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Rahul Kumar

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MAXIMA AND MINIMA , LOCAL MAXIMUM/ MINIMUM, GLOBAL MAXIMUM/ MINIMUM

MAXIMA AND MINIMA

LOCAL MAXIMA AND MINIMA:

      let y = f(x) be a function defined at x = a and also in the vicinity of the point x = a, then f(x) is said to have a local maximum at x =a if the value of the function at x = a is greater than the value of the function at the neighboring point of x = a.

Similarly, f(x) is said to have a local minimum at x = a, if the value of the function at x = a is less than the value of the function at the neighboring points of x = a.

A TEST FOR LOCAL MAXIMUM/ MINIMUM

Test for local maximum/minimum at x= a if f(x) is differentiable at x = a.

 

Let f(x) is differentiable at x= a and it is a critical point of function f' (a) =0 and if f '(x) function changes its sign while passing through the point x = a then,

  • f(x) would have a local maximum at x = a if f'(a - 0) > O and f’(a + 0) <0. It means that f'(x) should change its sign from positive to negative.

 

  • f(x) would have a local minimum at x = a  if f'(a - 0) <O and f'(a + 0) > 0. It means that f'(x) should change its sign from negative to positive.

 

  • If fix) doesn’t change its sign while passing through x = a then f(x) would have neither a maximum nor minimum at point x = a.

 

Second-order derivative test for maxima and minima

Let f(x) be a differentiable function on an interval I. let

and f''(x) is continuous at x = a then 

  • x = a is a point of local maximum if f'(a) = 0 and f''(a) < 0.

 

  • x = a is a point of local minimum if f'(a) = 0 and f''(a) > 0.

 

  • if f'(a) = f''(a) and f'''(a) =! 0 if exists  then x = a  is neither a point of local maximum nor a point of local minimum.

Test for local ma ximum/minimum at x= a if f(x) is not differentiable at x = a:

When f(x) is continuous at x = a and f'(a-h) and f'(a + h) exist and are non-zero, then f(x) has a local maximum or minimum at x= a if  f'(a-h) and f'(a + h) are of opposite signs.

When f(x) is continuous and f‘(a-h) and f(a + h) exist but one of them is zero, we should compare the information about the existence of local maximum/minimum from the basic definition of local maximum/minimum.

If f (x) is not continuous at x= a and f'(a—h) and/or f'(a + h) are not finite  then compare the values of f(x) at the neighbouring points of x=a.

 

Concept of Global maximum and minimum

let y = f(x) be a given function with Domain D.

Global maximum/minimum of f(x) in [a, b] is basically the greatest/least value of f(x) in [a, b]. Global maximum/minimum in [a, b] would always occure at critical points of f(x) with in [a, b] or at the end points of the interval.

Rolle’s Theorem

It states that if y = f(x) be a given function and satisfies the following conditions.

  • f(x) be continuous in [a,b]

 

  • f(x) be differentiable in (a, b)

 

  • f(a) = f(b) then f'(c) =0 at least once for some c E (a, b)

Lagrange’s Mean Value Theorem

It states that if if y = f(x) be a given function and satisfies the following conditions

  • f(x) be continuous in [a,b]

 

  • f(x) be differentiable in (a, b)

then 

Virus , STRUCTURE OF VIRUSES, Component of virus

VIRUS

A virus is a small infectious agent that typically consists of a segment of nucleic acid with a protein or lipoprotein coat. Viruses replicate only inside the living cells of other organisms as they require host resources for their replication. They can infect all types of life forms, from animals and plants to microorganisms, including bacteria and archaea.

STRUCTURE OF VIRUSES

Viruses usually range between 10-400 nm in size while few are exceptionally large. They can be observed only by electron microscopy and X-ray crystallography. A fully assembled infectious virus is called a virion. The simplest virions consist of two basic components - a core portion containing nucleic acid (either DNA or RNA) and a protein coat called the capsid. A capsid consists of numerous subunits called capsomeres. Viruses are grouped on the basis of size and shape, chemical composition, type of genome and mode of replication.

