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 Discussion Forums -> This Post 5 points    (1    in 1 votes )   [?]

f(x) = 6^ x+3^x+6^-x+3^-x + 2

under the limiting case when x tends to + infinity or - infinity f(x) tends to infinity

When x tends to ZERO then f(x) tensds to 6

Thus range is {6 to infinity)

 Discussion Forums -> This Post 5 points    (1    in 1 votes )   [?]

he moment of the inertia force on a single particle around an axis multiplies the mass of the particle by the square of its distance to the axis, and forms a parameter called the moment of inertia. The moments of inertia of individual particles in a body sum to define the moment of inertia of the body rotating about an axis. For rigid bodies moving in a plane, such as a compound pendulum, the moment of inertia is a scalar, but for movement in three dimensions, such as a spinning top, the moment of inertia becomes a matrix, also called a tensor.

 Discussion Forums -> This Post 10 points    (2    in 2 votes )   [?]

Sunlight interacting with the Earth's atmosphere makes the sky blue. In outer space the astronauts see blackness because outer space has no atmosphere.
Sunlight consists of light waves of varying wavelengths, each of which is seen as a different color. The minute particles of matter and molecules of air in the atmosphere intercept and scatter the white light of the sun. A larger portion of the blue color in white light is scattered, more so than any other color because the blue wavelengths are the shortest.

When the size of atmospheric particles are smaller than the wavelengths of the colors, selective scattering occurs-the particles only scatter one color and the atmosphere will appear to be that color. Blue wavelengths especially are affected, bouncing off the air particles to become visible.

This is why the sun looks yellow from Earth (yellow equals white minus blue). In space, the sun appears white because there is nothing in between to scatter its white light.

At sunset, the sky changes color because as the sun drops to the horizon, sunlight has more atmosphere to pass through and loses more of its blue wavelengths. The orange and red, having the longer wavelengths and making up more of sunlight at this distance, are most likely to be scattered by the air particles.

The scattering of visible light by atmospheric gases is most correctly called the Tyndall effect, but it is more commonly known to physicists as Rayleigh scattering after Lord Rayleigh, who studied it in more detail a few years later. Rayleigh Scattering is where red, orange, yellow, and green are passed through and blue, indigo, and violet are "scattered" out creating the color.

Whichever direction you look, some of this scattered blue light reaches you. Since you see the blue light from everywhere overhead, the sky looks blue.

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

Characteristic and Mantissa:

Consider a number N > 0.

Then, let the value of log 10 N consist of two parts:

One an integral part, the other – a proper fraction

The integral part is called the Characteristic and the fractional or the decimal part is called the Mantissa.

Now log 10 (379) = 2.578

So characteristic = 2

and mantissa = 0.578

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 Discussion Forums -> This Post 5 points    (1    in 1 votes )   [?]

In a cyclotron, the most important condition is that of the cyclotron frequency. The frequency of the square wave oscillator connected to the dees of the cyclotron must match the frequency of revolution of the charged particle being accelerated.

For ordinary ions, once the  frequency is set there is no need to change or adjust the frequency.

The equation for cyclotron frequency is

As it is clear from the above equation that the cyclotron frequency is inversely proportional to mass of the ion. The frequency of revolution is apparently constant for ordinary ions.

If an electron is accelerated in a cyclotron, it quickly picks up high-speed comparable to the speed of light because of its light mass. The speed comparable to the speed of light is called relativistic speed. At relativistic speeds, mass is not constant but varies according to the relation.

As per the equation as speed increases, relativistic mass increases. This will change the frequency of revolution and the revolution will go out of phase. The acceleration will stop.

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

VOLTAGE LOST ACROSS 0.5 OHM RESISTANCE = I R = 3 * 0.5 = 1.5V

THEREFORE,

PD ACROSS TERMINALS IS GIVEN BY

V = 2 + 1.5 = 3.5V

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

Displacement is a vector quantity by deifnition thus 4m displacement of a body is a vector quantity.

It means that the shortest distance from the intial position of an object is 4m.

But it does not mean that it is 4i as it can be 4j or 4k or any other combination of i, j and k basis vectors depending on the initial position of an object.

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

Transition of electron:

When an electron makes transition from higher energy level having energy E2(n2) to a lower energy level having energy E1 (n1) then a photon of frequency n is emitted

(ii) Frequency of emitted radiation :

(iii) Wave number/wavelength : Wave number is the number of waves in unit length

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

Noble gases combine with their own monotomic atoms to make gases. The electron affinity has to be greater than zero , otherwise the monotomic atoms wouldn't react at all. For instance Xenon atoms react with other xenon atoms to make xenon gas, Xe2. If it had no electron affinity , the gas would be impossible to covalently bond to itself. Noble gases have very low electron affinity, which is why they don't react with other substances. With the low affinity the noble gas compounds are easily dislocated which is why the noble gases are used in ionozing gases for lamps ,i.e neon and xenon bulbs

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

It is not exactly zero rather it is slightly greater than zero.

