CBSE NCERT Homework Help: Basic Integration:

MATHS BLOG 1:

Basic Integration:

A. An integral is basically the exact opposite of a derivative.

B. An integral gives the area under a curve from the x-axis to the curve from a specified range. An integral is expressed by the symbol: .

C. Below are some basic rules for computing integrals:

1. The integral of a constant, c, is cx.

2. The integral of xn is .

3. The integral of a*f(x), where a is a constant, is equal to a times the integral of f(x). (*Basically, you can factor out the constant*)

4. The integral of f(x) g(x) = .

a. In this problem, we can take each term separate.

b. The integral of 3x2 = x3; of 2x = x2; of 1 = x.

c. The answer is x3+x2+x+C

d. We usually add a C on the end because the integral of 0 is a constant.

D. You should notice, that if you take the derivative of the new function, you will end up with the original function.

E. Common Integrals:

1. The integral of sin(ax) = -1/a cos(ax)

2. The integral of cos(ax) = 1/a sin(ax)

3. The integral of sec2 x = tan (x)

4. The integral of e(ax) = 1/a e(ax).

5. The integral of 1/ax = 1/a ln(ax)

F. These are just some of the common integrals. There are lots of integrals in many different forms, which can be found on the front or back cover of most calculus books.

G. When calculating an integral in a specified range:

a. To compute this integral take each term separate and get: x3/3 - x2.

b. To find the value from 0 to 3, first plug in 3. Doing so gives: 33/3 - 32 = 9 - 9 = 0.

c. Next find the value for 0. This gives: 03/3 - 02 = 0.

d. The answer is 0 - 0 = 0.

a. The integral of cos 4x is sin 4x/4.

b. To find the value from 0 to we must first use . Doing so gives: sin 3(/2)/3 = sin(3/2)/3 = -1/3.

c. Next, find the value for 0. This is sin 0/4 = 0.

d. The answer is -1/3 - 0 = -1/3.

a. The integral of x2 - 2 is x3/3 - 2x.

b. Plug in 4 and get 64/3 - 8 = 40/3.

c. Plug in 1 and get 1/3 - 2 = -5/3.

d. The answer is 40/3 - (-5/3) = 45/3 = 15.

MATHS BLOG 2:

Derivatives:

A. A derivative is calculated the exact opposite to that of an integral.

B. A function's derivative is basically the equation for the slope of the original function. Derivatives are usually expressed by: f '(x) or y' or dy/dx. f '(x) or y' is the first derivative. f ''(x) or y'' is the second derivative and so on.

C. Below are the basic rules for computing derivatives.

1. The derivative of a constant is 0.

Ex [1] dy/dx 5 = 0.

2. The derivative of xn is n*xn-1.

Ex [2] dy/dx 3x2 = 6x.

3. The derivative of f(x) g(x) = f '(x) g'(x)

Ex [3] dy/dx 6x3 + 4x2 + 8x - 4 = _________

a. This rule means you can take each term separately.

b. So this becomes dy/dx 6x3 + dy/dx 4x2 + dy/dx 8x - dy/dx 4 = 18x2 + 8x + 8 - 0.

c. The answer is 18x2 + 8x + 8.

4. The derivative of [f(x)]n is f '(x)*[f(x)]n-1.

Ex [4] dy/dx (3x2+5x+2)3 = ________

a. You always want to work from the inside out.

b. The first step is to take the derivative of the inside first. So dy/dx 3x2 + 5x + 2 = 6x + 5. This represents f '(x).

c. Now, we need the derivative of the outside which is 3(3x2+5x+2)2.

d. Now, multiplying these two values together gives:

(6x+5)*3*(3x2+5x+2)2 or (18x+15)(3x2+5x+2)2.

5. The derivative of f(x)*g(x) = f '(x)g(x)+f(x)g'(x).

a. This type of problem will probably not be found on a number sense test.

Ex [5] dy/dx (3x-4)(x2-3) = _________

a. First, multiply the derivative of the first times the second. So we get: dy/dx 3x - 4 = 3. So 3(x2-3) is the first term.

b. Next, multiply the first term times the derivative of the second. So we get: dy/dx x2 - 3 = 2x. So we get 2x(3x-4) for the second term.

c. The answer is 3(x2-3)+2x(3x-4).

