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# Page:A Dynamical Theory of the Electromagnetic Field.pdf/33

491
PROFESSOR CLERK MAXWELL ON THE ELECTROMAGNETIC FIELD.

By the same demonstration as was used in the case of the mechanical action on a magnet, it may be shown that the mechanical force on a small body containing a quantity ${\displaystyle e_{2}}$ of free electricity placed in a field whose potential arising from other electrified bodies is ${\displaystyle \Psi _{1}}$, has for components

 ${\displaystyle \left.{\begin{array}{c}X=e_{2}{\frac {d\Psi _{1}}{dx}}=-P_{1}e_{2},\\\\Y=e_{2}{\frac {d\Psi _{1}}{dy}}=-Q_{1}e_{2},\\\\X=e_{2}{\frac {d\Psi _{1}}{dz}}=-R_{1}e_{2},\end{array}}\right\}}$ (D)

So that an electrified body is urged in the direction of the electromotive force with a force equal to the product of the quantity of free electricity and the electromotive force.

If the electrification of the field arises from the presence of a small electrified body containing ex of free electricity, the only solution of ${\displaystyle \Psi _{1}}$ is

 ${\displaystyle \Psi _{1}={\frac {k}{4\pi }}{\frac {e_{1}}{r}}}$ (43)

where ${\displaystyle r}$ is the distance from the electrified body.

The repulsion between two electrified bodies ${\displaystyle e_{2}}$, ${\displaystyle e_{2}}$ is therefore

 ${\displaystyle e_{2}{\frac {d\Psi _{1}}{dr}}={\frac {k}{4\pi }}{\frac {e_{1}e_{2}}{r^{2}}}}$ (44)

Measurement of Electrical Phenomena by Electrostatic Effects.

(80) The quantities with which we have had to do have been hitherto expressed in terms of the Electromagnetic System of measurement, which is founded on the mechanical action between currents. The electrostatic system of measurement is founded on the mechanical action between electrified bodies, and is independent of, and incompatible with, the electromagnetic system; so that the units of the different kinds of quantity have different values according to the system we adopt, and to pass from the one system to the other, a reduction of all the quantities is required.

According to the electrostatic system, the repulsion between two small bodies charged with quantities ${\displaystyle \eta _{1},\eta _{2}}$ of electricity is

${\displaystyle {\frac {\eta _{1}\eta _{2}}{r^{2}}}}$

where ${\displaystyle r}$ is the distance between them.

Let the relation of the two systems be such that one electromagnetic unit of electricity contains ${\displaystyle v}$ electrostatic units; then ${\displaystyle \eta _{1}=ve_{1}}$ and ${\displaystyle \eta _{2}=ve_{2}}$, and this repulsion becomes

 ${\displaystyle v^{2}{\frac {e_{1}e_{2}}{r^{2}}}={\frac {k}{4\pi }}{\frac {e_{1}e_{2}}{r^{2}}}}$ by equation (44), (45) (45)

whence ${\displaystyle k}$, the coefficient of "electric elasticity" in the medium in which the experiments are made, i. e. common air, is related to ${\displaystyle v}$, the number of electrostatic units in one electromagnetic unit, by the equation

 ${\displaystyle k=4\pi v^{2}}$ (46)


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# Converting From One Representation of a Field Element to Another

Let $A \subseteq B \subseteq C$ be fields and let $\alpha$, $\beta$, $\gamma$ be such that $A(\alpha) = B$, $B(\beta) = C$, $A(\gamma) = C$. Assume $B$ and $C$ have finite degree over $A$.

Let $m(\alpha,A)$ be the minimal polynomial of $\alpha$ over $A$, let $m(\beta,B)$ be the minimal polynomial of $\beta$ over $B$, let $m(\beta,A)$ be the minimal polynomial of $\beta$ over $A$, and let $m(\gamma,A)$ be the minimal polynomial of $\gamma$ over $A$.

Let $c \in C$. We have, on the one hand, $$c=\sum_{i = 1}^{[C:B]} b_i \beta^{i-1}, \quad b_i \in B$$ $$b_i=\sum_{j = 1}^{[B:A]} a_{ij} \alpha^{j-1}, \quad a_{ij} \in A$$ and on the other hand $$c=\sum_{k = 1}^{[C:A]} c_k \gamma^{k-1}, \quad c_k \in A$$ We also have $$\gamma = \sum_{i = 1}^{[C:B]} d_i \beta^{i-1}, \quad d_i \in B$$ $$d_i=\sum_{j = 1}^{[B:A]} e_{ij} \alpha^{j-1}, \quad e_{ij} \in A$$ If the polynomials $m(\alpha,A)$, $m(\beta,B)$, $m(\beta,A)$, $m(\gamma,A)$ and the numbers $a_{ij}$, $e_{ij}$ are known explicitly, how can I calculate the $c_i$?

ADDED: Now that Joriki and Jyriki have helped me to formulate the problem correctly, I see the solution is not so hard. Since $$\gamma = \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} e_{ij} \alpha^{j-1}\beta^{i-1}$$ we can find numbers $f_{ijk}$ such that $$\gamma^{k-1} = \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} f_{ijk} \alpha^{j-1}\beta^{i-1}$$ by using the polynomials $m(\alpha,A)$, $m(\beta,B)$ to express large powers of $\alpha$ and $\beta$ in terms of smaller powers. Then $$c = \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} a_{ij} \alpha^{j-1}\beta^{i-1}$$ and also $$c = \sum_{k=1}^{[C:A]} c_k \left( \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} f_{ijk} \alpha^{j-1}\beta^{i-1} \right) = \sum_{i = 1}^{[C:B]} \sum_{j = 1}^{[B:A]} \left( \sum_{k=1}^{[C:A]} c_k f_{ijk} \right) \alpha^{j-1}\beta^{i-1}$$ Therefore $$a_{ij} = \sum_{k=1}^{[C:A]} c_k f_{ijk}$$ So we are reduced to solving this system of system $[C:A] = [C:B] \cdot [B:A]$ linear equations for the $c_k$.

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With some difficulty, I think. Take a fairly simple example. Let $A$ be the rationals, $\alpha=\sqrt2$, $\beta=\sqrt3$, $\gamma=\sqrt2+\sqrt3$. You have $c=a_{11}+a_{12}\sqrt2+a_{21}\sqrt3+a_{22}\sqrt6$, and you want $c=c_1+c_2(\sqrt2+\sqrt3)+c_3(\sqrt2+\sqrt3)^2+c_4(\sqrt2+\sqrt3)^3$. Expressing the $c_i$ in terms of the $a_{ij}$ (and the minimal polynomials) looks mildly unpleasant. –  Gerry Myerson Sep 8 '11 at 6:30
I would have thought that you can't, since there may be several values of $\gamma$ with the same $A(\gamma)$ and $m(\gamma,A)$ that will generally have different $c_i$? For example, consider $A=\mathbb Q$, $\alpha=1$, $\beta=\sqrt2$ and $\gamma=\pm\sqrt2$. Then $A(\gamma)=\mathbb Q(\sqrt2)$ and $m(\gamma,A)=x^2-2$ for both signs, but the sign of $c_2$ is flipped. –  joriki Sep 8 '11 at 6:46
@Gerry Unpleasant or not, I need to find an algorithm do it. –  maxpower Sep 8 '11 at 9:20
@maxpower: I don't understand. How do the $a_{ij}$ help? They're the same for both signs in my example. –  joriki Sep 8 '11 at 9:29
@maxpower: I agree with joriki. The problem specification must include a way of identifying the element $\gamma$ in terms of $\alpha$ and $\beta$. Another example would be $\alpha=\sqrt2$, $\beta=i$, $\gamma$ any primitive eighth root of unity. All four sign combination are possible in $\gamma=\pm\alpha(1\pm\beta)/2$ in that all those numbers share the same minimial polynomial $x^4+1$. Unless you know which combination is $\gamma$, the problem cannot be solved. –  Jyrki Lahtonen Sep 8 '11 at 9:44

## 1 Answer

The $c_i$ are not determined by the given data. Different values of $\gamma$ can lead to identical $A(\gamma)$ and $m(\gamma,A)$ but different values of $c_i$, as in the examples given in comments by Jyrki and me.