Noble gases combine with their own monotomic atoms to make gases. The electron affinity has to be greater than zero , otherwise the monotomic atoms wouldn't react at all. For instance Xenon atoms react with other xenon atoms to make xenon gas, Xe2. If it had no electron affinity , the gas would be impossible to covalently bond to itself.

Noble gases have very low electron affinity, which is why they don't react with other substances. With the low affinity the noble gas compounds are easily dislocated which is why the noble gases are used in ionozing gases for lamps ,i.e neon and xenon bulbs

 Discussion Forums -> This Post 5 points    (1    in 1 votes )   [?]

# Depletion Region

When a p-n junction is formed, some of the free electrons in the n-region diffuse across the junction and combine with holes to form negative ions. In so doing they leave behind positive ions at the donor impurity sites.

# Depletion Region Details

 In the p-type region there are holes from the acceptor impurities and in the n-type region there are extra electrons. When a p-n junction is formed, some of the electrons from the n-region which have reached the conduction band are free to diffuse across the junction and combine with holes. Filling a hole makes a negative ion and leaves behind a positive ion on the n-side. A space charge builds up, creating a depletion region which inhibits any further electron transfer unless it is helped by putting a forward bias on the junction.

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

An electronic component is any physical entity in an electronic system used to affect the electrons or their associated fields in a manner consistent with the intended function of the electronic system. Components are generally intended to be connected together, usually by being soldered to a printed circuit board (PCB), to create an electronic circuit with a particular function (for example an amplifier, radio receiver, or oscillator). Components may be packaged singly, or in more complex groups as integrated circuits. Some common electronic components are capacitors, inductors, resistors, diodes, transistors, etc. Components are often categorized as active (e.g. transistors and thyristors) or passive (e.g. resistors and capacitors).

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

A parabola (plural "parabolas"; Gray 1997, p. 45) is the set of all points in the plane equidistant from a given line  (the conic section directrix) and a given point  not on the line (the focus). Thefocal parameter (i.e., the distance between the directrix and focus) is therefore given by , where  is the distance from the vertex to the directrix or focus. The surface of revolution obtained by rotating a parabola about its axis of symmetry is called a paraboloid.

The parabola was studied by Menaechmus in an attempt to achieve cube duplication. Menaechmus solved the problem by finding the intersection of the two parabolas  and . Euclid wrote about the parabola, and it was given its present name by Apollonius. Pascal considered the parabola as a projection of a circle, and Galileo showed that projectiles falling under uniform gravity follow parabolic paths. Gregory and Newton considered the catacaustic properties of a parabola that bring parallel rays of light to a focus (MacTutor Archive), as illustrated above.

For a parabola opening to the right with vertex at (0, 0), the equation in Cartesian coordinates is

 (1)
 (2)
 (3)
 (4)

The quantity  is known as the latus rectum. If the vertex is at  instead of (0, 0), the equation of the parabola is

 (5)

If the parabola instead opens upwards, its equation is

 (6)

Three points uniquely determine one parabola with directrix parallel to the -axis and one with directrix parallel to the -axis. If these parabolas pass through the three points , and, they are given by equations

 (7)

and

 (8)

In polar coordinates, the equation of a parabola with parameter  and center (0, 0) is given by

 (9)

(left figure). The equivalence with the Cartesian form can be seen by setting up a coordinate system  and plugging in  and  to obtain

 (10)

Expanding and collecting terms,

 (11)

so solving for  gives (?). A set of confocal parabolas is shown in the figure on the right.

In pedal coordinates with the pedal point at the focus, the equation is

 (12)

The parabola can be written parametrically as

 (13) (14)

or

 (15) (16)
 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

Surface area is an extensive property.

As Extensive properties are those that depend on the "extent" of the material ... that is, how much of it there is. Some other examples are Volume, shape, and size.

 Discussion Forums -> This Post 5 points    (1    in 1 votes )   [?]

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How do you obtain 40,000 volts across a sparkplug in an automobile when you have only 12 volts DC to start with? The essential task of firing the sparkplugs to ignite a gasolene-air mixture is carried out by a process which employs Faraday's Law.

The primary winding of the ignition coil is wound with a small number of turns and has a small resistance. Applying the battery to this coil causes a sizable DC current to flow. The secondary coil has a much larger number of turns and therefore acts as a step-up transformer. But instead of operating on AC voltages, this coil is designed to produce a large voltage spike when the current in the primary coil is interrupted. Since the induced secondary voltage is proportional to the rate of change of the magnetic field through it, opening a switch quickly in the primary circuit to drop the current to zero will generate a large voltage in the secondary coil according to Faraday's Law. The large voltage causes a spark across the gap of the sparkplug to ignite the fuel mixture. For many years, this interruption of the primary current was accomplished by mechanically opening a contact called the "points" in a synchronized sequence to send high voltage pulses through a rotary switch called the "distributer" to the sparkplugs. One of the drawbacks of this process was that the interruption of current in the primary coil generated an inductive back-voltage in that coil which tended to cause sparking across the points. The system was improved by placing a sizable capacitor across the contacts so that the voltage surge tended to charge the capacitor rather than cause destructive sparking across the contacts. Using the old name for capacitors, this particular capacitor was called the "condenser".