D. Common Derivatives:

1. The derivative of sin(f(x)) = f '(x)*cos(f(x)).

2. The derivative of cos(f(x)) = -f '(x)*sin(f(x)).

3. The derivative of tan(f(x)) = f '(x)*sec2(f(x)).

4. The derivative of ef(x) = f '(x)*ef(x).

5. The derivative of ln (f(x)) = f '(x)*1/f(x).

E. Examples

Ex [1] If f(x) = sin 4x, then f '() = _______

a. First, take the derivative of 4x which is 4.

b. The derivative of sin x is cos x, so the derivative of sin 4x = 4 cos 4x.

c. Plugging in x = , we get 4 cos = 4(-1) = -4.

d. The answer is -4.

Ex [2] Find the slope of the tangent line of y = (x-2)3 at the point x = 4.

a. Since a derivative is the slope of the tangent line, we need to find y' and use the value x = 4 to find the slope.

b. y' = 3(x-2)2, so plugging in x=4 we get 3(22) = 12.

c. The answer is 12.

Ex [3] If f(x) = 4x3 - 12x2 + 4x - 3, then f ''(2) = ______

a. For this problem we are looking for the second derivative. So we need to find the derivative of the derivative, or just take the derivative twice.

b. f '(x) = 12x2 - 24x + 4

c. So f ''(x) = 24x - 24. f ''(2) = 24(2) - 24 = 24.

d. The answer is 24.

SCIENCE BLOG 1:

Let us start with the question "What is Organic chemistry?".

The simple answer is: It is the chemistry of carbon containing compounds, which are otherwise known as organic compounds.

So it is pretty easy to recognize that we should start our journey of organic chemistry by exploring the chemical nature of carbon.

So the next question is: What is carbon?

* Carbon is an element with atomic number (Z) = 6.

* Its ground state electronic configuration can be represented as: 1s22s22p2 (or) 1s22s22px12py12pz0

electronic configuration of carbon atom

* It is the first element in Group-14 of Long form of Periodic table.

* It is a non metal.

* On Pauling's scale, its electronegativity value is around 2.5.

* It usually forms covalent bonds.

* Its valency is 4 since there are four electrons in the outer shell i.e., it can form four covalent bonds with other atoms.

simple representation of carbon atom

WHY THERE ARE MILLIONS OF ORGANIC COMPOUNDS?

It is well known that there are millions of organic compounds around, which are either originated from the nature or prepared synthetically.

Examples of organic compounds include carbohydrates, proteins, enzymes, vitamins, lipids, nucleic acids , synthetic polymers, synthetic fabrics, synthetic rubbers, plastics, medicines, drugs, organic dyes and so on.

Now the immediate question is: Why the carbon atom is so special and forms millions of compounds?

To answer this question, we should know about catenation.

Catenation is the ability of atoms of same element to bond covalently among themselves and form long chains or rings.

Carbon has a stronger tendency to catenate since it is a smaller atom and can form stronger covalent bonds with other carbons. The C-C bonds are stronger due to effective overlapping of atomic orbitals.

catenation of carbon to from chains & rings of differenct length

It also forms stronger bonds with other elements like hydrogen, oxygen, nitrogen, halogens, sulfur, phosphorus etc.

The organic compounds can also exhibit isomerism due to different structural and spatial arrangement of atoms or groups leading to formation of huge array of compounds.

These arguments explain why the carbon can form millions of compounds and organic chemistry is flourishing like nothing.

HOW DOES CARBON FORM CHEMICAL BONDS?

LEWI'S DOT MODEL

Carbon is an appreciably electronegative element and tends to form four covalent bonds by using all the four electrons in its valence shell i.e., the second shell for which the electronic configuration can be written as 2s22p2. Hence the combining power or the valency of carbon is 4. It can form 4 bonds with other atoms.

It is possible to understand the bonding in carbon compounds by using Lewi's dot model. According to this model, each atom participating in the bonding contributes one electron to form an electron pair which is shared between the two contributing atoms. Thus a covalent bond is formed. If atoms share two electron pairs, a double bond is formed. And a triple bond is formed when three electron pairs are shared.