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Wait a minute - you mean, Jyrki isn't just an alternate spelling of joriki? –  Gerry Myerson Sep 8 '11 at 11:40
@Gerry: No, one is Finnish and the other is Japanese :-) –  joriki Sep 8 '11 at 11:45
:-) :-) I don't know, whether the pronounciations would be similar though? My first name shares the first syllable with German 'Jürgen', and (according to my experience) is unpronouncable by people whose exposure is to English/Spanish/French only. Given that joriki is a Japanese name I may need to revise the way I mentally pronounce it <insert an emoticon indicating headache>. –  Jyrki Lahtonen Sep 8 '11 at 12:01
@Jyrki: It's not a Japanese name; it's a Japanese word that I use as a name :-) (As the White Knight might say, "that's what the name is called" :-) Here's what it means: the-wanderling.com/joriki.html. –  joriki Sep 8 '11 at 12:08
@joriki: Thanks. Sounds interesting! Doesn't really help me to pronounce it, though :-)? Other than that I think I know how the 'ki'-syllable is pronounced. –  Jyrki Lahtonen Sep 8 '11 at 12:23

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# Cotangent

The cotangent function is defined by the formula $$\cot(z)=\dfrac{1}{\tan z} \equiv \dfrac{\cos(z)}{\sin(z)},$$ where $\tan$ denotes the tangent function.


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# blockDiag¶

## Purpose¶

Creates a block-diagonal matrix from one or more input matrices.

## Format¶

d = blockDiag(A[, B, C ...])
Parameters

A (One or more matrices) –

Returns

d (matrix) – Diagonal matrix constructed from the input matrices.

## Examples¶

### Basic example¶

a = { 0.5 1.1,
2.0 0.3 };
b = 0.8;
c = { 0.2 1.0 0.7,
1.3 0.6 1.4 };

d = blockDiag(a, b, c);

After the above code, d should equal:

    0.5 1.1   0   0   0   0
2.0 0.3   0   0   0   0
d =   0   0 0.8   0   0   0
0   0   0 0.2 1.0 0.7
0   0   0 1.3 0.6 1.4

## Remarks¶

The input matrices may be square or rectangular. The matrices do not need to have the same dimensions.


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### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Consider two n x m matrices A and B. What can you say about the relationship among the quantities rank(A), rank(B), rank(A+B).

If we have two n x m matrices A and B then

$\mathrm{rank}\left(\mathrm{A}+\mathrm{B}\right)\le \mathrm{rank}\left(\mathrm{A}\right)+\mathrm{rank}\left(\mathrm{B}\right).$

See the step by step solution

## Step 1:   Mentioning concept

Since we know that

Dim (Im(A)) = rank(A) and dim (Im(B)) = rank(B)

So, dim (Im(A+B)) = rank(A+B)

## Step 2:   Finding the rank of (A+B)

Let $A=\left[{\stackrel{\to }{v}}_{1}{\stackrel{\to }{v}}_{2}...{\stackrel{\to }{v}}_{m}\right]\mathrm{and}B=\left[w{⃗}_{1}w{⃗}_{2}...w{⃗}_{m}\right]$

Where $v{⃗}_{i}andw{⃗}_{i}$ are column vectors of A and B respectively.

Then,

role="math" localid="1659355348377" $\mathrm{rank}\left(\mathrm{A}\right)=\mathrm{dim}\left(\mathrm{span}\left(\mathrm{v}{⃗}_{1},\mathrm{v}{⃗}_{2},...,\mathrm{v}{⃗}_{\mathrm{m}}\right)\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{and}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{rank}\left(\mathrm{B}\right)=\mathrm{dim}\left(\mathrm{span}\left(\mathrm{w}{⃗}_{1},\mathrm{w}{⃗}_{2},...\mathrm{w}{⃗}_{\mathrm{m}}\right)\right)$

$⇒rank\left(A+B\right)=dim\left(span\left(v{⃗}_{1}+w{⃗}_{1},v{⃗}_{2}+w{⃗}_{2},...v{⃗}_{m}+w{⃗}_{m}\right)\right)$

## Step 3: Finding the dimension of image of (A+B)

Let us claim,

$span\left(v{⃗}_{1}+w{⃗}_{1},\stackrel{\to }{v}{⃗}_{2}+w{⃗}_{2},...,v{⃗}_{m}+w{⃗}_{m}\right)\subset span\left(v{⃗}_{1},v{⃗}_{2},...,v{⃗}_{m}\right)+span\left(w{⃗}_{1},w{⃗}_{2},...,w{⃗}_{m}\right)$

Then any vector can be written as

$x⃗={a}_{1}\left(v{⃗}_{1}+w{⃗}_{1}\right)+{a}_{2}\left(v{⃗}_{2}+w{⃗}_{2}\right)+...+{a}_{m}\left(v{⃗}_{m}+w{⃗}_{m}\right)\phantom{\rule{0ex}{0ex}}⇒x⃗=\left({a}_{1}v{⃗}_{1}+{a}_{2}v{⃗}_{2}+...+{a}_{m}v{⃗}_{m}\right)+\left({a}_{1}w{⃗}_{1}+{a}_{2}w{⃗}_{2}+...+{a}_{m}w{⃗}_{m}\right)\phantom{\rule{0ex}{0ex}}⇒x⃗\in span\left(v{⃗}_{1},v{⃗}_{2},...,v{⃗}_{m}\right)+span\left(w{⃗}_{1},w{⃗}_{2},...,w{⃗}_{m}\right)$

for some scalars $a{⃗}_{1},a{⃗}_{2},...,{a}_{m}$.

$⇒span\left(v{⃗}_{1}+w{⃗}_{1},v{⃗}_{2}+w{⃗}_{2},...,v{⃗}_{m}+w{⃗}_{m}\right)\subseteq span\left(v{⃗}_{1},v{⃗}_{2},...,v{⃗}_{m}\right)+span\left(w{⃗}_{1},w{⃗}_{2},...,w{⃗}_{m}\right)\phantom{\rule{0ex}{0ex}}⇒Im\left(A+B\right)\subseteq Im\left(A\right)+Im\left(B\right)\phantom{\rule{0ex}{0ex}}⇒dim\left(Im\left(A+B\right)\right)\le dim\left(Im\left(A\right)\right)+dim\left(Im\left(B\right)\right)\phantom{\rule{0ex}{0ex}}⇒rank\left(A+B\right)\le rank\left(A\right)+rank\left(B\right)$

$⇒Im\left(A+B\right)\subseteq Im\left(A\right)+Im\left(B\right)\phantom{\rule{0ex}{0ex}}⇒dim\left(Im\left(A+B\right)\right)\le dim\left(Im\left(A\right)\right)+dim\left(Im\left(B\right)\right)\phantom{\rule{0ex}{0ex}}⇒rank\left(A+B\right)\le rank\left(A\right)+rank\left(B\right)$


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# HRRL Positron Rotating W Target

Q: Does the W target has to be in the vacuum?
What if we have it in the air, at the ends of the beam line we have very thin vacuum window?
In photofission experiment we had 1 mil vacuum window. under
15 MeV e- beam, only 1/1000 electron will interact with the window, produce bremsstrahlung photon.

Q: Can we make positron target that is inner part of the vacuum box, then we rotate all the box. Heat will be transfered to box,
box can cooled by water from outside.
May be box does not have to rotation movement? I can do other moments like up-down and right-left.

  I think we can make a Tungsten target as a part of the beam line. That means outer part Tungsten target will be beam pine.
Then we can use water cooling to cool our Tungsten target from outside.
Before we had all the electrons hitting Faraday Cup, which did not melt the copper even when there was not cooling water
circling around FC.
Tungsten has higher melting point, plus we run cooling water around Tungsten target, then Tungsten shouldn't melt.