More modern ignition systems use a transistor switch instead of the points to interrupt the primary current.

The transistor switches are contained in a solid-state Ignition Control Module. Modern coil designs produce voltage pulses up in the neighborhood of 40,000 volts from the interruption of the 12 volt power supplied by the battery.

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

# Lenz's Law

When an emf is generated by a change in magnetic flux according to Faraday's Law, the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the examples below, if the B field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant.

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Asymptotes

Definition of a horizontal asymptote: The line y = y0 is a "horizontal asymptote" of f(x) if and only if f(x) approaches y0 as x approaches + or - .

Definition of a vertical asymptote: The line x = x0 is a "vertical asymptote" of f(x) if and only if f(x) approaches + or -  as x approaches x0 from the left or from the right.on of a slant asymptote: the line y = ax + b is a "slant asymptote" of f(x) if and only if lim (x-->+/-) f(x) = ax + b.

Concavity

Definition of a concave up curve: f(x) is "concave up" at x0 if and only iff '(x) is increasing at x0

Definition of a concave down curve: f(x) is "concave down" at x0 if and only if f '(x) is decreasing at x0

The second derivative test: If f ''(x) exists at x0 and is positive, then f ''(x) is concave up at x0. If f ''(x0) exists and is negative, then f(x) is concave down at x0. If f ''(x) does not exist or is zero, then the test fails.

Critical Points
Definition of a critical point: a critical point on f(x) occurs at x0 if and only if either f '(x0) is zero or the derivative doesn't exist.
Extrema (Maxima and Minima)
Local (Relative) Extrema

Definition of a local maxima: A function f(x) has a local maximum at x0 if and only if there exists some interval I containing x0 such that f(x0) >= f(x) for all x in I.

Definition of a local minima: A function f(x) has a local minimum at x0 if and only if there exists some interval I containing x0 such that f(x0) <= f(x) for all x in I.

Occurrence of local extrema: All local extrema occur at critical points, but not all critical points occur at local extrema.

The first derivative test for local extrema: If f(x) is increasing (f '(x) > 0) for all x in some interval (a, x0] and f(x) is decreasing (f '(x) < 0) for all x in some interval [x0, b), then f(x) has a local maximum at x0. If f(x) is decreasing (f '(x) < 0) for all x in some interval (a, x0] and f(x) is increasing (f '(x) > 0) for all x in some interval [x0, b), then f(x) has a local minimum at x0.

The second derivative test for local extrema: If f '(x0) = 0 and f ''(x0) > 0, then f(x) has a local minimum at x0. If f '(x0) = 0 and f ''(x0) < 0, then f(x) has a local maximum at x0.

Absolute Extrema

Definition of absolute maxima: y0 is the "absolute maximum" of f(x) on I if and only if y0 >= f(x) for all x on I.

Definition of absolute minima: y0 is the "absolute minimum" of f(x) on I if and only if y0 <= f(x) for all x on I.

The extreme value theorem: If f(x) is continuous in a closed interval I, then f(x) has at least one absolute maximum and one absolute minimum in I.

Occurrence of absolute maxima: If f(x) is continuous in a closed interval I, then the absolute maximum of f(x) in I is the maximum value of f(x) on all local maxima and endpoints on I.

Occurrence of absolute minima: If f(x) is continuous in a closed interval I, then the absolute minimum of f(x) in I is the minimum value of f(x) on all local minima and endpoints on I.

Alternate method of finding extrema: If f(x) is continuous in a closed interval I, then the absolute extrema of f(x) in I occur at the critical points and/or at the endpoints of I.
(This is a less specific form of the above.)

Increasing/Decreasing Functions
Definition of an increasing function: A function f(x) is "increasing" at a point x0 if and only if there exists some interval I containing x0 such that f(x0) > f(x) for all x in I to the left of x0 and f(x0) < f(x) for all x in I to the right of x0.

Definition of a decreasing function: A function f(x) is "decreasing" at a point x0 if and only if there exists some interval I containing x0 such that f(x0) < f(x) for all x in I to the left of x0 and f(x0) > f(x) for all x in I to the right of x0.

The first derivative test: If f '(x0) exists and is positive, then f '(x) is increasing at x0. If f '(x) exists and is negative, then f(x) is decreasing at x0. Iff '(x0) does not exist or is zero, then the test tells fails.

Inflection Points
Definition of an inflection point: An inflection point occurs on f(x) at x0 if and only if f(x) has a tangent line at x0 and there exists and interval I containing x0 such that f(x) is concave up on one side of x0 and concave down on the other side.

Reference: http://www.math.com/tables/derivatives/extrema.htm

 Discussion Forums -> This Post 0 points    (0    in 0 votes )   [?]

Since we know the derivative:  ex = ex
we can use the Fundamental Theorem of calculus:
ex dx =   (ex) dx = ex + C
Q.E.D.

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