This learning solution provides a clear exposition of essential tools of Algebra, Trigonometry, Co-Ordinate Geometry, Differential Calculus, Applications Of The Differentiation.
Logarithms - Logarithms– Example problems – Common vs natural logarithm - Exponential function - Logarithmic Functions - Engineering calculations. Partial Fractions Type I - Partial Fractions- Rational, Proper and Improper Fractions of Functions - Methods to Resolve a Rational Fraction into Partial Fractions – Example problems. Partial Fractions Type II - The Denominator g(x) contains repeated linear (ax + b)n and non-repeated linear factors (px + q) - Example problems. Partial Fractions Type III - The Denominator g(x) contains irreducible quadratic factor (ax2 + bx + c) - Example problems. Partial Fractions Type IV - The Denominator g(x) contains irreducible repeated quadratic factor – Example problem. Introduction of Matrices - Introduction of Matrices – Application of Matrices. Types of Matrices - Types of Matrices. Equality, Addition and Subtraction of Matrices - Equality, Addition and Subtraction of Matrices- Example problems. Multiplication of Matrices - Multiplication of Matrices – Example problems. Properties of multiplication of matrices - Properties of multiplication of matrices - Example problems. Transpose of Matrix- Introduction to transpose of matrix - Properties of Transpose - Symmetric and skew-symmetric matrices - Related matrices – Example problems. Determinant of a square matrix - Introduction - Minor and co-factor of an element - Determinant of Matrix – Example problems. Properties of Determinant - Properties of Determinants - Example problems. Singular and Non singular matrix - Singular and Non singular matrix – Example problems. Square matrix - Adjoint of a square matrix - Inverse of square matrix. – Example problem. Solution of simultaneous linear equations by Cramer’s rule - Solution of simultaneous linear equations by cramers rule - Example programs. Solution of linear equations by matrix inversion method - Matrix Inversion Method – Example problems. Solution of simultaneous equations by gauss-Jordan method - Solution of simultaneous equations by Gauss – Jordan method – Example problems.
Trigonometric functions of any angles - Fundamental relations between trigonometric functions of an angle - Trigonometric function of some important angles - Working principle of D.C. generator Sign of trigonometric functions - Graphs of trigonometric functions - Periodic function – Example problems. Compound Angles - Known to Unknown - Addition Formulae for Two Angles - Subtraction Formulae for Two Angles - Properties - Trigonometrically Ratios of A + B + C. Example problems of Compound Angles –Example problems. Multiple and Sub-Multiple Angles - Introduction - Trigonometrically Ratios of 3A - Trigonometrically Ratios of Sub-Multiple Angles - The values of sin 18, cos 18, sin 36, cos 36. Example Problems of Multiple and Sub-Multiple Angles – Example problems. Transformations – Transformations of Product into Sum or Difference - Transformation of Sums or Difference into Product - Conditional Identities. Example Problems of Transformations Inverse – Example programs. Trigonometric Functions – Introduction - Domain and Range of Trigonometric Functions - Properties of Inverse Trigonometric Functions. Example Problems of Inverse Trigonometric Functions - Example problems. Trigonometric Equations - Introduction – Theorems. Example Problems of Trigonometric Equations – Example problems. Properties of Triangles - Introduction - Relations Between Sides and Angles of a Triangle - Cosine Rule - Projection Rule - The Law of Tangents - Expressions for Half Angles in Terms of Sides - Area of a Triangle. Example Problems for Properties of Triangles – Example problems. Solution of Triangles – Introduction – Example problems. Hyperbolic Functions in Terms of Logarithm Functions – Definitions of hyperbolic functions - Inverse hyperbolic functions in terms of logarithmic functions - Example problems. Complex Numbers – Introduction - Algebra of Complex Numbers - Polar form (or) Modulus – Amplitude form of a Complex Number - Properties of Conjugate, Modulus and Arguments of Complex Numbers - Representation of Complex Number in Exponential form (Euler form) - Demoivre's Theorem. Example Problems of Complex Numbers – Example problems.
