CHAPTER 1 Functio, Limits and Continuity
1.1 Mathematical Sign Language
1.1.1 Sets
1.1.2 Number
1.1.3 Intervals
1.1.4 Implication and Equivalence
1.1.5 Inequalities and Numbe
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 Functio
1.2.1 Definition of a Function
1.2.2 Properties of Functio
1.2.3 Invee and Composite Functio
1.2.4 Combining Functio
1.2.5 Elementary Functio
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 Functio
1.3.6 Calculating Limits Using Limit Laws
1.3.7 Two Important Limit Results
1.3.8 Asymptotic Functio and Small o Notation
1.4 Continuous and Discontinuous Functio
1.4.1 Definitio
1.4.2 Building Continuous Functio
1.4.3 Theorems on Continuous Functio
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 Numbe 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 Weietrass
1.7 Exercises
1.7.1 Evaluating Limits
1.7.2 Continuous Functio
1.7.3 Questio to Guide Your Revision
CHAPTER 2 Differential Calculus
2.1 The Derivative
2.1.1 The Tangent to a Curve
2.1.2 Itantaneous Velocity
2.1.3 The Definition of a Derivative
2.1.4 Notatio for the Derivative
2.1.5 The Derivative as a Function
2.1.6 One-sided,Derivatives
2.1.7 Continuity of Differentiable Functio
2.1.8 Functio with no Derivative
2.2 Finding the Derivatives
2.2.1 Derivative Laws
2.2.2 Derivative of an Invee Function
2.2.3 Differentiating a Composite Funetion--The Chain Rule
2.3 Derivatives of Higher Orde
2.4 Implicit Differentiation
2.4.1 Implicitly Defined Functio
2.4.2 Finding the Derivative of an Implicitly Defined Function
2.4.3 Logarithmic Differentiation
2.4.4 Functio Defined by Parametric Equatio
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 Questio to Guide Your Revision
3 The Mean Value Theorem and Applicatio of the
CHAPTER 3 The Mean Value Theorem and Applicatio of the Derivative
3.1 The Mean Value Theorem
3.2 L''Hospital''s Rule and Indeterminate Forms
3.3 Taylor Series
3.4 Monotonic and Concave Functio and Graphs
3.4.1 Monotonic Functio
3.4.2 Concave Functio
3.5 Maximum and Minimum Values of Functio
3.5.1 Global Maximum and Global Minimum
3.5.2 Curve Sketching
3.6 Solving Equatio Numerically
3.6.I Decimal Search
3.6.2 Newton''s Method
3.7 Additional Materia
3.7.1 Fermat
3.7.2 L''Elospital
3.8 Exercises
3.8.l The Mean Value Theorem
3.8.2 L''Hospital''s Rules
3.8.3 Taylor''s Theorem
3.8.4 Applicatio of the Derivative
3.8.5 Questio to Guide Your Revision
CHAPTER 4 Integral Calculus
4.1 The Indefinite Integral
4.1.1 Definitio 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 Fractio in Integration
4.1.8 Rationalizing Substitutio
4.2 Definite Integrals and, the Fundamental Theorem of Calculus
4.2.1 Introduction
4.2.2 The Definite Integral
4.2.3 Interpreting ∫fx dx as an Area
4.2.4 Interpreting ∫ft 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 Applicatio 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 Questio to Guide Your Revision
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