The first version of a positron converter target will be designed to distribute the heat load by rotating the tungsten target.

Calculate for 1 mm and 2 mm thick Tungsten

Look for Tungsten disks to attach to brushless motor and fit into beam pipe

Accelerator Settings
 beam energy 10 MeV Rep Rate: 300 Hz I_peak: 200 mA/pulse pulse width: 0.1 s (100 ns) Vacuum: Torr

Time on for one second is 30 us. and Time off is 0.999999 s The average current 0.999999 * 0 + 0.000001*300*200mA = 60A.

P = (10 MeV) (60 A) = 600 Watts.

# Properties of Tungsten

• Melting Point = 3695 K.
• Heat Capacity =

• Heat loss due to radiation: ( \sigma T^4)

# Tungsten Temperature as a function of heat load

• IAC beamline pressure = Torr
• The Tungsten heats up when an MeV energy electron impinges its surface. The imperfect vacuum inside the beam pipe does allow some radiative cooling.

## 1.Calculating number of particles per second

We have electron beam of:

Frequency: f=1000Hz

Peak current: I=10~mAmp=0.01 Amp

Pulse width: seconds

So, how many electrons we have in each second?

By Q=It, we have

Where N is the total electron numbers hits target per second, e is electron charge and f, I and ∆t are given above. Number of particle per second is:

## 2.Calculating Energy deposited per second

If we find the energy deposited by each electron and multiply to the total number of electrons in each second, we will find the total energy per second deposited in radiator.

To find energy deposited by each electron, we need to use formula

Where is is energy deposited by one electron, is mean energy loss (also stopping power) by collision of electron and is thickness of the radiator.

Actually, energy loss of a electron comes from two parts: 1) the emission of electromagnetic radiation arising from scattering in the electric field of a nucleus (bremsstrahlung); 2) Collisional energy loss when passing through matter. Bremsstrahlung will not contribute to the temperature, because it is radiation.

Stopping power can be found from nuclear data tables and thickness is 0.001 times of radiation length. From Particle Data group we got radiation length and average total stopping powers around 15MeV for electrons in these materials from National Institute of Standards and Technology: Tungsten Stopping Power.

Note:These data is from Particle Data group,Link: [1].

### Table of energy calculations

For the thickness of 0.001 Radiation Length (0.0001RL) of radiators. Note: is from National Institute of Standards and Technology. Link: [[2]])

 Elements ) (MeV) (MeV/s) (J/s) W 1.247 0.00676 0.00842972

In above table,we took the total numbers of electrons per second and multiply it to Energy deposited by one electron,get total energy deposited per second (which is power).

## 3.Calculating equilibrium temperature using Stefan–Boltzmann law

If we assume that there is no energy conduction and total energy is just radiated from two surfaces of the radiators which are as big as beam spot,in our case beam spot is 2mm in diameter. According to Stefan–Boltzmann law, this total power radiated will be

Where T is radiating temperature P is the radiating power, A is surface area that beam incident and σ is Stefan–Boltzmann constant or Stefan's constant. To reach equilibrium temperature, Power deposited in and power radiated should be. So

so

# Vendors

## Brushless Motor

rick@empiremagnetics.com:

The VX grade motors, with dry lubes have been used in deep space (Wake Shield Module) where I'm told they were trying to get 10^-12 torr.

We need 10^{-8 }Torr

PositronsGo Back


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CBSE Sample Papers for Class 12 Economics Compartment Outside Delhi -2011

Time allowed : 3    hours                                                                                Maximum marks 100

GENERAL INSTRUCTIONS

1.  All questions in both the sections are compulsory.
2. Marks for questions are indicated against each.
3. Questions No. 1-5 and 17-21 are very short-answer questions carrying 1 nick each. They are required to be answered in one sentence each.
4.  Questions No. 6-10 and 22-26 are? short-answer questions carrying 3 marks each. Answers to them should normally not exceed 60 words each.
5.  Questions No. 11-13 and 27-29 are also short-answer questions carrying 4 marks each. Answers to them should normally not exceed 70 words each.
6.  Questions No. 14-16 and 30-32 are long-answers questions carrying 6 marks each. Answers to them should normally not exceed 100 words each.
7.  Answers should be brief and to the point and the above word limit should be adhered to as far as possible.

SET I

SECTION A
Question.1. Why is the study of consumer’s equilibrium a subject matter of microeconomics?
Answer. Consumer’s equilibrium is a subject matter of microeconomics because it relates to the problem of an individual unit and not of the entire economy.

Question.2. Define marginal utility.
Answer. Marginal utility may be defined as the change in total utility that takes place as a result of consuming an additional unit of a commodity.

Question.3. What causes an upward movement along a demand curve?
Answer. There will be an upward movement on a demand curve of a commodity when its price increases.

Question.4. Give the meaning of marginal revenue.
Answer. Marginal revenue is the change in total revenue that takes place by selling an additional unit of a commodity.  $MR={ TR }_{ (n) }-{ TR }_{ (n-1) }$

Question.5. Which market form has the least number of producers?

Question.6. Explain the problem of ‘for whom to produce’.
Answer. The main objective of producing a commodity in a country is its consumption by the people of the country. However, even after employing all the resources of a country, it is not possible to produce all the commodities which are required by the people. Therefore, an economy has to decide as to for whom goods should be produced. This problem is the problem of distribution of produced goods and services. Therefore, what goods should be consumed and by whom depends on how national product is distributed among various people.

Question.7. Define an indifference curve. Why is it convex to the origin?
Answer. Indifference Curve: There are many combinations of two commodities (say X1 and X2) which give equal level of satisfaction to a consumer. When these combinations are graphically, represented, the curve so formed is known as an indifference curve.
An indifference curve is convex to the origin because of the diminishing marginal rate of substitution. Marginal rate of substitution is the amount of commodity X2 which a consumer is walling to surrender for obtaining one unit of commodity X1. Because of the law of diminishing utility the amount of commodity X2 for obtaining a successive unit of commodity X1 keeps on declining. Because of this an indifference curve is convex to the origin.
Or ‘
What are the conditions of consumer’s equilibrium under utility analysis?
Answer. (i) Consumer’s equilibrium, according to utility analysis, is established at a point where marginal utility of money is equal to the price of the commodity.
Marginal utility goes on declining but price remains the same. Therefore in such a situation a consumer shall go on consuming a commodity till he reaches a point where marginal utility is equal to the price.
Consumer’s equilibrium in purchase of single good is attained when MU in money terms = Price i.e..,
$\frac { MUofaproduct }{ MUofarupee } =PriceofaProduct$
(ii) In the case of two commodities the consumer’s equilibrium will be established where marginal utilities of both the goods he consumes are equal, MU of good X = MU of good Y, assuming the price of each good is the same.
If the prices of the two goods are different, then consumer’s equilibrium is when,
[/latex]\frac { { MU }_{ X } }{ PriceofX } =\frac { { MU }_{ Y } }{ PriceofY } =MUofmoney[/latex]

Question.8. How is the demand of a commodity affected by a fall in the prices of related goods?
Answer. Related goods may be substitute goods or complementary goods. Substitute goods are those which can be used in place of one another to satisfy the same want such as pen and pencil and complementary goods are a pair of goods which are used together to satisfy a given want such as car and petrol.
If the price of a substitute good falls it becomes cheaper in comparison to the commodity in question and therefore its demand will fall. For example, fall in the demand for pen because of the fall in the price of the pencil. In case of complementary goods a fall in the price of petrol will lead to an increase in its demand and therefore the demand for cars will also increase. Therefore when there is a fall in the price of a complementary good the demand for related goods increases.

Question.9. Explain the implications of the homogeneous products feature of a perfectly competitive market.
Answer. Because of products being homogeneous in perfect competition, the price of the commodity is the same and remains unchanged. This is so because if a firm charges a higher price nobody will buy the product from that firm because the same product is available at a lesser price. On the other hand, if a firm charges lesser price than the market price, it will be an unwise decision on the part of the firm because it can get the market price which is higher than the price it fixes. In short, a competitive firm is a price taker and has no role to play in fixing the price of the commodity.

