| CHAPTER 1 Functions， Limits and Continuity 1. 1 Mathematical Sign Language 1.1.1 Sets 1.1.2 Numbers 1.1.3 Intervals 1.1.4 Implication and Equivalence 1.1.5 Inequalities and Numbers 1.1.6 Absolute Value of a Number 1.1.7 Summation Notation 1.1.8 Factorial Notation 1.1.9 Binomial Coefficients 1 2 Functions 1.2.1 Definition of a Function 1.2.2 Properties of Functions 1.2. 3 Inverse and Composite Functions 1.2.4 Combining Functions 1. 2.5 Elementary Functions 1.3 Limits I. 3.1 The Limit of a Sequence 1.3.2 The Limits of a Function 1.3.3 One-sided Limits 1.3.4 Limits Involving the Infinity Symbol 1.3.5 Properties of Limits of Functions 1.3.6 Calculating Limits Using Limit Laws 1.3.7 Two Important Limit Results 1.3.8 Asymptotic Functions and Small o Notation 1.4 Continuous and Discontinuous Functions 1.4.1 Definitions 1.4.2 Building Continuous Functions 1.4.3 Theorems on Continuous Functions 1.5 Further Results on Limits 1.5.1 The Precise Definition of a Limit 1.5.2 Limits at Infinity and Infinite Limits 1.5.3 Real Numbers and Limits 1.5.4 Asymptotes 1.5.5 Uniform Continuity 1.6 Additional Material 1.6.1 Cauchy 1.6.2 Heine 1.6.3 Weierstrass 1.7 Exercises 1.7.1 Evaluating Limits 1.7.2 Continuous Functions 1.7.3 Questions to Guide Your Revision CHAPTER 2 Differential Calculus 2.1 The Derivative 2.1.1 The Tangent to a Curve 2.1.2 Instantaneous Velocity 2.1.3 The Definition of a Derivative 2.1.4 Notations for the Derivative 2.1.5 The Derivative as a Function 2.1.6 One sided Derivatives 2.1.7 Continuity of Differentiable Functions 2.1.8 Functions with no Derivative 2.2 Finding the Derivatives 2.2.1 Derivative Laws 2.2.2 .Derivative of an Inverse Function 2.2.3 Differentiating a Composite Function——The Chain Rule 2.3 Derivatives of Higher Orders a 2.4 Implicit Differentiation 2.4.1 Implicitly Defined Functions 2.4.2 Finding the Derivative of an Implicitly Defined：Function 2.4.3 Logarithmic Differentiation 2.4.4 Functions Defined by Parametric Equations 2.5 Related Rates of Change r 2.6 The Tangent Line Approximation and the Differential 2.7 Additional Material 2.7.1 Prellminary result needed to prove the Chain Rule 2.7.2 Proof of the Chain Rule …… CHAPTER 3 The Mean Value Theorem and Applications of the Derivative CHAPTER 4 Integral Calculus Answers Reference Books |
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