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Table of Contents 1. FunctionsSymbols of Some Sets of Specific Type NumbersIntervals as SetsOrdered Pairs Cartesian Product of Two SetsRelationsFunctions Representation of Function Vertical Line Test to Identify whether a Given Relation is a Function or NotTypes of Functions Based on Mapping (A) One -One Function or Injective (B) Many-One Function (C) Onto Function or Surjective (D) Into Function (E) One-One Onto Function or Bijection (F) One-One Into Function (G) Many One-Onto Function (H) Many One-Into Function Number of Functions Domain, Co-domain and Range of a Function Rules for Finding the Domain of a FunctionRules for Solving Problems on Domain of a Function Rules for Finding the Range of a Function Equal or Identical Functions Arithmetic Combinations of FunctionsComposition of FunctionsCommon Functions (A) Real Value Functions (B) Even and Odd Functions (C) Continuous and Discontinuous Functions (D) Homogenous functions (E) Bounded Functions (F) Implicit and Explicit Functions (G) Periodic Functions (H) Invertible Functions (I) Inverse FunctionsSpecial Type Functions (A) Algebraic Functions (B) Transcendental FunctionsDomains and Ranges of Common FunctionsGraphing New FunctionsGraphing Made Easy Via Transformations Reflections of the Basic Curves X-axis Reflection : Y-axis Reflection :Reading GraphsUseful TipsMiscellaneous Solved ExamplesExercise 1, 2, 3 & 4 2. Limits Limit of a FunctionNeighbourhood of a PointGeneral Definition of the Limit of a FunctionBounded and Unbounded Functions Comparison between the Value of a Function at a Point and The Limit of a Function at the Same PointProperties of LimitsLimit in Case of Composite FunctionsSandwich TheoremThe Limit of as x ? Some Standard Results on Limits Expansion of Some Standard FunctionsL-Hospital�s Rule Standard Methods of Evaluation of Limits (A) When x ? ? (B) When x ? a , a R Limit of Miscellaneous Forms (A) ? 0 form (B) Use of Newton-Leibnitz�s formula in evaluating the limits (C) Summation of series using definite integral as the limit Miscellaneous Solved Examples Exercise 1, 2, 3 & 4 3. Continuity of Functions Continuity and Discontinuity of a Function Continuity of a Function at a point Continuity of a Function in an Interval Single Point Continuity Examining the Continuity of a Function in a Given Domain Basic Elementary Functions Which are Everywhere Continuous Basic Elementary Functions Which are Continuous in their Domain Basic Results on Continuous Functions Types of Discontinuity at a Point Discontinuity of Special Types of Functions Discontinuity of Some More Functions Intermediate Value Theorem Miscellaneous Solved ExamplesExercise 1, 2, 3 & 44. Differentiability Differentiability of a Function at a Point Differentiability of a Function in a Set Differentiability in an open interval : Differentiability on a closed interval : Some Standard Results on Differentiability Useful Tips Miscellaneous Solved ExamplesExercise 1, 2, 3 & 45. Differentiation Introduction Derivative of a Function Derivative of a Power Function Derivative of a Constant Function Derivative of a Sum of Functions Derivative of a Product of Functions (Product Rule) Derivative of a Quotient of two Functions (Quotient Rule) Derivative of Trigonometric Functions Derive of Composite Functions (Chain Rule) Derivative of an Inverse Functions Derivative of Logarithmic Function Derivative of an Exponential Function Derivative of Inverse Trigonometric Functions Derivative of a Function Represented Parametrically Differentiation of One Function w.r.t. other Function Derivative of an Implicit Function Logarithmic Differentiation Differentiation of a Determinant Differentiation by Trigonometrical Substitution Differentiation of Infinite Series Successive Differentiation nth Derivatives of Some Standard Functions Leibnitz�s Theorem [nth Derivative of Product of Two Specific Type Functions] Partial Derivatives List of Differentiation of Some Standard Functions List of Important Theorems Miscellaneous Solved ExamplesExercise 1, 2, 3 & 46. Mean values theorems and applications of derivatives Introduction Rolle�s Theorem Langrange�s Mean Value Theorem Useful Results, While Applying Rolle�s and Lagrange�s Mean Value Theorems Monotonicity (Strictly Increasing / Strictly Decreasing Function) Differentials Approximation of Error(s) Derivative as a Rate Measurer Slopes of the Tangent & The Normal Equations of Tangent and Normal Length of Intercepts made on Axes by the Tangent Length of Perpendicular from Origin to the Tangent Angle of Intersection of Two Curves Orthogonal Curves Length of Tangent, Normal, Sub Tangent & Sub Normal L� Hospital�s Rule Application of Derivative In Determining The Nature of Roots of a Cubic Polynomial Useful Tips Miscellaneous Solved ExamplesExercise 1, 2, 3 & 4