Calculus

Description

"Here are the notes for my Calculus courses that I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn Calculus or needing a refresher in Calculus. This document contains the majority, if not all, of the topics that are typically taught in full set of Calculus courses (i.e. Calculus I, Calculus II and Calculus III)." Contents Preface Outline 1. Review 1.1 Functions 1.2 Inverse Functions 1.3 Trig Functions 1.4 Solving Trig Equations 1.5 Solving Trig Equations with Calculators, Part I 1.6 Solving Trig Equations with Calculators, Part II 1.7 Exponential Functions 1.8 Logarithm Functions 1.9 Exponential and Logarithm Equations 1.10 Common Graphs 2. Limits 2.1 Tangent Lines and Rates of Change 2.2 The Limit 2.3 One-Sided Limits 2.4 Limit Properties 2.5 Computing Limits 2.6 Infinite Limits 2.7 Limits at Infinity, Part I 2.8 Limits at Infinity, Part II 2.9 Continuity 2.10 The Definition of the Limit 3. Derivatives 3.1 The Definition of the Derivative 3.2 Interpretation of the Derivative 3.3 Differentiation Formulas 3.4 Product and Quotient Rule 3.5 Derivatives of Trig Functions 3.6 Derivatives of Exponentials & Logarithms 3.7 Derivatives of Inverse Trig Functions 3.8 Derivatives of Hyperbolic Functions 3.9 Chain Rule 3.10 Implicit Differentiation 3.11 Related Rates 3.12 Higher Order Derivatives 3.13 Logarithmic Differentiation 4. Derivative Applications 4.1 Rates of Change 4.2 Critical Points 4.3 Minimum and Maximum Values 4.4 Finding Absolute Extrema 4.5 The Shape of a Graph, Part I 4.6 The Shape of a Graph, Part II 4.7 The Mean Value Theorem 4.8 Optimization 4.9 More Optimization 4.10 L'Hospital's Rule 4.11 Linear Approximations 4.12 Differentials 4.13 Newton's Method 4.14 Business Applications 5. Integrals 5.1 Indefinite Integrals 5.2 Computing Indefinite Integrals 5.3 Substitution Rule for Indefinite Integrals 5.4 More Substitution Rule 5.5 Area Problem 5.6 Definition of the Definite Integral 5.7 Computing Definite Integrals 5.8 Substitution Rule for Definite Integrals 6. Applications of Integrals 6.1 Average Function Value 6.2 Area Between Curves 6.3 Volume with Rings 6.4 Volume with Cylinders 6.5 More Volume Problems 6.6 Work 7. Integration Techniques 7.1 Integration by Parts 7.2 Integrals Involving Trig Functions 7.3 Trig Substitutions 7.4 Partial Fractions 7.5 Integrals Involving Roots 7.6 Integrals Involving Quadratics 7.7 Integration Strategy 7.8 Improper Integrals 7.9 Comparison Test for Improper Integrals 7.10 Approximating Definite Integrals 8. More Applications of Integrals 8.1 Arc Length 8.2 Surface Area 8.3 Center Of Mass 8.4 Hydrostatic Pressure and Force 8.5 Probability 9. Parametric and Polar 9.1 Parametric Equations 9.2 Tangents with Parametric Equations 9.3 Area with Parametric Equations 9.4 Arc Length with Parametric Equations 9.5 Surface Area with Parametric Equations 9.6 Polar Coordinates 9.7 Tangents with Polar Coordinates 9.8 Area with Polar Coordinates 9.9 Arc Length with Polar Coordinates 9.10 Surface Area with Polar Coordinates 9.11 Arc Length and Surface Area Revisited 10. Series and Sequences 10.1 Sequences 10.2 More on Sequences 10.3 Series - Basics 10.4 Convergence & Divergence of Series 10.5 Special Series 10.6 Integral Test 10.7 Comparison & Limit Comparison Test 10.8 Alternating Series Test 10.9 Absolute Convergence 10.10 Ratio Test 10.11 Root Test 10.12 Strategy for Series 10.13 Estimating the Value of a Series 10.14 Power Series 10.15 Power Series and Functions 10.16 Taylor Series 10.17 Applications of Series 10.18 Binomial Series 11. Vectors 11.1 Basic Concepts 11.2 Vector Arithmetic 11.3 Dot Product 11.4 Cross Product 12. 3 Dimensional Space 12.1 The 3-D Coordinate System 12.2 Equations of Lines 12.3 Equations of Planes 12.4 Quadric Surfaces 12.5 Functions of Several Variables 12.6 Vector Functions 12.7 Calculus with Vector Functions 12.8 Tangent and Normal Vectors 12.9 Arc Length with Vector Functions 12.10 Curvature 12.11 Velocity and Acceleration 12.12 Cylindrical Coordinates 12.13 Spherical Coordinates 13. Partial Derivatives 13.1 Limits 13.2 Partial Derivatives 13.3 Interpretations of Partial Derivatives 13.4 Higher Order Partial Derivatives 13.5 Differentials 13.6 Chain Rule 13.7 Directional Derivatives 14. Applications of Partial Derivatives 14.1 Tangent Planes 14.2 Gradient Vector 14.3 Relative Extrema 14.4 Absolute Extrema 14.5 Lagrange Multipliers 15. Multiple Integrals 15.1 Double Integrals 15.2 Iterated Integrals 15.3 Double Integrals over General Regions 15.4 Double Integrals in Polar Coordinates 15.5 Triple Integrals 15.6 Triple Integrals in Cylindrical Coordinates 15.7 Triple Integrals in Spherical Coordinates 15.8 Change of Variables 15.9 Surface Area 15.10 Area and Volume Revisited 16. Line Integrals 16.1 Vector Fields 16.2 Line Integrals - Part I 16.3 Line Integrals - Part II 16.4 Line Integrals of Vector Fields 16.5 Fundamental Theorem for Line Integrals 16.6 Conservative Vector Fields 16.7 Green's Theorem 17. Surface Integrals 17.1 Curl and Divergence 17.2 Parametric Surfaces 17.3 Surface Integrals 17.4 Surface Integrals of Vector Fields 17.5 Stokes' Theorem 17.6 Divergence Theorem A. Calculus I Extras A.1 Proof of Various Limit Properties A.2 Proof of Various Derivative Properties A.3 Proof of Trig Limits A.4 Proofs of Derivative Applications Facts A.5 Proof of Various Integral Properties A.6 Area and Volume Formulas A.7 Types of Infinity A.8 Summation Notation A.9 Constant of Integration Index