Question.10. From the following data, calculate the price at each level of output:

Question.11. Explain the effect of the following on price elasticity of demand of a good:
(i) Number of substitutes of the good
(ii) Proportion of income spent on the good.
Answer. (i) Number of substitutes. The demand for a Commodity will be very elastic if some other commodities can be used as a substitute for it. A small rise in the price of such a commodity will decrease its demand as its substitutes are available at a lesser price e.g. in case of firewood and coal, coal will be used more as fuel if the price of wood increases. On the other hand, the demand of such commodities is inelastic which have no substitutes, such as salt.
(ii) Proportion of total income spent on the product. If a small proportion of total income is spent on a commodity, its demand will be inelastic such as demand for salt. On the other hand, if a major portion of total income is spent on a commodity, its demand will be more or highly elastic such as demand for luxury goods.

Or
Give the meaning of price elasticity of demand. Explain the relationship between it and total expenditure.
Answer. Price elasticity of demand may be defined as the ratio of percentage change in the quantity demanded to percentage change in price of a commodity. In short, it is the degree of responsiveness of demand for a change in the price of a commodity. According to total expenditure the elasticity of demand will be as follows when price and total expenditure, change.
(i) If price and total expenditure move in opposite directions, elasticity of demand will be greater than 1.
(ii) If total expenditure remains the same whether price increases or decreases, elasticity of demand will be equal to 1.
(iii) If price and total expenditure change in the same direction, price elasticity of demand wiU be less than 1.

Question.12. State the different phases of returns to a factor in terms of total product. Represent the same on a diagram.
Answer. In the short run, units of variable factors change (or increase) for increasing the total product. In this context, three phases or situations can be possible. All the three phases have been exhibited in the given diagram. These are:
(i) In the beginning,4 when additional units of a variable
factor are employed, marginal product increases and Y total product also increases. This is phase I. I st phase ends where AP is highest or from where MP starts declining;
(ii) After this point, marginal product of the additional units of variable factor employed will start declining.
Therefore, total product increases but at a decreasing rate. This is phase II.
(iii) If additional units of variable factor are still used, marginal product becomes negative and hence total product starts declining. This is phase III.
A producer always operates in the second stage. ‘
For Blind Candidates only in lieu of Q. No. 12 State the different phases of returns to a factor in terms of total product. Represent the same in a schedule.

Question.13. The price of a commodity is 110 per unit and total revenue from it is Rs 1,000. Its price elasticity of supply is 0.8. Its price falls by 10 percent. Calculate the total revenue at the reduced price.

Question.14. Giving reasons, state whether the following statements are true or false:
(i) Marginal utility can never be negative. .
(ii) A budget set is the collection of all bundles of goods that a consumer wants to buy.
(iii) The demand for a commodity always increases with increase in the price of other goods.
Answer. (i) The statement is false. Marginal utility can be negative also. Marginal utility keeps on declining on using additional units of a commodity successively. Therefore if we go on using additional units, a stage comes, when marginal utility becomes zero and then negative.
(ii) The statement is false. The set of bundles of two goods that are available to a consumer is a budget set. This signifies that the bundles constituting budget set are available to the consumer and does not mean that these are the bundles which consumer wants to buy.
(iii) The statement is false. Increase in the prices of other goods leads to an increase in the demand for a commodity but not always.

Question.15. Complete the following table:

Answer. Total Fixed Cost = AFC x No. of Units = 18 x 4 = Rs 72

Question.16. Explain the sequence of changes that will take place when there is excess demand of a commodity.
Answer. There will be excess demand of a commodity when price is less than the equilibrium price. In the given diagram, OP is the equilibrium price where quantity demanded and supplied are equal. However, when price is reduced to OP1, demand exceeds supply, with the result that there is excess demand = Q2Q1 In such a case the demand should be reduced and supply increased. So that both these may be equal. These would be equal, when price is increased from OP1 to OP, where quantity demanded and supplied are equal.

Or
X and Y are complementary goods. Explain the sequence of effects of a fall in the price of X on the equilibrium price and quantity of Y.
Answer. X and Y are complementary goods. This means both goods are demanded simultaneously. In other words, X cannot be used without Y and Y cannot be used without X. If the price of X falls, its demand will increase because of the application of law of demand.
When demand of X increases, demand of Y will also increase, with the result that there will be a rightward shift in the demand curve of Y.
As a result of this, the equilibrium point will shift to the right from E to F, where price increases from OP to OP1 and demand increases from OQ to OQ1
Therefore, as a result of fall in the price of X, the equilibrium price and quantity of commodity Y both increase. This is clear from the diagrams given below.

SECTION B
Question.17. Define domestic product.
Answer. Domestic product is the value of final goods and services produced within the domestic territory of a country in an accounting year

Question.18. What is transfer payment?
Answer. Transfer payments are unilateral payments which recipients get without adding anything to the current flow of goods and services.

Question.19. What are demand deposits?
Answer. Demand deposits are those bank deposits which can be withdrawn by the depositors whenever they want.

Question.20. How is primary deficit calculated?
Answer. Primary deficit is the difference of fiscal deficit and interest payments .
Primary Deficit = Fiscal Deficit – Interest Payment.

Question.21. What is meant by cash reserve ratio?
Answer. Cash Reserve Ratio (CRR) is the portion of total deposits which commercial banks are required to keep as reserve with the central bank of the country.

Question.22. From the following data calculate “net value added at factor cost”:

Question.23. Explain the relationship between average propensity to consume and average propensity to save. Which of these can have a negative value and when?
Answer. Average Propensity to Consume (APC) is the ratio of total consumption expenditure to total income. Average Propensity to Save (APS) is the ratio of total savings to total income. In short,

Question.24. Explain the meaning and implications of revenue deficit.
Answer. See Q. 29, 2008 (I Outside Delhi)

Question.25. In an economy aggregate demand is less than aggregate supply. Explain the changes that will take place in this economy.
Answer. If equilibrium is established before the full employment level of output, it is a situation where aggregate demand is less than aggregate supply. The gap between the two is deflationary gap as shown by EF in the adjacent diagram.
This gap can be rectified by increasing the level of aggregate demand. In this case autonomous investment needs to be increased. This situation can be exhibited with the help of a diagram.

Or
Explain the meaning and implications of deflationary gap.
Answer. When aggregate demand and aggregate supply are in equilibrium at less than the full employment level of output, this is a situation of deficient demand.
This happens when aggregate demand in the economy is not sufficient to ensure that level of output which can give employment to all those who are willing to work.
Aggregate demand falls short of aggregate supply at full employment level of output by a certain amount, which is called deflationary gap. The extent of deficiency in demand (deflationary gap i.e., EF) at the full employment level is shown in the adjacent diagram.

Question.26. Explain why there is a rise in demand for foreign exchange when its price falls.
Answer. See Q. 23, 2008 (I Outside Delhi)

Question.27. How can distribution of income be a limitation of using gross domestic product as an index of welfare? Explain.
Answer. It is to be seen as to how the increase in national income is distributed. If national income rises, it is not necessary that income of each individual rises in the same proportion. Individual incomes may rise in different proportions. In some cases, it may fall. In other words, inequalities may rise. Rise in inequalities in incomes may adversely affect the economic welfare of the society. Therefore, for ascertaining the effect of increase in national income, it has to be seen whether it increases income inequalities or reduces income inequalities.
Or
Giving reason, categories the following into stocks and flows:
(i) Profits (ii) Capital (iii) Savings (iv) Balance in a bank account
Answer. (i) Profits are flows because they relate to a period of time.
(ii) Capital is a stock as it relates to a point of time.
(iii) Savings are flows as they relate to a period of time.
(iv) Balance in a bank account is a stock as it relates to a point of time.

