Mathematics

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Mathematics

Skype Online Tution 

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Mathematics

Skype Online Tution 

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Mathematics

Prof. Rajeev Tripathi 

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Mathematics

Ordering of Decimals

Ordering Decimals

“Could I have a 3.65 and an 0.8, please … ?”
NO, not THAT type of ordering. I mean putting them in order

Ordering Decimals can be tricky. Because often we look at 0.42 and 0.402 and say that 0.402 must be bigger because there are more digits. But no!

We can use this method to see which decimals are bigger:

  • Set up a table with the decimal point in the same place for each number.
  • Put in each number.
  • Fill in the empty squares with zeros.
  • Compare using the first column on the left
  • until one number If the digits are equal move to the next column to the right wins.

If you want ascending order you always pick the smallest first

If you want descending order you always pick the largest first

Example: Put the following decimals in ascending order:

1.506, 1.56, 0.8

In a table they look like this:

Units Decimal
Point
Tenths Hundredths Thousandths
1 . 5 0 6
1 . 5 6
0 . 8

Fill in the empty squares with zeros:

Units Decimal
Point
Tenths Hundredths Thousandths
1 . 5 0 6
1 . 5 6 0
0 . 8 0 0

Compare using the first column (Units)

Two of them are “1”s and the other is a “0”. Ascending order needs smallest first, and so “0” is the winner:

Answer so far: 0.8

Now we can remove 0.8 from the list:

Units Decimal
Point
Tenths Hundredths Thousandths
1 . 5 0 6
1 . 5 6 0
.

Compare the Tenths

Now there are two numbers with the same “Tenths” value of 5, so move along to the “Hundredths” for the tie-breaker

Compare the Hundredths

One of those has a 6 in the hundredths, and the other has a 0, so the 0 wins (remember we are looking for the smallest each time). In other words 1.506 is less than 1.56:

Answer so far: 0.8, 1.506

Remove 1.506 from the list:

Units Decimal
Point
Tenths Hundredths Thousandths
.
1 . 5 6 0
.

Only one number left, it must be the largest:

Answer: 0.8, 1.506, 1.56

Done!

Example: Put the following decimals in DESCENDING order:

0.402, 0.42, 0.375, 1.2, 0.85

In a table they look like this:

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 4 0 2
0 . 4 2  
0 . 3 7 5
1 . 2    
0 . 8 5  

And we want to go from highest to lowest (descending).

Fill in the empty squares with zeros:

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 4 0 2
0 . 4 2 0
0 . 3 7 5
1 . 2 0 0
0 . 8 5 0

Compare using the first column (Units):

There is a 1, all the rest are 0. Descending order needs largest first, so 1.2 must be the highest. (Write it down in your answer and cross it off the table).

Answer so far: 1.2

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 4 0 2
0 . 4 2 0
0 . 3 7 5
0 . 8 5 0

Compare the Tenths.

The 8 is highest, so 0.85 is next in value.

Answer so far: 1.2, 0.85

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 4 0 2
0 . 4 2 0
0 . 3 7 5

Now there are two numbers with the same “Tenths” value of 4, so move along to the “Hundredths” for the tie-breaker

One number has a 2 in the hundredths, and the other has a 0, so the 2 wins. So 0.42 is bigger than 0.402:

Answer so far: 1.2, 0.85, 0.42, 0.402

Units Decimal
Point
Tenths Hundredths Thousandths
0 . 3 7 5

Only 0.375 left, so the answer is:

Answer: 1.2, 0.85, 0.42, 0.402, 0.375

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Mathematics

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Mathematics

What is Function?

What is a Function?

A function relates an input to an output.

It is like a machine that has an input and an output.

And the output is related somehow to the input.

 

                                  f(x)   f(x) = … ” is the classic way of writing a function.
And there are other ways, as you will see!

Input, Relationship, Output

We will see many ways to think about functions, but there are always three main parts:

  • The input
  • The relationship
  • The output

Example: “Multiply by 2” is a very simple function.

Here are the three parts:

Input Relationship Output
0 × 2 0
1 × 2 2
7 × 2 14
10 × 2 20

For an input of 50, what is the output?

Some Examples of Functions

  • x2 (squaring) is a function
  • x3+1 is also a function
  • Sine, Cosine  and Tangent are functions used in trigonometry
  • and there are lots more!

Names

First, it is useful to give a function a name.

The most common name is “f“, but we can have other names like “g” … or even “marmalade” if we want.

But let’s use “f”:

f(x)

We say “f of x equals x squared”

what goes into the function is put inside parentheses () after the name of the function:

So f(x) shows us the function is called “f“, and “x” goes in

And we usually see what a function does with the input:

f(x) = x2 shows us that function “f” takes “x” and squares it.

