Topics
Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
Algebra
Calculus
Matrices
- Introduction of Matrices
- Order of a Matrix
- Types of Matrices
- Equality of Matrices
- Introduction of Operations on Matrices
- Addition of Matrices
- Multiplication of a Matrix by a Scalar
- Properties of Matrix Addition
- Properties of Scalar Multiplication of a Matrix
- Multiplication of Matrices
- Properties of Multiplication of Matrices
- Transpose of a Matrix
- Properties of Transpose of the Matrices
- Symmetric and Skew Symmetric Matrices
- Invertible Matrices
- Inverse of a Matrix by Elementary Transformation
- Multiplication of Two Matrices
- Negative of Matrix
- Subtraction of Matrices
- Proof of the Uniqueness of Inverse
- Elementary Transformations
- Matrices Notation
Determinants
- Introduction of Determinant
- Determinants of Matrix of Order One and Two
- Determinant of a Matrix of Order 3 × 3
- Area of a Triangle
- Minors and Co-factors
- Inverse of a Square Matrix by the Adjoint Method
- Applications of Determinants and Matrices
- Elementary Transformations
- Properties of Determinants
- Determinant of a Square Matrix
- Rule A=KB
Vectors and Three-dimensional Geometry
Linear Programming
Continuity and Differentiability
- Concept of Continuity
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions - Chain Rule
- Derivatives of Implicit Functions
- Derivatives of Inverse Trigonometric Functions
- Exponential and Logarithmic Functions
- Logarithmic Differentiation
- Derivatives of Functions in Parametric Forms
- Second Order Derivative
- Derivative - Exponential and Log
- Proof Derivative X^n Sin Cos Tan
- Infinite Series
- Higher Order Derivative
- Continuous Function of Point
- Mean Value Theorem
Applications of Derivatives
- Introduction to Applications of Derivatives
- Rate of Change of Bodies or Quantities
- Increasing and Decreasing Functions
- Maxima and Minima
- Maximum and Minimum Values of a Function in a Closed Interval
- Simple Problems on Applications of Derivatives
- Graph of Maxima and Minima
- Approximations
- Tangents and Normals
Probability
Integrals
- Introduction of Integrals
- Integration as an Inverse Process of Differentiation
- Some Properties of Indefinite Integral
- Methods of Integration: Integration by Substitution
- Integration Using Trigonometric Identities
- Integrals of Some Particular Functions
- Methods of Integration: Integration Using Partial Fractions
- Methods of Integration: Integration by Parts
- Fundamental Theorem of Calculus
- Evaluation of Definite Integrals by Substitution
- Properties of Definite Integrals
- Definite Integrals
- Indefinite Integral Problems
- Comparison Between Differentiation and Integration
- Geometrical Interpretation of Indefinite Integrals
- Indefinite Integral by Inspection
- Definite Integral as the Limit of a Sum
- Evaluation of Simple Integrals of the Following Types and Problems
Sets
- Sets
Applications of the Integrals
Differential Equations
- Differential Equations
- Order and Degree of a Differential Equation
- General and Particular Solutions of a Differential Equation
- Linear Differential Equations
- Homogeneous Differential Equations
- Solutions of Linear Differential Equation
- Differential Equations with Variables Separable Method
- Formation of a Differential Equation Whose General Solution is Given
- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
Vectors
- Introduction of Vector
- Basic Concepts of Vector Algebra
- Direction Cosines
- Vectors and Their Types
- Addition of Vectors
- Properties of Vector Addition
- Multiplication of a Vector by a Scalar
- Components of Vector
- Vector Joining Two Points
- Section Formula
- Vector (Or Cross) Product of Two Vectors
- Scalar (Or Dot) Product of Two Vectors
- Projection of a Vector on a Line
- Geometrical Interpretation of Scalar
- Scalar Triple Product of Vectors
- Position Vector of a Point Dividing a Line Segment in a Given Ratio
- Magnitude and Direction of a Vector
- Vectors Examples and Solutions
- Introduction of Product of Two Vectors
Three - Dimensional Geometry
- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Relation Between Direction Ratio and Direction Cosines
- Equation of a Line in Space