The purpose of participating in bond formation is to get the nearest inert gas configuration and thus by getting stability. Most of the atoms try to get eight electrons or octet configuration in the valence shell. This is also called as octet rule.

The structures of some simple organic molecules are explained as shown below.

Methane, CH4: The carbon atom contributes four valence electrons to make four bonds with hydrogen atoms. Each hydrogen also contributes one electron for the bond formation.

Thus there are 4 C-H bonds in the methane molecule and carbon gets octet configuration in the valence shell.

Note that the valency of hydrogen atom is one. It can form only one bond since there is only one electron in this atom. It also gets Helium's configuration during bond formation.

Also note that in Lewi dot models, only the valence electrons are shown. The bond pairs can also be shown by lines.

Lewi dot model of methane molecule

Ethane, C2H6: In Ethane molecule each carbon forms 4 bonds again. Among them three are C-H bonds, while the fourth one is a C-C bond.

Lewi dot model of ethane molecule

Ethylene, C2H4: In this molecule, there is a double bond between two carbon atoms due to sharing of two pairs of electrons. Each carbon also forms two bonds with hydrogen atoms.

Lewis dot model of ethylene molecule

Acetylene, C2H2: There is a triple bond between two carbon atoms in acetylene molecule. It is formed due to sharing of three electron pairs. Each carbon also forms a single bond with hydrogen atom.

lewis dot model for acetylene molecule

Methyl fluoride, CH3F: Since there are 7 electrons in the valence shell of Fluorine, it require one electron to complete octet. Hence it contributes one electron for bond formation with carbon as shown below.

Note that the only the bond pairs are shown as lines in the second representation.

There are three lone pairs and one bond pair around fluorine atom. The bond pair is shown as a line.

lewis dot model for methyl fluoride molecule

In the same way, carbon atom forms bonds with other halogen atoms.

Formaldehyde, CH2O: There are six electrons in the valence shell of oxygen. It forms two bonds by contributing two of its valence electrons and thus by completing the octet. In formaldehyde, the oxygen atom forms two bonds with a carbon atom.

lewis dot structure of formaldehyde

Methyl alcohol, CH3OH: However, the oxygen atom can also form just one bond with the carbon as in case of methyl alcohol. It forms the second bond with hydrogen.

lewis dot structure of methyl alcohol

Hydrogen cyanide, HCN: The carbon atom can also form bonds with nitrogen. In hydrogen cyanide there is a triple bond between carbon and nitrogen. The nitrogen atom contributes three of its valence electrons for the formation of this triple bond.

lewis dot structure of hydrogen cyanide

Methyl amine, CH3NH2: In this molecule, the carbon and nitrogen atoms are sharing only one pair of electrons.

lewi dot model for methyl amine

However this model could not explain the exact geometry of organic molecules. For example, methane molecule is tetrahedral, whereas ethylene is a planar molecule. These structures with exact bond angles cannot be explained by this model. Therefore it is necessary to understand the structures of these molecules by using valence bond theory as explained in the next section.

VALENCE BOND THEORY

According to valence bond theory, four unpaired electrons are required to form four covalent bonds. But there are only 2 unpaired electrons in the valence shell of carbon in the ground state.

However it is possible to get 4 unpaired electrons by transferring one of the electrons from 2s orbital into the empty 2p orbital. This process is called excitation and carbon is said to be in the excited state. Now the electronic configuration of carbon in the excited state becomes 2s12p3.

A small amount of energy, which is available during the chemical bond formation, is sufficient for a carbon atom to undergo excitation.

excitation of carbon atom

It is now possible for carbon atom to form 4 bonds in the excited state.

However, carbon undergoes hybridization before forming actual chemical bonds with other atoms.

Hybridization is the process of intermixing of two or more pure atomic orbitals of almost same energy to form same number of identical and degenerate new orbitals known as hybrid orbitals.

The carbon atom can undergo three types of hybridizations i.e., sp3 or sp2 or sp.

SP3 HYBRIDIZATION OF CARBON

In sp3 hybridization, one 2s and three 2p orbitals of excited carbon intermix together and form 4 hybrid orbitals which are oriented in tetrahedral geometry in space around the carbon atom. Each sp3hybrid orbital is occupied by one electron.

sp3 hybridization of carbon

Each of these sp3 orbitals can form σ-bond with other atom. Thus carbon is forming four single bonds with other atoms in tetrahedral geometry. The bond angles are usually equal to or nearer to 109o28'.

