Solve your IGNOU Doubts

Question:

What do ássimilation’ and áccommodation’ mean? Explain, the terms in the context of learning measurement of time.

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Question:

) Deepa had to introduce the concept of ‘half’ to her class of 7-yar olds. She began by asking children if they could show half of anything. Most children came up with examples by breaking a chalk piece or tearing a sheet of paper into two. Deepa allowed children to talk to each other and share their examples with the whole class. More examples emerged, such as half of a glass of water. Deepa did not emphasize that the size of the two pieces must be equal for each to be one-half of the whole. Through the entire class she did not introduce the symbol ½.

i) Do you think Deepa should have started by giving examples of ‘half’ herself? Justify your answer.

ii) Was Deepa justified in not introducing the symbol? Explain why?

iii) Suggest an activity to carry this forward on the following day.

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Question:

If E is a measurable set and f is a simple function such that $a \leq f(x) \leq b \text{ } \forall x \in E$ , show that

$$am(E) \leq \int_{E} f \, dm \leq m(E)b.$

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Question:

Give an example of an algebra which is not a $\sigma$ -algebra. Justify your choice of examples

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Question:

Let (X, d) be a metric space and A be a subset of X. Show that $bdy(A) = \emptyset$ if and only if A is both open and closed.

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Question:

Verify the hypothesis and conclusions of the Fatou’s lemma for the sequence $\{f_n\}$ given by

$$f_n(x) = 2n \text{ for } x \in \left( \frac{1}{2n}, \frac{1}{n} \right)$

$$= 0 \text{ for } x \in \left( 0, \frac{1}{2n} \right] \cup \left[ \frac{1}{n}, 1 \right]$

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Question:

Give an example of a family f_i of subsets of a set X which has finite intersection property. Justify your choice of example.

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Question:

Find $B \left[ 2, \frac{1}{2} \right]$ in $\mathbb{R}$ where d is the metric given by $d(x, y) = \min\{1, |x - y|\}$ .

Based on the image provided, here is the transcription of the final questions:

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Question:

Suppose that $f : \mathbb{R} \to \mathbb{R}^2$ is given by $f(t) = (t, t^2)$ and $g : \mathbb{R}^2 \to \mathbb{R}^3$ is given by

$g(x, y) = (x^2, xy, y^2 - x^2)$ . Compute the derivative of $g \circ f$ .

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Question:

Find the directional derivative of the function $f : \mathbb{R}^4 \to \mathbb{R}^4$ defined by

$$f(x, y, z, w) = (x^2y, xyz, x^2 + y^2, zw^2)$

at the point (1, 2, -1, -2) in the direction $v = (1, 0, -2, 2)$ .

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Question:

Is the continuous image of a Cauchy sequence a Cauchy sequence? Justify.

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Question:

) Show that the function f defined by
$$f(x, y) = \begin{cases} \frac{xy}{x^2 + y^2}, & \text{if } (x, y) \neq (0, 0) \\ 0, & \text{if } (x, y) = (0, 0) \end{cases}$
is not differentiable at (0,0). Do the partial derivatives of f exist at (0,0)? or at any other point in $\mathbb{R}^2$ ? Justify your answer.

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Question:

Let F be the function from $\mathbb{R}^2$ to $\mathbb{R}^2$ defined by

$$F(x, y) = (x^2 + y^2, xy).$

Show that F is differentiable at (1, 2). Find the differential matrix of F.

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Question:

Find the interior and closure of the set $\mathbb{Q}$ of rationals in $\mathbb{R}$ with standard metric.

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Question:

Show that a set A in a metric space is closed if and only if every convergent sequence in A converges to a point of A.

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Question:

If E is a subset of $\mathbb{R}$ with standard metric, then show that $(\bar{E})^c = (E^c)^\circ$ .

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Question:

Which of the following subsets of $\mathbb{R}$ are compact w.r.t. the metric given against them. Justify your answer.
i) $A = (0, 1)$ in $\mathbb{R}$ of $-\mathbb{R}$ with standard metric.
ii) $A = [3, 4] - \mathbb{R}$ with discrete metric.
iii) $\{(x, y) \in \mathbb{R}^2 \quad y > 0\} - \mathbb{R}^2$ with standard metric.

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Question:

Prove that if an open set U can be written as the union of pariwise disjoint family V of open connected subsets, then these subsets must be the components of U. Use this theorem to find the components of the set $D \cup E$ where

$$D = \{(x, y) \in \mathbb{R}^2 : \|(x+1, y)\| = 1\} \text{ and}$

$$E = \{(x, y) \in \mathbb{R}^3 : \|x-1, y\| = 1\}.$

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Question:

Find the Fourier series of $f(t) = t^2$ on $[-\pi, \pi]$ .

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Question:

For the equation $x^2 + y^2 + z^3 = 0$ , at which points on its solution set, can we assured that there is a neighbourhood of the point in which the surface given by the equation can be described by an equation of the form $z = f(x, y)$ .