{\begin{matrix} 2, &if\: x\; is\; irrational \\ -2,& if\; if\; x\; is\; rational \end{matrix}\right.$

is discontinuous, ∀ x ∈ $\mathbb{R}$ , using the sequential definition of continuity.

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Question:

The function $f$ (x) = ln(1+x) is to be tabulated at equispaced points in the interval [2, 3] using linear interpolation. Find the largest step size h that can be used so that the error $f\leq 5 \times 10^{-4}$ in magnitude.

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Question:

Let f [: − 3,3 ]→ $\mathbb{R}$ be defined by f (x ) = 5 (x ) + x³ where [x] denotes the greatest integer ≤ x. Show that this function is integrable.

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Question:

Determine the constants $a , \: \: \beta \: \: ,\gamma$ in the differentiation formula

$Y^{{}'} (x_{0} )= ay( x _{0} -h) =\beta y ( x _{0}) +\gamma y ( x_{0} + h)$

so that the method is of the highest possible order. Find the order and the error term of the method.

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Question:

Show that there is no real number, k for which the equation, x⁴ − 3x² + k = 0 has two distinct roots in the interval [2,3].

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Question:

Using the principle of mathematical induction, prove that $\frac{n^{5}}{5}+\frac{n^{3}}{3}+\frac{7n}{15}$ is a natural number, ∀n∈ N.

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Question:

Solve the system of equations

$3 x + 2y + 4z= 7$

$2 x + y + z= 7$

$x + 3y + 5z= 2$

with partial pivoting. Store the multipliers and also write the pivoting vectors.

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Question:

$_{Find\ lim}^{x\rightarrow 0}\frac{1-cos^{2}}{x^{2}sin\: x^{2}}$

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Question:

Find the dominant eigenvalue and the corresponding eigenvector for the matrix

$A = \begin{bmatrix} -4& 14 & 0\\ -5&13 &0 \\ -1 &0 & 2 \end{bmatrix}$

using five iterations of the power method and taking $y^{0} = [ 111 ]^{7}$ as the initial vector

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Question:

Find the dominant eigenvalue and the corresponding eigenvector for the matrix

$A = \begin{bmatrix} -4& 14 & 0\\ -5&13 &0 \\ -1 &0 & 2 \end{bmatrix}$

using five iterations of the power method and taking $y^{0} = [ 111 ]^{7}$ as the initial vector

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Question:

Examine the function f : $\mathbb{R}\rightarrow \mathbb{R}$ defined by

$f(x)=\left\{\begin{matrix} \frac{1}{6}(x+1)^{3} &x\neq 0 \\ \frac{5}{6}&\; \; x=0 \end{matrix}\right.$

for continuity on $\mathbb{R}$ . If it is not continuous at any point of $\mathbb{R}$ , find the nature of discontinuity there.

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Question:

For the linear system of equations

$\begin{bmatrix} 1 &2 &-2 \\ 1& 1&1 \\ 2&2 & 1 \end{bmatrix}$ $\begin{bmatrix} x_{_{1}} \\x _{2} \\x _{_{3}} \end{bmatrix}=$ $\begin{bmatrix} 1 \\3 \\5 \end{bmatrix} \\$

set up the Gauss-Jacobi and Gauss-Seidal iteration schemes in matrix form. Also check the convergence of the two schemes.

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Question:

Solve the system of equations

$0. 6 x+ 0.8y + 0.1z =1$

$1. 1 x+ 0.4y + 0.3z =0.2$

$x + y +2z = 0.5$

by LU decomposition method and find the inverse of the coefficient matrix

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Question:

Find the inverse of the matrix

$A = \begin{bmatrix} 1 &-1 &1 \\ 1& -2 &4 \\ 1& 2& 2 \end{bmatrix}$

using Gauss Jordan method.

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Question:

Estimate the eigenvalues of the matrix

$\begin{bmatrix} 1 &-2 &3 \\ 6 &-13 &18 \\ 4 & -10 & 14 \end{bmatrix}$

using the Gershgorin bounds. Draw a rough sketch of the region where the eigenvalues lie.

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Question:

Prove that the union of two closed sets is a closed set. Give an example to show that union of an infinite number of closed sets need not be a closed set.

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Question:

a) − 2 is a limit point of the interval ] −3,2 ].

b) The series $\frac{1}{2}-\frac{1}{6}+\frac{1}{10}-\frac{1}{4}+...$ is divergent.

c) The function, f (x) = sin² x is uniformly continuous in the interval .

d) Every continuous function is differentiable.

e) The function f defined on $\mathbb{R}$ by

$f(x)=\left\{\begin{matrix} 0, &x\: is \: rational \\ 2, &2,is\; irrational \end{matrix}\right.$

Is integrable in the interval [2,3].

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Question:

The equation $x^{2} + ax +b =0$ has two real roots p and q such that | p | $<$ | q | . If we use the fixed point iteration $x_{k+1} = \frac{-b}{x_{k} +a }$ , to find a root then to which root does it converge?