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Relations and Functions
Relations and Functions
Inverse Trigonometric Functions
Algebra
Matrices
- Concept of Matrices
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- Negative of Matrix
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- Proof of the Uniqueness of Inverse
- Elementary Transformations
Calculus
Vectors and Three-dimensional Geometry
Determinants
- Determinants of Matrix of Order One and Two
- Determinant of a Matrix of Order 3 × 3
- Minors and Co-factors
- Inverse of a Square Matrix by the Adjoint Method
- Applications of Determinants and Matrices
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- Determinant of a Square Matrix
- Rule A=KB
- Overview of Determinants
- Geometric Interpretation of the Area of a Triangle
Linear Programming
Continuity and Differentiability
- Concept of Continuity
- Algebra of Continuous Functions
- Concept of Differentiability
- Derivatives of Composite Functions - Chain Rule
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- 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
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- Continuous Function of Point
- Mean Value Theorem
- Overview of Continuity and Differentiability
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
- Overview of Applications of Derivatives
Probability
Sets
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
- Overview of Integrals
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
- Overview of Differential Equations
Vectors
- Vector
- Basic Concepts of Vector Algebra
- Direction Cosines
- Vector Operations>Addition and Subtraction of Vectors
- Properties of Vector Addition
- Vector Operations>Multiplication of a Vector by a Scalar
- Components of Vector
- Vector Joining Two Points
- Section Formula
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- Scalar (Or Dot) Product of Two Vectors
- Projection of a Vector on a Line
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- 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
- Overview of 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
- Forms of the Equation of a Straight Line
- 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
- Overview of Three Dimensional Geometry
Linear Programming
Probability
Notes
The equation of a straight line passing through a given point `(x_0, y_0)` having finite slope m is given by
y - `y_0` = m (x - `x_0`)
Note that the slope of the tangent to the curve y = f(x) at the point `(x_0, y_0)` is given by `(dy)/(dx)]_(x_0,y_0)` `(=f'(x_0))`.
So the equation of the tangent at `(x_0,y_0)` to the curve y = f(x) is given by `y-y_0 = f'(x_0)(x - x_0)`Fig.
Also, since the normal is perpendicular to the tangent, the slope of the normal to the curve y = f(x) at `(x_0, y_0)` is `(-1)/(f'(x_0))` , if f'(x_0) ≠ 0 . Therefore, the equation of the normal to the curve y = f(x) at `(x_0, y_0)` is given by
`y – y_0 = (-1)/(f'(x_0))(x - x_0)`
i.e. `(y-y_0) f'(x_0) + (x-x_0) = 0`
Particular cases:
(i) If slope of the tangent line is zero, then tan θ = 0 and so θ = 0 which means the tangent line is parallel to the x-axis. In this case, the equation of the tangent at the point `(x_0, y_0)` is given by `y = y_0`.
(ii) If `theta →pi / 2 `, then tan θ → ∞, which means the tangent line is perpendicular to the x-axis, i.e., parallel to the y-axis. In this case, the equation of the tangent at `(x_0, y_0)` is given by `x = x_0`
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