Question.28. Distinguish between revenue receipts and capital receipts. Give two examples of each.
Answer. Revenue Receipts: Any receipt which does not either create a liability or leads to reduction in the value of assets is a revenue receipt. Revenue receipts may be tax revenue such as income tax and may be non-tax revenue such as fees and fines.
Capital Receipts: Capital receipts are those receipts which, the government gets either by incurring liability or disposing off some assets. Its examples are mobilising funds through loans from the public. Government may also get funds by disinvestment such as selling a public sector unit. These are examples of public receipts
Corresponding to revenue receipts, there is no liability whereas in the case of capital receipts a liability is created or an asset is sold.

Question.29. Giving reasons, state whether the following statements are true or false:
(i) Current account of Balance of Payments account records only exports and imports of goods and services.
(ii) Foreign investments are recorded in the capital account of balance of payments.
Answer. (i) The statement is false. In current account of balance of payments, besides recording export and import of services, unilateral receipts and payments are also included.
(ii) The statement is true. Foreign investments are recorded in the capital account of balance of payments.

Question.30. How is “bank rate” used by central bank in influencing credit creation by commercial banks? Explain.
Answer. Bank rate is the rate at which central bank discounts first class bills and securities of the commercial banks. In other words, the rate of interest which a central bank charges from the commercial banks, to give them credit can be called as bank rate.
(i) When bank rate is higher, commercial banks will charge higher rate of interest from their customers. Therefore, when bank rate increases, rate of interest on loans will also be more and so credit becomes expensive and the demand for credit is reduced. This . results in credit control.
(ii) With decrease in bank rate, interest rate on loans is also decreased, 50 credit becomes cheap and demand for credit increases. It is called credit expansion. ,
In short, bank rate and volume of credit are inversely related and the effects of changes in the bank rate on the volume of credit is brought about through the changes in the rate of interest.
Or
How does money overcome the problems of barter system? Explain briefly.
Answer. Money overcomes the problems of barter system by the following functions:
Medium of exchange: It means that money acts as an intermediary for the exchange of goods and services. It has helped a lot in promoting trade among people and countries. Money has removed the difficulty of double coincidence of wants.
Measure of value: Money serves as a measure of value. All the goods and services have their prices. This helps in measuring the exchange values of commodities. So value of all goods can be expressed in terms of money.
Standard of deferred payment: The value of money is more stable in comparison to the value of other commodities. It has general acceptability also. It is more durable as compared to other commodities.
Store of value: Money serves as a store of value because it can be stored without loss in its value and it is also convenient to store. Moreover, goods are perishable, whereas money is not perishable in the same sense.

Question.31. Explain the steps taken in derivation of the Consumption Curve from the Saving Curve.Use diagram.
Answer. Total income is the sum of = Consumption Expenditure + Saving. Consumption is the function of income. However, some consumption is always there even if there is no income i.e. zero level of income. This is called as autonomous consumption. Saving is that part of income which is not consumed. Therefore, if we subtract consumption from income, we get saving. When consumption is greater than income, saving is negative when consumption and income are equal, saving is zero; when consumption is less than income, saving is positive.
If we subtract consumption from income, we get saving. In the same way, if we subtract saving from income, we get consumption. In short,
Saving Income – Consumption , Consumption = Income – Saving Thus consumption and saving function can be stated as below: Y C = a + c Y S = a + cY.

In the diagram, CC is consumption curve. OY is 45 ° line. E is the point of intersection where consumption and income are equal. At this level, saving is equal to zero. Up to OQ level of income, consumption is greater than income and therefore saving is negative. Beyond OQ level of income, consumption is less than income and therefore saving is positive. Because of this SS saving curve does not start from the point of origin but from the point S.
For Blind Candidates only in lieu of Q. No. 31.
Explain the steps taken in deriving the consumption function from the Saving function.

Question.32. Calculate: (i) GroSs National Product at Market Price and (ii) Net National Disposable Income from the following data:

SET II

Note : Except for the following questions, all the remaining questions have been asked in Set I.
SECTION A
Question.8. Explain the effect of rise in income of the buyers of a commodity on its demand.
Answer. When income of the buyers increases, they demand more quantity at the same price. As a result of this, there is a shift in demand curve towards the right. This phenomenon can be explained with the help of a diagram. As a result of increase in the income of buyers, the point of equilibrium, which was at E, shifts to F, where the, quantity demanded at price OP increases from OQ to OQ1.

Question.9. Explain any two features of monopoly.
Answer. The two features of monopoly are:
(i) There is a single seller of the commodity in the market.
The monopoly firm has full control over supply of the commodity.
(ii) There are no close substitutes of the commodity sold by the monopolist.

Question.13. When the price of a commodity falls from Rs 10 per unit to Rs 9 per unit, total revenue from it falls from Rs 1,200 to Rs 918. Calculate its elasticity of supply.

Question.15. Complete the following table:

Answer. Total Fixed Cost = AFC x No. of Units = 18 x 5 = Rs 90

SECTION B
Question.17. Define intermediate goods.
Answer. Intermediate goods are those which are used during production process. These goods are not included in the estimation of national income.

Question.21. Define money.
Answer. Money may be defined as anything which is used as a medium of exchange and has general acceptability.

Question.24.Distinguish between direct tax and indirect tax.
Answer. Direct Tax: When the liability to pay a tax and the burden of that tax falls on the same person, the tax is called a direct tax. The burden cannot be shifted on others. For example, Income tax. Indirect Tax: When the liability to pay a tax is on one person and the burden of it falls on another person, it is called an indirect tax.’The burden of such a tax is shifted others. For example, series tax.

Question.26. Give the meaning of foreign exchange rate. How is it determined under flexible exchange rate regime?
Answer. The rate at which the currency of a country is exchanged for another currency is known as the rate of exchange. Rate of exchange may be fixed or flexible. Fixed exchange rate is that, – where the exchange rate is fixed by the government of the country deliberately. On the other hand, the flexible exchange rates is that which is fixed by the forces of demand for and supply of foreign exchange. These days, rates of exchange are flexible.
There is an inverse relationship between demand of a foreign currency and its price in terms of domestic currency. On the other hand, there is a direct relationship between file supply of a foreign currency and its price in terms of domestic currency.
The equilibrium exchange rate is determined at a point where demand for and supply of foreign currency is equal. As shown in the diagram, the exchange rate will be OP because at this level the demand for and supply of foreign currency is equal.

SET III

Note : Except for the following questions, all the remaining questions have been asked in Set I and Set II.
SECTION A
Question.6. Explain the problem of ‘what to produce’
Answer. The problem of ‘what to produce’ signifies what goods should be produced and in what quantities. This problem arises when due to scarcity of resources we cannot produce each and every thing which we want. Therefore, a decision has to be taken as to what goods should be produced and in what quantities. In short, we have to see whether we should produce consumer goods or producer goods or defense goods or all the goods in some quantity combination. So, on the basis of importance of various goods, an economy has to decide which goods should be produced and in what quantities.

Question.9. Explain the implication of the product differentiation feature of monopolistic competition.
Answer. Under monopolistic competition, artificial differences are created in the product by different producers on the basis of size, packing, brand name, colour, quantity, smell, etc These differences are created with a view to project the product as a separate product from the point of view of consumers. Different brands of toothpaste, such as Colgate, Meswak, Pepsodent are the examples of product differentiation. As a result of product differentiation, if a producer increases the price of his product, he does not lose all the consumers. On the other hand, if he reduces the price of his product, the number of consumers buyers of his product does not increase.

Question.13. From the following data, calculate elasticity of supply:

Question.16. Complete the following table:

Answer. Fixed cost = AFC x No. of units = 6×5= Rs 30

SECTION B
Question.17. Define capital goods.
Answer. Capital goods are those goods which form the capital stock of a country at the end of an accounting year and which are used in the production process.

Question.21. What is statutory liquidity ratio?
Answer. Statutory liquidity ratio is the proportion 6f total deposits which commercial banks are legally required to keep with them.

