Calculus and Analytical Geometry

Course Content:

1: Sets and Types of Sets

2: Functions

3: Interval

4: Inequalities

5: Rules of Inequalities

6: Notation of Functions

7: Types of Functions

8: L' Hospital Rules

9: Continuity and discontinuity

10: Rolle's Theorem

11: Increasing Function

12: Decreasing Function

13: Central Point

14: Limit

15: Derivation

16: Trigonometric Functions

17: Inverse Function

18: Mean value

19: Minima & Maxima

20: Chain Rule

21: Composition of Function

Set

"A collection of well-defined distinct objects is called a set."

Examples

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A= {1,2,3}

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B={a,b,c,d,e,f}

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Video:

Well-Defined

"The sets which can be defined properly are called well-defined sets"

Examples

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Set of whole numbers / natural numbers

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Set of students in class, etc.

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Not Well-Defined/ Undefined

"The sets which can not be defined properly are called undefined sets."

Examples

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Set of humans in class (LOL)

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Set of sharpness of boys

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Set of cunningness of fox

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Types of Sets

Empty or Null or Void Set

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"Any Set that does not contain any element is called an empty or null or void set."

The symbol used to represent an empty set is – {} or φ.

Examples:

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Let A = {x: 8 < x < 9, x is a natural number} will be a null set because there is no natural number between numbers 8 and 9. Therefore, A = {} or φ

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Let W = {d: d > 8, d is the number of days in a week} will also be a void set because there are only 7 days in a week.

Finite and Infinite Sets

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"Any set which is empty or contains a definite and countable number of elements is called a finite set."

Sets defined otherwise,

"for uncountable or indefinite numbers of elements are referred to as infinite sets."

Examples:

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A = {a, e, i, o, u} is a finite set because it represents the vowel letters in the English alphabetical series.

B = {x: x is a number appearing on a dice roll} is also a finite set because it contains – {1, 2, 3, 4, 5, 6} elements.

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C = {p: p is a prime number} is an infinite set.

D = {k: k is a real number} is also an infinite set.

Equal and Unequal Sets

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"Two sets X and Y are said to be equal if they have exactly the same elements (irrespective of the order of appearance in the set)."

Equal sets are represented as X = Y. Otherwise, the sets are referred to as unequal sets, which are represented as X ≠ Y.

Examples:

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If X = {a, e, i, o, u} and H = {o, u, i, a, e} then both of these sets are equal.

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If C = {1, 3, 5, 7} and D = {1, 3, 5, 9} then both of these sets are unequal.

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If A = {b, o, y} and B = {b, o, b, y, y} then also A = B because both contain the same elements.

Equivalent Sets

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"Equivalent sets are those which have an equal number of elements irrespective of what the elements are. "

Examples:

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A = {1, 2, 3, 4, 5} and B = {x: x is a vowel letter} are equivalent sets because both these sets have 5 elements each.

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S = {12, 22, 32, 42, …} and T = {y : y2 ϵ Natural number} are also equal sets.

Singleton Set

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"These are those sets that have only a single element."

Examples:

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E = {x : x ϵ N and x3 = 27} is a singleton set with a single element {3}

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W = {v: v is a vowel letter and v is the first alphabet of English} is also a singleton set with just one element {a}.

Universal Set

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"A universal set contains ALL the elements of a problem under consideration."

It is generally represented by the letter U.

Example:

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The set of Real Numbers is a universal set for ALL natural, whole, odd, even, rational and irrational numbers.

Power Set

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"The collection of ALL the subsets of a given set is called a power set of that set under consideration."

Example:

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A = {a, b} then Power set – P (A) = φ, {a}, {b} and {a, b}. If n (A) = m then generally, n [P (A)] = 2m

Thus, these are the different types of sets.

Functions

"An expression that defines a relationship between one variable(the independent variable) and the other variable (dependent variable)

is called a function."

VIdeo:

Two sets A and B will find a function if:

Every value of the domain should have a relation with range.
Every value of the domain should have a unique or single relation with range.(Each domain can't be connected with two ranges but the domain can connect to a single range.)

f: A--->B

Notation of a function

The function can be denoted as:

f: A-->B

y= f(x)

h= g(p)

Where 'h' and 'y' are dependent variables and 'x' and 'p' are independent variables.

Domain

"The set of all possible inputs is called domain."

Note: "To find the domain of a function in fraction we have to put the denominator as zero."

Range

"The set of all possible outputs is called range."

Core-Domain/ Codomain

"All the elements of range are called core-domain."

Core-Domain and range are equal if fall the elements of range are used by the domain.

Types of Function

One-to-One Function

A function f: A → B is One to One if for each element of A there is a distinct element of B. It is also known as an Injective. Consider if a1 ∈ A and a2 ∈ B, f is defined as f: A → B such that f (a1) = f (a2)

Many to One Function

It is a function that maps two or more elements of A to the same element of set B. Two or more elements of A have the same image in B.

Onto Function

If there exists a function for which every element of set B there is (are) pre-image(s) in set A, it is an Onto Function. Onto is also referred to as a Surjective Function.

Polynomial Function: