Solve your IGNOU Doubts

Question:

Using the sequential definition of the continuity, prove that the function f , defined by:

$f(x)=\left\{\begin{matrix} 3,if\; x\: is\: irrational & \\ -3,if\; x\; is\; rational \end{matrix}\right.$

is discontinuous at each real number.

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Question:

Use modified Euler’s method to find the approximate solution of IVP $y{}' = 2xy ,y (1) =1 at \: \: x= 1.5$ with h = 0.1

If the exact solution is $y (x) = e^{x^{2 -1} }$ , find the error.

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Question:

Let {a_n } an be a sequence defined as $a_{1}=3,a_{n+1}=\frac{1}{5}a_{n}$ show that {a_n} an converges to zero.

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Question:

Compute the values of

$1 =\int_{0}^{1}\frac{dx}{1+ x^{2}}$

by using the trapezoidal rule with h = 0.5, 0.25, 0.125 . Improve this value by using the Romberg’s method. Compare your result with the true value.

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Question:

For the function, f (x) = x² −2 defined over [5,1], verify : L(P, f ) ≤ U(−P, f ) where P is the partition which divides[5,1 ] into four equal intervals.

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Question:

Using the classical R-K method of $0 (h^{4})$ calculate approximate solution of the IVP, $y{}'= 1 -x +4y\: \: \: ,y (0) =1 \: \: at \: \: x =0.6$ , taking $h =0.1$ and 0.2 . Use extrapolation technique to improve the accuracy

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Question:

$(ii) _{x\rightarrow \frac{5}{3}}^{lim}\; \frac{1}{(3x+5)^{2}}=\infty$

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Question:

A table of values is to be constructed for the function f (x) given by $f (x)= \frac{1}{1+x}$ in the interval [1, 4] with equal step length. Determine the spacing h such that quadratic interpolation gives result with accuracy $1\times 10^{-6}$ .

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Question:

$(i)\; _{x\rightarrow \infty }^{lim}\left ( \frac{x-3}{x-1} \right )^{x}=\frac{1}{e^{2}}$

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Question:

Show that $\sqrt{1 + \mu^{2}\delta^{2} } =1+ \frac{\delta^{2} }{2}$ where $\mu$ and $\delta$ are the average and central differences operators, respectively.

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Question:

Give an example of an infinite set with finite number of limit points, with proper justification.

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Question:

Show that the series $\sum_{n=1}^{\infty }\frac{x}{1+n^{2}\; x^{2}}$ is uniformly convergent in [α, 1] for any α > .0

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Question:

Examine the function, f (x) = (x +1)³ (x − 3)² for extreme values.

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Question:

Take 10 figure logarithm to bases 10 from x =300 to x = 310 by unit increment. Calculate the first derivative of log_10^x when x =310

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Question:

Prove that continuous function of a continuous function is continuous.

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Question:

Prove that the sequence $\left \{ \frac{a_{n}}{n} \right \}$ is convergent where {a_n } an is a bounded sequence.

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Question:

Prove that

$_{n\rightarrow \infty }^{lim}\left [ \frac{1}{\sqrt{2n-1}}+\frac{1}{\sqrt{4n-2^{2}}} +\frac{1}{\sqrt{6n-3^{2}}}+...+\frac{1}{n}\right ]=\frac{\pi }{2}$

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Question:

The position f (x) of a particle moving in a line at various times x_k is given in the following table. Estimate the velocity and acceleration of the particle at x =1.2.