- Angle Between Two Lines
- Shortest Distance Between Two Lines
- Three - Dimensional Geometry Examples and Solutions
- Equation of a Plane Passing Through Three Non Collinear Points
- Intercept Form of the Equation of a Plane
- Coplanarity of Two Lines
- Distance of a Point from a Plane
- Angle Between Line and a Plane
- Angle Between Two Planes
- Vector and Cartesian Equation of a Plane
- Equation of a Plane in Normal Form
- Equation of a Plane Perpendicular to a Given Vector and Passing Through a Given Point
- Distance of a Point from a Plane
- Plane Passing Through the Intersection of Two Given Planes
Linear Programming
Probability
- Introduction of Probability
- Conditional Probability
- Properties of Conditional Probability
- Multiplication Theorem on Probability
- Independent Events
- Bayes’ Theorem
- Variance of a Random Variable
- Probability Examples and Solutions
- Random Variables and Its Probability Distributions
- Mean of a Random Variable
- Bernoulli Trials and Binomial Distribution
| Derivatives | Integrals (Anti derivatives) |
| `d/(dx) (x^(n+1)/(n+1)) = x^n` | `int x^n dx = x^(n+1)/(n+1) + "C"`, n ≠ –1 |
| `d/(dx)`(x) = 1 | `int dx` = x + C |
| `d/(dx)`(sin x) = cos x | `int` cos x dx = sin x +C |
| `d/(dx)` (-cos x) = sin x | `int`sin x dx = -cos x +C |
| `d/(dx)` (tan x) = sec2x | `int sec^2 x` dx = tanx + C |
| `d/(dx)`(-cot x) = `cosec^2x ` | `int cosec^2x` dx = -cot x +C |
| `d/(dx)` (sec x) = sec x tan x | `int` sec x tan x dx = sec x +C |
| `d/(dx)` (-cosecx) = cosec x cot x | `int` cosec x cot x dx = -cosec x +C |
| `d/(dx) (sin^-1) = 1/(sqrt(1-x^2))` | `int (dx)/(sqrt(1-x^2))= sin^(-1) x +C ` |
| `d/(dx) (-cos^(-1)) = 1/(sqrt (1-x^2))` | `int (dx)/(sqrt (1-x^2))= -cos^(-1) x + C ` |
| `d/(dx) (tan^(-1) x) = 1/(1+x^2)` | `int (dx)/(1+x^2)= tan^(-1) x + C ` |
| `d/(dx) (-cot^(-1) x) = 1/(1+x^2)` | `int (dx)/(1+x^2)= -cot^(-1) x + C ` |
| `d/(dx) (sec^(-1) x) = 1/(x sqrt (x^2 - 1))` | `int (dx)/(x sqrt (x^2 - 1))`= `sec^(-1)` x + C |
| `d/(dx) (-cosec^(-1) x) = 1/(x sqrt (x^2 - 1))` | `int (dx)/(x sqrt (x^2 - 1))=-cosec^(-1) x + C ` |
| `d/(dx)(e^x) = e^x` | `int e^x dx = e^x + C` |
| `d/(dx) (log|x|) = 1/x` | `int 1/x dx = log|x| +C` |
| `d/(dx) ((a^x)/(log a)) = a^x` | `int a^x dx = a^x/log a` +C |
Notes
Integration is the inverse process of differentiation. The derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation.
Let us consider the following examples:
`d/(dx) sin x = cos x`
we observ that ,the function cos x is the derived function of sin x and also we say that sin x is an anti derivative of cos x .
There is a function F such that
`d/(dx) F(x)` = f(x) , ∀ x ∈ I (interval), then for any arbitrary real number C, (also called constant of integration)
`d/(dx) [F(x) + C] = = f(x), x ∈ I`
Thus, {F + C, C ∈ R} denotes a family of anti derivatives of f.
We already know the formulae for the integrals of these functions :
| Derivatives | Integrals (Anti derivatives) |
| `d/(dx) (x^(n+1)/(n+1)) = x^n` | `int x^n dx = x^(n+1)/(n+1) `+C`, n ≠ –1 |
| `d/(dx)`(x) = 1 | `int dx` = x + C |
| `d/(dx)`(sin x) = cos x | `int` cos x dx = sin x +C |
| `d/(dx)` (-cos x) = sin x | `int`sin x dx = -cos x +C |
| `d/(dx)` (tan x) = `sec^2x` | `int sec^2 x` dx = tanx +C |
| `d/(dx)`(-cot x) = `cosec^2x ` | `int cosec^2x` dx = -cot x +C |
| `d/(dx)` (sec x) = sec x tan x | `int` sec x tan x dx = sec x +C |
| `d/(dx)` (-cosecx) = cosec x cot x | `int` cosec x cot x dx = -cosec x +C |
| `d/(dx) (sin^-1) = 1/(sqrt(1-x^2))` | `int (dx)/(sqrt(1-x^2))= sin^(-1) x +C ` |
| `d/(dx) (-cos^(-1)) = 1/(sqrt (1-x^2))` | `int (dx)/(sqrt (1-x^2))= -cos^(-1) x + C ` |
| `d/(dx) (tan^(-1) x) = 1/(1+x^2)` | `int (dx)/(1+x^2)= tan^(-1) x + C ` |
| `d/(dx) (-cot^(-1) x) = 1/(1+x^2)` | `int (dx)/(1+x^2)= -cot^(-1) x + C ` |
| `d/(dx) (sec^(-1) x) = 1/(x sqrt (x^2 - 1))` | `int (dx)/(x sqrt (x^2 - 1))`= `sec^(-1)` x + C |
| `d/(dx) (-cosec^(-1) x) = 1/(x sqrt (x^2 - 1))` | `int (dx)/(x sqrt (x^2 - 1))=-cosec^(-1) x + C ` |
| `d/(dx)(e^x) = e^x` | `int e^x dx = e^x + C` |
| `d/(dx) (log|x|) = 1/x` | `int 1/x dx = log|x| +C` |
| `d/(dx) ((a^x)/(log a)) = a^x` | `int a^x dx = a^x/log a` +C |
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