  1. Helical or Cylindrical Symmetry

Helical morphology is seen in many filamentous and pleomorphic viruses. The capsomeres and nucleic acids are wined together to form a helical or spiral tube-like structure. The rod-shaped helical capsid of Capsid these viruses consists of numerous identical capsomeres wrapped around the helical filament, e.g., Tobacco Mosaic Virus.

 

Helical or Cylindrical Symmetry

2.  icosahedral Symmetry

Many viruses appear as spherical, cuboidal or polygonal in shape which is actually icosahedral. An icosahedron is a polyhedron that has 20 equilateral triangular faces and 12 vertices. An icosahedral capsid comprises of both pentamers (pentagonal capsomeres at the vertices) and hexamers (hexagonal capsomeres at the vertices). The icosahedral capsid is the most efficient way to enclose a space. The total number of capsomeres of different icosahedral viruses varies greatly, e.g., Turnip Yellow Mosaic Virus (TYMV) has 32 capsomeres, Papilloma Virus has 72 capsomeres, Adenovirus have 252 capsomeres, etc.

icosahedral Symmetry

3.Complex Symmetry Head

Although most of the viruses have either icosahedral or helical capsids, many viruses do not fit into either of the two categories, they combine both polygonal and filamentous shapes. E.g., Poxviruses, Bacteriophages. T4, bacteriophage comprises of the polygonal head which contains DNA genome and rod-shaped tail of long fibers.

 

Complex Symmetry Head

 

Components of Virus


Four components of viruses are


(a) Nucleoid: Viral genome is made of a single molecule of nucleic acid. It may be linear or circular with
various degrees of coiling. The nucleic acid is the infective part of the virus. The nucleic acid is either DNA or RNA but never both. Viruses have all four possible nucleic acid types - single-stranded DNA, double-stranded DNA, single-stranded RNA, and double-stranded RNA.
(i) Double-stranded or dsDNA occurs in T, , Tg bacteriophages, Coliphage Lambda, Pox Virus, Adenovirus, etc.


(ii) Single-stranded or ssDNA occurs in Coliphage @ x 174, Coliphage fd, etc. The single strand of DNA is called a plus strand. A complementary or negative DNA strand is synthesized to produce DNA duplex for replication.


(iii) Double-stranded or dsRNA is found in Reovirus and Tumour Virus. Both are linear type.


(iv) Single-stranded or ssRNA is more common in riboviruses. The single stranded RNA is generally linear, @.9., Poliomyelitis virus, influenza virus, etc, Retroviruses have two copies of ssRNA, @g., HIV, Rous Sarcoma Virus of the mouse.


(b) Capsid (Sheath, Coat): It is the proteinaceous covering around the virus which protects the nucleoid from damage by physical and chemical agents. The capsid consists of a number of subunits called capsomeres or capsomers. The capsid of TMV has 2130 capsomeres. In binal bacteriophages, the capsid sheath of the tail is contractile.


(c) Envelope: Some animal viruses, a few plant, and bacterial viruses are bounded by an outer loose membranous layer called an envelope. In contrast to enveloped viruses, the viruses without an envelope are called naked. The envelope consists of proteins (from virus), lipids and carbohydrates (from the host). It has subunits called telomeres or peplomers. The surface of the envelope can be smooth or have outgrowths called spikes. Common enveloped viruses are HIV, Herpes Virus, Vaccinia Virus, etc.


(d) Enzymes: They are occasional. Enzyme lysozyme is present in the region that comes in contact with host cell in bacteriophages. Other enzymes are neuraminidase (in Influenza Virus), RNA polymerase, RNA transcriptase, reverse transcriptase, etc In some cases, enzymes are associated with the envelope or capsid but most viral enzymes are located within the capsid.

Joint Entrance Examination (Main) - 2020

IIT Entrance Test -2018

The entrance examination for the Indian Institutes of Technology (IITs) will go completely online from 2018 to make logistics and evaluations easier, the Joint Admission Board (JAB) decided on Sunday.

The JAB, which is the policy-making body on IIT admissions, took the decision at a meeting in Chennai.

 

In the online mode, students will take up the examination at designated centres where they need to answer the questions on a computer instead of using pen and paper.