Example: with f(x) = x2:

  • an input of 4
  • becomes an output of 16.

In fact we can write f(4) = 16.

 

The “x” is Just a Place-Holder!

Don’t get too concerned about “x”, it is just there to show us where the input goes and what happens to it.

It could be anything!

So this function:

f(x) = 1 – x + x2

Is the same function as:

  • f(q) = 1 – q + q2
  • h(A) = 1 – A + A2
  • w(θ) = 1 – θ + θ2

The variable (x, q, A, etc.) is just there so we know where to put the values:

f(2) = 1 – 2 + 22 = 3

 

Sometimes There is No Function Name

Sometimes a function has no name, and we see something like:

y = x2

But there is still:

  • an input (x)
  • a relationship (squaring)
  • and an output (y)

Relating

At the top we said that a function was like a machine. But a function doesn’t really have belts or cogs or any moving parts – and it doesn’t actually destroy what we put into it!

A function relates an input to an output.

Saying “f(4) = 16” is like saying 4 is somehow related to 16. Or 4 → 16

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Sets/Relation/Function

Sets/Relation/Function

1.1 Introduction

 

  • The theory of sets was developed by German mathematician Georg Cantor.
  • Sets are used to define the concepts of relations and functions.
  • The study of geometry, sequences, probability, etc. requires the knowledge of sets.

1.2 Sets And Their Representations

  • A set is a well-defined collection of objects.
  • Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
  • The elements of a set are represented by small letters a, b, c, x, y, z, etc.
1.2.1 Representation Of Sets
  • There are two methods of representing a set:
  • Roster or tabular form
  • Set – builder form.
1.2.1.1 Roster Form:
  • In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.
  • Example: The set of all even positive integers less than 7 is described in roster form as { 2, 4, 6 }.
1.2.1.2 Set-Builder Form
  • In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
  • Example: In the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property.
  • Denoting this set by K, we write,

K = { x : x is a vowel in English alphabet }

Example:
  • Write the set L = {1, 4, 9, 16, 25, . . . }in set-builder form.
Solution:
  • We may write the set L as

L = {x : x is the square of a natural number}

  • Alternatively, we can write

L = {x : x = n2 , where n  N}

1.3 The Empty Set 

  • A set which does not contain any element is called the empty set or the null set or the void set.
  • The empty set is denoted by the symbol  or { }.
Example:
  • D = {x : x2 = 4, x is odd}. Then D is the empty set because the equation x2 = 4 is not satisfied by any odd value of x.

1.4 Finite and Infinite Sets

Finite Sets
  • A set which is empty or consists of a definite number of elements is called finite.
Example:
  • Let W be the set of the days of the week. Then W is finite.
Infinite Sets
  • An infinite set is a set with an infinite number of elements.
Example:
  • The set of all integers, {….., -1, 0, 1, 2, …..}

1.5 Equal Sets 

 

  • Two sets A and B are said to be equal if they have exactly the same elements and we write A = B.
Example:
  • Let A be the set of prime numbers less than 6 and P the set of prime factors of 30.
  • Then A and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6.

1.6 Subsets

  • A set P is said to be a subset of a set Q if every element of P is also an element of Q.
  • It is expressed as P  Q;The symbol  stands for ‘is a subset of’.
Example:
  • Let A = { a, e, i, o, u } and B = { a, b, c, d }. Is A a subset of B?
Solution:
  • No, A is not a subset of B as every element of A is not an element of B.

1.7 Power Set

  • The collection of all subsets of a set A is called the power set of A.
  •  It is denoted by P(A).
Example:
  • If A = { 1, 2 }, then P(A) = { , {1}, {2}, {1, 2} }
Note:

Cardinal Number :- The total no. of elements are known as Cardinal no.

  • If A is a set with n(A) = m, then it can be shown that n[P(A)] = 2m .

1.8 Universal Set

  • A Universal Set is the set of all elements under consideration, denoted by capital U.
  • All other sets are subsets of the universal set.
Example:
  • Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets.
  • A = {x : x is a factor of 60}
  • B = {x : x is a prime number}
Solution:
  • The elements of sets A and B can only be selected from the given universal set U.
  • A = {5, 6, 10, 12}
  • B = {5, 7, 11}

1.9 Venn Diagrams

  • A Venn diagram is a drawing in which circular areas represent groups of items which share a common property.
  • The drawing consists of two or more circles, each representing a specific group or set.

