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

1.3.1 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

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

2.6 The Tangent Line Approximation and the Differential

2.7 Additional Material

2.7.1 Preliminary result needed to prove the Chain Rule

2.7.2 Proof of the Chain Rule

2.7.3 Leibnitz

2.7.4 Newton

2.8 Exercises

2.8.1 Finding Derivatives

2.8.2 Differentials

2.8.3 Questions to Guide Your Revision

CHAPTER 3 The Mean Value Theorem and Applications of the Derivative

3.1 The Mean Value Theorem

3.2 L'Hospital's Rule and Indeterminate Forms

3.2.1 The Indeterminate Forms 0/0,∞/∞,∞-∞ and ∞·0

3.2.2 The Indeterminate Forms 00,∞0, 0∞ and 1∞

3.3 Taylor Series

3.4 Monotonic and Concave Functions and Graphs

3.4.1 Monotonic Functions

3.4.2 Concave Functions

3.5 Maximum and Minimum Values of Functions

3.5.1 Global Maximum and Global Minimum

3.5.2 Curve Sketching

3.6 Solving Equations Numerically

3.6.1 Decimal Search

3.6.2 Newton's Method

3.7 Additional Material

3.7.1 Fermat

3.7.2 L'Hospital

3.8 Exercises

3.8.1 The Mean Value Theorem

3.8.2 L'Hospital's Rules

3.8.3 Taylor's Theorem

3.8.4 Applications of the Derivative

3.8.5 Questions to Guide Your Revision

CHAPTER 4 Integral Calculus

4.1 The Indefinite Integral

4.1.1 Definitions and Properties of Indefinite Integrals

4.1.2 Basic Antiderivatives

4.1.3 Properties of Indefinite Integrals

4.1.4 Integration By Substitution

4.1.5 Further Results Using Integration by Substitution

4.1.6 Integration by Parts

4.1.7 Partial Fractions in Integration

4.1.8 Rationalizing Substitutions

4.2 Definite Integrals and the Fundamental Theorem of Calculus

4.2.1 Introduction

4.2.2 The Definite Integral

4.2.3 Interpreting ∫b a f(x) dx as an Area

4.2.4 Interpreting ∫b a f(t) dt as a Distance

4.2.5 Properties of the Definite Integral

4.2.6 The Fundamental Theorem of Calculus

4.2.7 Integration by Substitution

4.2.8 Integration by Parts

4.2.9 Numerical Integration

4.2.10 Improper Integrals

4.3 Applications of the Definite Integral

4.3.1 The Area of the Region Between Two Curves

4.3.2 Volumes of Solids of Revolution

4.3.3 Arc Length

4.4 Additional Material

4.4.1 Riemann

4.4.2 Lagrange

4.5 Exercises

4.5.1 Indefinite Integrals

4.5.2 Definite Integrals

4.5.3 Questions to Guide Your Revision

Answers

Reference Books