At present optical mark reading (OMR) sheets are used which need to be filled using pen or pencils and are evaluated by machines.

In a statement, Director, IIT-Madras, and Chairman JAB 2017, Prof Bhaskar Ramamurthi said, "It has been decided that the JEE (Advanced) will be conducted in online mode from 2018 onwards. Further information regarding the examination will be provided by the JAB in due course."

 

 

The Ministry of Human Resource Development had earlier introduced the option of taking the Joint Entrance Examination (JEE) Mains online.

The JEE-Mains is the entrance examination for admission to engineering courses offered across the country and a qualifying exam for JEE-Advanced which is required for admission to the IITs and NITs.

 


"In order to make logistics and evaluations easier it was decided today that the JEE-Advanced should be made online," a JAB member said.

"The concept was being discussed for many years, but it was necessary to have adequate infrastructure to conduct the exam online," the member added.

More than 13 lakh students took the JEE-Mains this year, with less than 10 per cent of them going online. Around 2.2 lakh students were eligible to write the JEE (Advanced). 

Source : http://www.ndtv.com/india-news/iit-entrance-exam-to-go-completely-online-from-2018-official-1739846

Derivatives (Differential Calculus)

The Derivative is the "rate of change" or slope of a function.

What is Organic Chemistry

Organic chemistry is the study of the structure, properties, composition, reactions, and preparation of carbon-containing compounds, which include not only hydrocarbons but also compounds with any number of other elements, including hydrogen (most compounds contain at least one carbon–hydrogen bond), nitrogen, oxygen, halogens, phosphorus, silicon, and sulfur. This branch of chemistry was originally limited to compounds produced by living organisms but has been broadened to include human-made substances such as plastics. The range of application of organic compounds is enormous and also includes, but is not limited to, pharmaceuticals, petrochemicals, food, explosives, paints, and cosmetics.

What is Cuboid?

In geometry, a cuboid is a convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube. While mathematical literature refers to any such polyhedron as a cuboid,[1] other sources use "cuboid" to refer to a shape of this type in which each of the faces is a rectangle (and so each pair of adjacent faces meets in a right angle); this more restrictive type of cuboid is also known as a rectangular cuboidright cuboidrectangular boxrectangular hexahedronright rectangular prism, or rectangular parallelepiped.

 

 

General cuboids

By Euler's formula the numbers of faces F, of vertices V, and of edges E of any convex polyhedron are related by the formula F + V = E + 2. In the case of a cuboid this gives 6 + 8  = 12 + 2; that is, like a cube, a cuboid has 6 faces, 8 vertices, and 12 edges.

Along with the rectangular cuboids, any parallelepiped is a cuboid of this type, as is a square frustum (the shape formed by truncation of the apex of a square pyramid).

 

 

Rectangular cuboid

Rectangular cuboid

 

In a rectangular cuboid, all angles are right angles, and opposite faces of a cuboid are equal. By definition this makes it a right rectangular prism, and the terms rectangular parallelepiped or orthogonal parallelepiped are also used to designate this polyhedron. The terms "rectangular prism" and "oblong prism", however, are ambiguous, since they do not specify all angles.

The square cuboidsquare box, or right square prism (also ambiguously called square prism) is a special case of the cuboid in which at least two faces are squares. It has Schläfli symbol {4} × { }, and its symmetry is doubled from [2,2] to [4,2], order 16.

The cube is a special case of the square cuboid in which all six faces are squares. It has Schläfli symbol {4,3}, and its symmetry is raised from [2,2], to [4,3], order 48.

If the dimensions of a rectangular cuboid are ab and c, then its volume is abc and its surface area is 2(ab + ac + bc).

The length of the space diagonal is

 

Cuboid shapes are often used for boxescupboardsrooms, buildings, etc. Cuboids are among those solids that can tessellate 3-dimensional space. The shape is fairly versatile in being able to contain multiple smaller cuboids, e.g. sugar cubes in a box, boxes in a cupboard, cupboards in a room, and rooms in a building.

A cuboid with integer edges as well as integer face diagonals is called an Euler brick, for example with sides 44, 117 and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists.Rectangle Cuboid

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