Question.24. Distinguish between revenue expenditure and capital expenditure with one example of each.
Answer. Revenue Expenditure: It is the expenditure of a government which does not create any asset nor reduce liabilities. Its example is the salaries of employees.
Capital Expenditure: An expenditure which results in the creation of assets or reduction in liability, such as construction of roads and bridges.

Question.26. Explain why is there a direct relationship between price and supply of a foreign currency.
Answer. There is a direct relationship between price of a foreign currency and its supply. As such, when the price of a foreign currency increases, its supply also increases and when its price declines it supply also declines. When the price of a foreign currency falls its demand increases. Naturally more and more foreign currency will be required for buying foreign goods, this would lead to a fall in the supply of the foreign currency. On the other hand, when the price of foreign currency increases its supply will also increase.

Question.30. Calculate: (i) Net Domestic Product at Factor Cost and (ii) Gross National Disposable Income from the following data:


text:
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# Formal and alternate form of the derivative

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.B (LO)
,
CHA‑2.B.2 (EK)
,
CHA‑2.B.3 (EK)
,
CHA‑2.B.4 (EK)

## Video transcript

so let's think about how we could find the slope of the tangent line to this curve right over here so what I've drawn in red at the point x equals a and we've already seen this with the definition of the derivative we could try to find a general function that gives us the slope of the tangent line at any point so let's say we have some arbitrary point let me define some arbitrary point X right over here then this would be the point X comma f of X and then we could take some x + H so let's say that this right over here is the point X plus h and so this point would be X plus h f of X plus h we can find the slope of the secant line that goes between these two points so that would be your change in your vertical which would be f of X plus h f of X plus h minus f of X minus f of X over the change in the horizontal which would be X plus h minus X X plus h minus X minus X and these two x's cancel so this would be the slope of this secant line and then if we want to find the slope of the tangent line at X the slope of the tangent line at X we would just take the limit the limit of this expression as H approaches 0 as H approaches 0 this point moves towards X and that slope of the secant line between these two is going to approximate the slope of the tangent line at X and so this right over here this we would say is equal to this is equal to F prime of X this is still this is a function of X you give me an X you give me an arbitrary X that where the derivative is defined I'm going to plug it into this whatever this ends up being it might be some nice clean algebraic expression that I'm going to give you a number so for example if you wanted to find you you could calculate this somehow or you could even leave it in this form and then if you wanted F prime of a if you wanted F prime of a you would just substitute a into your function definition and you would say well that's going to be the limit as H approaches zero of every place you see an X replace it with an A f of I'll stay in this color for now blank plus h minus minus F minus F of blank minus F of blank all of that over all of that over H and I left those blanks so I could write the a I could write the a in red notice every place where I had an X before it's now an A so this is the derivative evaluated at a so this is one way to find the slope of the tangent line when x equals a another way and this is often used as the alternate form of the derivative would be to do it directly so this is the point a comma F of a let's just take another arbitrary point some here someplace so let's say this is the value X this point right over here on the function would be X comma f of X and so what's the slope of the secant line between these two points what would be change in the vertical which would be f of X minus f of a minus F of a over a change in the horizontal over X minus a so let me do that purple color over X minus a now how can we get a better and better approximation for the slope of the tangent line here well we could take the limit as X approaches a as X gets closer and closer and closer to a the secant line slope is going to better and better and better approximate the slope of the tangent line this tangent line that I have in red here so we would want to take the limit the limit as as X approaches a here either way we're doing a very similar we're doing the exact same thing we're finding that we're taking this we're have an expression for the slope of a secant line and then we're bringing those those X values of those points closer and closer together closer and closer together so the slopes of those secant lines better and better and better approximate that slope of the tangent line and at limit it does become the slope of the tangent line that is the derivative of that's the definition of the derivative so this is the clot that kind of the more standard definition of a derivative it would give you your derivative as a function of X and then you can then input your X that you want your the the particular value of x or you could use the alternate form of the derivative if you know that hey look I'm just looking to find the derivative exactly at a I don't need a general function of F then you could do this but they're doing the same thing
AP® is a registered trademark of the College Board, which has not reviewed this resource.


text:
# Why are coils of electric toasters and electric irons

Question.
Why are coils of electric toasters and electric irons made of an alloy rather than a pure metal ?

solution:
This is because

(i) resistivity of an alloy is generally higher than that of pure metals

(ii) an alloy does not oxidise at high temperatures.


text:
### How to fix v-cloak not working in Vue.js?

Sometimes, we want to fix v-cloak not working in Vue.js.

In this article, we’ll look at how to fix v-cloak not working in Vue.js.

### How to fix v-cloak not working in Vue.js?

To fix v-cloak not working in Vue.js, we’ve to set the styles for `[v-cloak]` to hide the items.

For instance, we write

``````<template>
<div id="app">
<div v-cloak>
{{ message }}
</div>
</div>
</template>
``````

to add the `v-cloak` directive.

Then we hide the content of the div before it’s rendered with

``````<style>
[v-cloak] {
display: none;
}
</style>
``````

### Conclusion

To fix v-cloak not working in Vue.js, we’ve to set the styles for `[v-cloak]` to hide the items.


text:
1. ## implicit differentiation

I tried doing this myself but I'm really not sure my answer makes sense. :/

find dy/dx by implicit differentiation
square root (x+y)=1 +x^2 y^2

2. Hello,
Originally Posted by midsummer
I tried doing this myself but I'm really not sure my answer makes sense. :/

find dy/dx by implicit differentiation
square root (x+y)=1 +x^2 y^2
Consider y as a function with respect to x.

So you have $\sqrt{x+y}$
What is its derivative ? It's $\frac{(x+y)'}{2 \sqrt{x+y}}$ (using chain rule)
What is $(x+y)'$ ? It's $(x)'+(y)'=1+y'$
So $(\sqrt{x+y})'=\frac{1+y'}{2 \sqrt{x+y}}$

You have $1+x^2 y^2$. What I advise you is to write $1+(xy)^2$
What is its derivative ? Using chain rule, we have : $(1)'+2(xy)'(xy)=2(xy)'(xy)$
What is $(xy)'$ ? Using product rule, we have : $y+xy'$
So $(1+(xy)^2)'=2(y+xy')(xy)$

$\boxed{\frac{1+y'}{2 \sqrt{x+y}}=2(y+xy')(xy)}$

Solve for $y'=\frac{dy}{dx}$

******** chain rule says : $f(g(x))=g'(x) f'(g(x))$

3. Great explanation. You are awesome!


text:
# Do the proton PDFs change much with Q?

Specifically, the second moments, do they change much from say 100 to 1000 GeV? Why or why not?

-

You can check directly yourself using the Durham HEPData page, which lets you plot PDFs as a function of $x$ or $Q^2$.

At large $x = 0.1$, they're fairly constant over a range of $Q^2$.

But at small $x = 10^{-4}$, they rise quickly with $Q^2$:

This is an illustration of Bjorken scaling at large $x$ and its breaking at small $x$.

I'd also mention that the $Q^2$-dependence of the PDFs is predicted from QCD by the DGLAP equations, so you can look at those to further understand the behavior.

-


text:
# 2011-02-24 Wavy lines as delimiters (en)

Last edit

Summary: rewording

Changed:

< (The {{{\wavylrbox}}} macro takes one argument and sets it surrounded from //l//eft and //r//ight---hence its name---by wavy lines in display math style.)

to

> (The {{{\wavylrbox}}} macro takes one argument and sets it in display math style surrounded from //l//eft and //r//ight---hence its name---by wavy lines.)

Well, it’s been a while since I posted here; but today, I’ve got another real-life problem solution.

In Poland, it is customary to use “wavy lines” as delimiters when writing down integration by substitution or by parts. (This description is rather vague, I admit, so below is an example.)

The problem is: how to obtain such an effect in LaTeX?