Straight Lines - Straight Lines - Two Point Form - Point Slope Form - Slope Intercept Form - Intercept Form. Straight Lines (Example Problems) - General Form – Example problems. Circle - Circle - Standard Form - Points to Remember. Circle(Example Problems) – Example problems. Tangent and Normal at a Point on the Circle - Tangents and Normal - Theorem 5 and 6 - Chord of contact, poles and polar – Example problems. Conic Section - Conic Section - Theorem - Some Real Life Applications of Conic Sections. Parabola - Some Elements of Parabola - Different Forms of Parabola and Some Important Results – Example problems. Find the Equation of Parabola – To find the parabola equation with given focus and directrix – Example problems. Engineering problems in simple cases of Parabola - Example problems. Ellipse - Ellipse - Some Important properties of Ellipse - To Find Length of Latus Rectum of Ellipse - To Find the Equation of Ellipse – Theorem – Example problems. To Find the Equation of Ellipse Using Given Data - Example problems. Engineering problems in simple cases of Ellipse – Example problems. Hyperbola - Hyperbola - Standard form of Hyperbola - To Find Length of Latus Rectum (LLR) of a hyperbola - Equation of Hyperbola –Example problems. To Find the Equation of Hyperbola Using Given Data – Example problems.
Functions - Introduction to Calculus – Functions - Types of Functions. Limits - Properties of Limits - Definition of the Limits - Concept of Limits - Standard Limits - Example problems. One Side Limits - One Side Limits – Example problems. Continuity - Concept of Continuity – Example problems. Methods of Differentiation - Introduction on types of functions. Definition of Derivative (First Principle) - Definition of Derivative (First Principle) – Fundamental Theorems of Differentiation - Theorems - Example problems. Elementary Properties - Elementary Properties – Example problems. Derivative of a Function of a Function (Chain Rule) - Derivative of a Function of a Function (Chain Rule) – Example problems. Derivatives of Inverse Functions - Derivatives of Inverse Functions - Derivatives of Inverse Trigonometric functions – Example problems. Differentiation of One Function With Respect to Another Function - Differentiation of One Function With Respect to Another Function - Example problems. Derivative of hyperbolic function - Derivative of hyperbolic function - Derivative of inverse hyperbolic functions - Example problems. Parametric Differentiation - Parametric Differentiation – Example problems. Implicit Differentiation - Implicit Differentiation – Example problems. Logarithmic Differentiation - Logarithmic Differentiation – Example problems. Differentiation of Infinite Series - Differentiation of Infinite Series – Example problems. Second Order Derivatives - Second and Higher Order Derivatives – Example problems. Partial Differentiation - Partial Differentiation - Second Order Partial Derivatives – Example problems. Homogeneous Function and Euler's Theorem - Homogeneous Function - Euler's Theorem –Example problems.
Geometrical Applications of Derivatives - Geometrical Applications of Derivatives - Geometrical Interpretation of the Derivative – Example problems. Equation of the Tangent and Normal to a Curve - Equation of the Tangent and Normal to a Curve – Example problems . Length of the Tangent, Normal, Sub tangent and Subnormal - Length of the Tangent, Normal, Sub tangent and Subnormal – Example problems. Angle Between Two Curves - Angle Between Two Curves – Example problems. Physical Applications of Derivatives - Physical Applications of Derivatives .Velocity and Acceleration of a Particle – Example problems. Rate Measure (Dependent on Time) - Rate Measure (Dependent on Time) - Some Useful Formulas – Example problems. Rate Measure(Example Problems) – Example problems. Increasing and Decreasing Function - Increasing and Decreasing Functions –Example problems. Find Maximum and Minimum Value - Find Maximum and Minimum Value – Example problems. Maximum and Minimum (Geometrical and Physical Applications) - Some Useful Formulas – Example problems. Maximum and Minimum - Conditions for finding the maximum or minimum values of a function y = f(x) – example problems. Errors and Approximations – Errors - Approximate Error – Relative error. Errors and Approximations (Example Problems) – Example problems