Example:
  • Given U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 5, 6 } and B = { 3, 9 }, draw a Venn diagram to represent these sets.
Solution:
  • Given
  • U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 },
  • A = { 1, 2, 5, 6 } and
  • B = { 3, 9 }.

1.10 Operations on Sets 

  • There are some operations which when performed on two sets give rise to another set. They are:
  • Union of sets
  • Intersection of sets
  • Difference of sets
  • Complement of a set
1.10.1 Union of Sets:
  • The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both).
  • The symbol ‘’ is used to denote the union.
  • Symbolically, we write A  B = {x : x  A or x  B}
  • The union of two sets can be represented by a Venn diagram as shown in figure.

Example:
  • Let A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }. Find A  B?
Solution:
  • We have A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }. Then A  B = { 2, 4, 6, 8, 10, 12 }
Properties of the Operation of Union:
  • Commutative law : A  B = B  A
  • Associative law :  (A  B) C = A  (B C)
  • Law of identity element : A   = A
  • Idempotent law : A  A = A
  • Law of U : U  A = U

 

1.10.2 Intersection of Sets:
  • The intersection of sets A and B is the set of all elements which are common to both A and B.
  • The symbol  is used to denote the intersection.
  • Symbolically we write it as A  B = {x : x  A and x  B}
  • The shaded portion in the figure indicates the intersection of A and B.

Properties of Operation of Intersection:
  • Commutative law: A  B = B  A
  • Associative law: (A  B) C = A  (B  C)
  • Law of  and U:   =
  • Idempotent law: A  A = A
  • Distributive law:
1.10.3 difference of Sets:-
  • The difference of the sets A and B in this order is the set of elements which belong to A but not to B.
  • Symbolically, we write A – B and read as “A minus B”.
Example:
  • Let A = { 1, 2, 3, 4, 5, 6 }, B = { 2, 4, 6, 8 }. Find A – B and B – A.
Solution:
  • A – B = { 1, 3, 5 }, since the elements 1, 3, 5 belong to A but not to B
  • B – A = { 8 }, since the element 8 belongs to B and not to A.

1.11 Complement of a Set

  • Let U be the universal set and A is a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A.
  • Symbolically, we write A’ to denote the complement of A with respect to U. Thus,

A’ = {x : x  U and x  A}

Example:
  • Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } and A = { 1, 3, 5, 7, 9 }. Find A’.
Solution:
  • 2, 4, 6, 8, 10 are the only elements of U which do not belong to A.
  • Hence A’ = { 2, 4, 6, 8, 10 }
Properties of Complement Sets:
  • Complement laws:
  • A  A’ = U
  • A  A’ =
  • De Morgan’s law:
  • (A  B)’ = A’  B’
  • (A  B)’ = A’  B’
  • Law of double complementation:
  • (A’)’ = A
  • Law of empty set and universal set
Formulae:

n(A  B ) = n(A) +  n(B)  –  n(A  B)

n(A  B  C) = n(A) + n(B) + n(C) – n(A  B) – n(B  C) – n(A  C) + (A  B  C)

Example:
  • In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach both physics and mathematics. How many teach physics?
Solution:
  • Let M denote the set of teachers who teach mathematics and P denote the set of teachers who teach physics.

n (M  P) = 20,

n(M) = 12

n(M  P) = 4

  • To find n(P)

n (M  P) = n(M) + n(P) – n(M  P)

20 = 12 + n(P) – 4

n(P) =12

 

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Limit and Continuity

Limit and Continuity

In mathematics, a limit is the value that a function or sequence “approaches” as the input or index approaches some value.[1] Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.

In formulas, a limit is usually written as

 \lim_{n \to c}f(n) = L

and is read as “the limit of f of n as n approaches c equals L“. Here “lim” indicates limit, and the fact that function f(n) approaches the limit L as n approaches c is represented by the right arrow (→), as in

f(n) \to L \ . 
For example, if

 f(x) = \frac{x^2 - 1}{x - 1}

then f(1) is not defined (see division by zero), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2

In other words,  \lim_{x \to 1} \frac{x^2-1}{x-1} = 2

This can also be calculated algebraically, as \frac{x^2-1}{x-1} = \frac{(x+1)(x-1)}{x-1} = x+1 for all real numbers x ≠ 1.

Now since x + 1 is continuous in x at 1, we can now plug in 1 for x, thus \lim_{x \to 1} \frac{x^2-1}{x-1} = 1+1 = 2

That means if you want to solve a Limit based question, you have to Factorise given function, then put the value.

You will get your answer…..

Calculas

Calculas

Calculus is the mathematical study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus, and integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Leibniz. Today, calculus has widespread uses in science, engineering and economics and can solve many problems that elementary algebra alone cannot