The problem is, surprisingly, not so easy. There is the MnSymbol font & package, which contains “wavy variable-size delimiters”, but it replaces all other math fonts with variants someone might not like (and is incompatible with amssymb/amsfonts).

Of course, tikz comes to rescue. Here is the code I used to obtain the above example:

\documentclass{article}

\usepackage{amsmath}

\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}
\newcommand{\wavylrbox}[1]{%
\begin{tikzpicture}[baseline=-0.5ex,decoration={snake,amplitude=0.75pt,segment length=5pt}]
\node (b) {$\displaystyle #1$};
\draw[decorate] (b.north west) -- (b.south west);
\draw[decorate] (b.north east) -- (b.south east);
\end{tikzpicture}
}

\begin{document}

\begin{equation*}
\int 2x\cos x^2\,\textup{d}x=\wavylrbox{
\begin{aligned}
t&=x^2\\
\textup{d}t&=2x\,\textup{d}x\\
\end{aligned}
}=\int \cos t\,\textup{d}t=\sin t+C=\sin x^2+C
\end{equation*}

\end{document}

(The \wavylrbox macro takes one argument and sets it in display math style surrounded from left and right—hence its name—by wavy lines.)

Unfortunately, this quick solution has a few problems.

1. If the height of the formula in “wavy lines” is not an integer multiple of 5pt, the waves have short straight-line segments at the bottom.
2. We assume here that the “central line” of the formula is at the height of 0.5ex. This might not be true in some fonts.
3. If the “wavy box” has an odd number of lines, it would be much better if the middle line were at the height of the rest of the formula. This seems to be tricky to obtain—most probably, instead of aligned or friends, one would have to use tikz’s matrices or something like that.

Still, tikz (as usually) does the trick nicely.

I’d be more than happy to see other/better solutions, especially ones that don’t have the above disadvantages!


text:
# Thread: Rearranging Surds / Proof

1. ## Rearranging Surds / Proof

Hey..

Sorry for the horrible appearence of this question but here it goes..

Show that:
17(1-(1/17^2))^0.5 = n(sqrt(2))

Where n is an integer, whose value is to be found

the closest ive gotten so far is
17(sqrt(1-17^-2))
or
17(sqrt((17^2-1)/(17^2))

... but I'm not sure where to start since 17 is prime and I don't think the brackets can be expanded because the sqrt() surrounding the right-most part of the expression doesn't allow this as far as I'm aware

I'd prefer to have the steps to complete the problem since the value for n is given at the back of the textbook (n=12).

Kwah =]

2. Originally Posted by kwah
Hey..

Sorry for the horrible appearence of this question but here it goes..

Show that:
17(1-(1/17^2))^0.5 = n(sqrt(2))

Where n is an integer, whose value is to be found

the closest ive gotten so far is
17(sqrt(1-17^-2))
or
17(sqrt((17^2-1)/(17^2))

... but I'm not sure where to start since 17 is prime and I don't think the brackets can be expanded because the sqrt() surrounding the right-most part of the expression doesn't allow this as far as I'm aware

I'd prefer to have the steps to complete the problem since the value for n is given at the back of the textbook (n=12).

Kwah =]
one way to do this:

$17 \sqrt{1 - \frac 1{17^2}} = \sqrt{17^2 - 1}$

$= \sqrt{288}$

$= \sqrt{2(144)}$

$= \sqrt{2} \sqrt{144}$

$= 12 \sqrt{2}$

got it?

3. You missed ENTIRELY where I was having trouble with - getting

$17 \sqrt{1 - \frac 1{17^2}} = \sqrt{17^2 - 1}$

Everything after that I would have been entirely happy with ...

After sitting down at it for another 5 minutes i remembered that it becomes

$\sqrt{289} * \sqrt{1 - \frac {1}{289}}$

which is then

$\sqrt{289 - 289(\frac 1{289})}$

$\sqrt{289 - 1}$

$\sqrt{288}$

.. then the steps you outlined =]

4. Originally Posted by kwah
You missed ENTIRELY where I was having trouble with - getting

$17 \sqrt{1 - \frac 1{17^2}} = \sqrt{17^2 - 1}$

Everything after that I would have been entirely happy with ...

After sitting down at it for another 5 minutes i remembered that it becomes

$\sqrt{289} * \sqrt{1 - \frac {1}{289}}$

which is then

$\sqrt{289 - 289(\frac 1{289})}$

$\sqrt{289 - 1}$

$\sqrt{288}$

.. then the steps you outlined =]
the trick is this: if x is positive, then $\sqrt{x^2} = x$. so if we square a positive number and then square root it, we simply undo the square root, so that makes sense. furthermore, it is a law of surds that: $\sqrt{x} \sqrt{y} = \sqrt{xy}$

so, $17 \sqrt{1 - \frac 1{17^2}} = \sqrt{17^2} \sqrt{1 - \frac 1{17^2}}$ ............by the first thing i said

$= \sqrt{17^2 \left( 1 - \frac 1{17^2}\right)}$ .........by the law of surds that i mentioned

the rest follows

5. Originally Posted by Jhevon
the trick is this: if x is positive, then $\sqrt{x^2} = x$. so if we square a positive number and then square root it, we simply undo the square root, so that makes sense. furthermore, it is a law of surds that: $\sqrt{x} \sqrt{y} = \sqrt{xy}$

so, $17 \sqrt{1 - \frac 1{17^2}} = \sqrt{17^2} \sqrt{1 - \frac 1{17^2}}$ ............by the first thing i said

$= \sqrt{17^2 \left( 1 - \frac 1{17^2}\right)}$ .........by the law of surds that i mentioned

the rest follows
uhh... that is simply a less-clear explanation of the steps i did and naming it as the law of surds lol (dull)

btw, forgive me if ive misunderstood, but none of this quote from your post is accurate...
the trick is this: if x is positive, then $\sqrt{x^2} = x$.
For $\sqrt{x^2} = x$, it does not matter if x is positive since it gets squared first...
Obviously, if it gets rooted first then squared, $\sqrt{x} ^2 = x$ x has to be positive

so if we square a positive number and then square root it, we simply undo the square root, so that makes sense.
As for this second sentence, it does not make sense that squaring a number THEN rooting it = undoing the root ((in fact, applying a root is undoing the square - which came first))

regards,
Kwah =]

6. Originally Posted by kwah
uhh... that is simply a less-clear explanation of the steps i did and naming it as the law of surds lol (dull)
it is a law of surds. and my explanation is more general than yours was. you used 17 as the specific example, my x could be anything.

btw, forgive me if ive misunderstood, but none of this quote from your post is accurate...

For $\sqrt{x^2} = x$, it does not matter if x is positive since it gets squared first...
Obviously, if it gets rooted first then squared, $\sqrt{x} ^2 = x$ x has to be positive
you are mistaken. in general, $\sqrt{x^2} = |x|$

by the way, x cannot be rooted first if it was negative. you'd get a complex number in that case
As for this second sentence, it does not make sense that squaring a number THEN rooting it = undoing the root ((in fact, applying a root is undoing the square - which came first))
the square is under the square root, so we apply it first. $\sqrt{x^2} = (x^2)^{1/2}$, you do what is in the brackets first.

7. Originally Posted by kwah
As for this second sentence, it does not make sense that squaring a number THEN rooting it = undoing the root ((in fact, applying a root is undoing the square - which came first))
yes, i meant to type square, not square root. but that typo seems obvious, as you pointed out. my apologies

8. I'm really confused where to begin with replying to this ..

first off, i don't agree that $\sqrt{x^2} = |x|$
regardless of the value of $x$, positive or negative, it results in $\sqrt{positive number}$, yeah?

roots can have a positive OR negative value, therefore $\sqrt{x^2}$ does not necessarily equate to $|x|$

If, as I stated in my last post, the $^2$ is OUTSIDE the root, then $x$ is forced to become positive for two reasons..
$\sqrt{x}^2 = (\sqrt{x})^2 = |x|$ is true (note: the $^2$ outside the root)

for the reasons we have both stated, you cannot root negative numbers therefore x is required to be positive..
AND, squaring it afterwards would force it to become positive regardless..