E.g. In methane molecule, CH4, the carbon atom undergoes sp3 hybridization and forms four σ-bonds with hydrogen atoms.

Note that whenever carbon atom undergoes sp3 hybridization, it forms 4 σ-bonds i.e., 4 single bonds.

SP2 HYBRIDIZATION OF CARBON

In sp2 hybridization, there is intermixing of one 2s and two of the 2p orbitals of carbon in the excited state to form three hybrid orbitals. These are oriented in trigonal planar geometry. Each sp2 hybrid orbital is occupied by one electron. The remaining pure 2p orbital with one electron lies at right angle to the plane of hybrid orbitals.

sp2 hybridization of carbon atom

The sp2 hybrid orbital form 3 σ-bonds in trigonal planar geometry. Thus the bond angles are about 120o. The remaining pure 'p' orbital will form a π-bond. Thus carbon forms total four bonds i.e., three σ-bonds and one π-bond.

E.g. In ethylene molecule, C2H4, each carbon atom undergoes sp2 hybridization. Each carbon forms 2 σ-bonds with hydrogens and one σ-bond with another carbon. The remaining pure 'p' orbitals on two carbons overlap sidewise to form a π-bond. Thus there is a double bond between two carbons.

structure of ethylene molecule

Note that whenever carbon atom undergoes sp2 hybridization, it forms 3 σ-bonds and 1 π-bond i.e., two single bonds and one double bond.

SP HYBRIDIZATION OF CARBON

In sp hybridization, one 2s and one 2p orbitals of excited carbon intermix to form two sp-hybrid orbitals in linear geometry. Each sp hybrid orbital is occupied by one electron. The remaining pure 2p orbitals ( for our convenience, let us say: 2py and 2pz) orient at right angles to the sp-hybrid orbitals. These are also occupied by one electron each.

sp hybridization in carbon atom

The two sp hybrid orbitals form 2 σ-bonds in linear geometry. Thus the bond angle will be about 180o. The remaining pure 'p' orbitals will form two π-bonds. Thus carbon again forms total four bonds i.e., two σ-bonds and two π-bonds.

E.g., In acetylene molecule, C2H2, each carbon undergoes sp hybridization and forms one σ-bond with a hydrogen atom and one σ-bond with another carbon. The two carbon atoms also form two π-bonds with each other due to sidewise overlapping of pure p-orbitals. Thus a triple bond is formed between two carbon atoms in acetylene molecule.

structure of acetylene

Note that whenever carbon atom undergoes sp hybridization, it forms 2 σ-bonds and 2 π-bonds. It may either form one triple bond as in case of acetylene or two double bonds e.g. allenes.

The following diagram summarizes the bonding pattern by carbon atom in different hybridizations.

Hybridization sp3 hybridized carbon sp2 hybridized carbon sp hybridized carbon
Geometry
sp3 carbon

sp2 carbon

sp carbon

Bond angle 109o28' 120o 180o
Type of bonds 4 single bonds

i.e., 4 σ bonds 2 single & 1 double bonds

i.e., 3 σ bonds & 1 π-bond 1 triple & 1 single bonds
or
2 double bonds

i.e., 2 σ bonds & 2 π-bonds

SCIENCE BLOG 2:

Inorganic chemistry deals with the chemistry of all the elements, in the periodic table, and their compounds except that of carbon. It is a broad and complex field which overlaps with other branches of science and is growing at a rapid pace.

The study of inorganic chemistry includes the extraction or synthesis, chemical properties and applications of metals, metalloids and non metals as well as their compounds. It involves interpreting, correlating, and predicting the properties and structures of an enormous range of materials. We some time lend the physical principles from other areas of science to understand the structure, characterization and properties of inorganic substances.

The word "inorganic" means non-living. However the study of inorganic chemistry encompasses the role of inorganic compounds both in material and biological sciences. Some of the important branches of modern inorganic chemistry are: coordination chemistry, bioinorganic chemistry, organometallics etc.,

You can find various topics from inorganic chemistry from the list given below.

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