$\sqrt{x^2} = (x^2)^0.5 = x^1$
$\sqrt{(-10)^2} = ((-10)^2)^0.5 = (-10)^1$ (negative)

There was more that I wanted to say but it has gotten lost in my mind lol..

regards,
Kwah =]

9. Originally Posted by kwah
I'm really confused where to begin with replying to this ..

first off, i don't agree that $\sqrt{x^2} = |x|$
regardless of the value of $x$, positive or negative, it results in $\sqrt{positive number}$, yeah?

roots can have a positive OR negative value, therefore $\sqrt{x^2}$ does not necessarily equate to $|x|$

If, as I stated in my last post, the $^2$ is OUTSIDE the root, then $x$ is forced to become positive for two reasons..
$\sqrt{x}^2 = (\sqrt{x})^2 = |x|$ is true (note: the $^2$ outside the root)

for the reasons we have both stated, you cannot root negative numbers therefore x is required to be positive..
AND, squaring it afterwards would force it to become positive regardless..

$\sqrt{x^2} = (x^2)^0.5 = x^1$
$\sqrt{(-10)^2} = ((-10)^2)^0.5 = (-10)^1$ (negative)
$\sqrt{x^2} = |x|$ by definition. it is not something i made up. there are two standard definitions for |x|, and that is one of them.

(exercise: try graphing $\sqrt{x^2}$ and see what you get. it will be the absolute value function for x NOT the line y = x as you are claiming)

counter example to what you said.

let x = -2

compute $\sqrt{x^2}$

we get: $\sqrt{(-2)^2} = \sqrt{4} = 2 \ne x$

also, i never said anything about $(\sqrt{x})^2$, so i do not see why you are arguing with me about it. yes indeed, the domain of this function is $x \ge 0$. that is if we are considering real numbers. i have no problem with you saying that x has to be nonnegative here for real numbers.

There was more that I wanted to say but it has gotten lost in my mind lol..

try to remember and come back

10. Originally Posted by Jhevon
counter example to what you said.

let x = -2

compute $\sqrt{x^2}$

we get: $\sqrt{(-2)^2} = \sqrt{4} = 2 \ne x$
jeez lol ..

$\sqrt{(-2)^2} = \sqrt{4} = 2 \ne x$ is NOT correct ...
$\sqrt{(-2)^2} = \sqrt{4} = 2 OR -2$ (im not sure how to get the plus minus symbol)

Think about it .. how did you get $-2^2 = 4$ ??

(exercise: try graphing $\sqrt{x^2}$ and see what you get. it will be the absolute value function for x NOT the line y = x as you are claiming)
err.. graphing $\sqrt{x^2}$ would give the two lines y = x and y = -x

also, i never said anything about $(\sqrt{x})^2$, so i do not see why you are arguing with me about it.
I'm merely offering an alternative where it DOES have to equal |x|

11. Originally Posted by kwah
I'm really confused where to begin with replying to this ..

first off, i don't agree that $\sqrt{x^2} = |x|$
regardless of the value of $x$, positive or negative, it results in $\sqrt{positive number}$, yeah?

roots can have a positive OR negative value, therefore $\sqrt{x^2}$ does not necessarily equate to $|x|$
i thought i might respond to this specifically.

$\sqrt{\mbox{positive number}} = \mbox{a positive number}$ ALWAYS

it is never true that $\sqrt{\mbox{positive number}} = \mbox{negative number}$

i think i see where your misunderstanding comes from

for instance, if you are solving $x^2 = 1$ the solution is $x = \pm 1$ (remark here: how did i get +/- 1? if you take sqrt(x^2) to be |x| it makes sense! if we use say sqrt(x^2) = x, then the only solution would be x = 1, which we know is false)

however, that is not the same as saying $\sqrt{1} = \pm 1$ which is in fact what you are claiming by saying the root of a positive number can be negative or positive.

do you understand what i am saying, or is there something else you're not clear on?

12. Originally Posted by kwah
jeez lol ..

$\sqrt{(-2)^2} = \sqrt{4} = 2 \ne x$ is NOT correct ...
$\sqrt{(-2)^2} = \sqrt{4} = 2 OR -2$ (im not sure how to get the plus minus symbol)

Think about it .. how did you get $-2^2 = 4$ ??
i never said -2^2 = 4, i said (-2)^2 = 4. that is correct

-2*-2 = +4

also, what you have is not correct. the square root function does not return negative values! look at the graph of square root x, the range is y greater than or equal to zero

err.. graphing $\sqrt{x^2}$ would give the two lines y = x and y = -x
no it will not. it will give the graph for |x|

I'm merely offering an alternative where it DOES have to equal |x|
you are mistaken my friend

13. Originally Posted by Jhevon
i thought i might respond to this specifically.

$\sqrt{\mbox{positive number}} = \mbox{a positive number}$ ALWAYS

it is never true that $\sqrt{\mbox{positive number}} = \mbox{negative number}$

i think i see where your misunderstanding comes from

for instance, if you are solving $x^2 = 1$ the solution is $x = \pm 1$ (remark here: how did i get +/- 1? if you take sqrt(x^2) to be |x| it makes sense! if we use say sqrt(x^2) = x, then the only solution would be x = 1, which we know is false)

however, that is not the same as saying $\sqrt{1} = \pm 1$ which is in fact what you are claiming by saying the root of a positive number can be negative or positive.

do you understand what i am saying, or is there something else you're not clear on?
it is not me that is misunderstanding ..

use this numerical example ..

$(x)^2 = (-x)^2 = xx$
$(10)^2 = (-10)^2 = 100$

now root everything

$((10)^2)^{0.5} OR ((-10)^2)^{0.5} = \sqrt{100}$
$(10)^{2*0.5} OR (-10)^{2*0.5} = \sqrt{100}$
$(10)^1 OR (-10)^1 = \sqrt{100}$
$(10) OR (-10) = \sqrt{100}$
$\sqrt{100} = (10) OR (-10)$ ///// EDIT: Just added this line in so that if can be directly compared to its algebraic equivalent contained in the "therefore"

therefore

$\sqrt{(x)^2} = (x)$ or $(-x)$

EDIT: included the brackets to make it absolutely clear what I am intending here ..
and by "OR", i mean that either value is true - i do not know the plus OR minus symbol (a + sign with a - placed on top or below it)

Is this step-by-step proof clear enough?

regards
kwah =]

14. Originally Posted by kwah
it is not me that is misunderstanding ..

use this numerical example ..

$x^2 = -x^2 = xx$
$10^2 = -10^2 = 100$
now root everything
$(10^2)^0.5 OR (-10^2)^0.5 = \sqrt{100}$
$10^(2*0.5) OR -10^(2*0.5) = \sqrt{100}$
$10^1 OR -10^1 = \sqrt{100}$
$10 OR -10 = \sqrt{100}$
therefore
$\sqrt{x^2} = x or -x$

Is this step-by-step proof clear enough?

regards
kwah =]
first of all $x^2 \ne -x^2$ so nothing you said after that is correct (in the sense that the proof started with an incorrect premise). you started with an incorrect assumption. if we take f(x) = x^2, then -x^2 = - f(x) not f(x)

it is true that $x^2 = (-x)^2$ which is completely different from what you are saying

it is funny though, you last statement agrees with me. |x| returns x or minus x depending on the sign of x

15. Originally Posted by Jhevon
i never said -2^2 = 4, i said (-2)^2 = 4. that is correct

-2*-2 = +4
yeah .. that is what i meant .. for (-2) to be squared..

also, what you have is not correct. the square root function does not return negative values! look at the graph of square root x, the range is y greater than or equal to zero
HAVING THE RANGE GREATER THAN OR EQUAL TO ZERO IS LIMITING TO NOT SHOW ANY VALUES LESS THAN ZERO WHICH IS WHY YOU ARE ONLY SEEING ONE VALUE, DUH!!

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