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Calculus With Analytic Geometry
We do not plan to review this book.
CHAPTER 1: Numbers, Functions, and Graphs
1-1 Introduction
1-2 The Real Line and Coordinate Plane: Pythagoras
1-3 Slopes and Equations of Straight Lines
1-4 Circles and Parabolas: Descartes and Fermat
1-5 The Concept of a Function
1-6 Graphs of Functions
1-7 Introductory Trigonometry
1-8 The Functions Sin O and Cos O
CHAPTER 2: The Derivative of a Function
2-0 What is Calculus ?
2-1 The Problems of Tangents
2-2 How to Calculate the Slope of the Tangent
2-3 The Definition of the Derivative
2-4 Velocity and Rates of Change: Newton and Leibriz
2-5 The Concept of a Limit: Two Trigonometric Limits
2-6 Continuous Functions: The Mean Value Theorem and Other Theorem
CHAPTER 3: The Computation of Derivatives
3-1 Derivatives of Polynomials
3-2 The Product and Quotient Rules
3-3 Composite Functions and the Chain Rule
3-4 Some Trigonometric Derivatives
3-5 Implicit Functions and Fractional Exponents
3-6 Derivatives of Higher Order
CHAPTER 4: Applications of Derivatives
4-1 Increasing and Decreasing Functions: Maxima and Minima
4-2 Concavity and Points of Inflection
4-3 Applied Maximum and Minimum Problems
4-4 More Maximum-Minimum Problems
4-5 Related Rates
4-6 Newtons Method for Solving Equations
4-7 Applications to Economics: Marginal Analysis
CHAPTER 5: Indefinite Integrals and Differential Equations
5-1 Introduction
5-2 Differentials and Tangent Line Approximations
5-3 Indefinite Integrals: Integration by Substitution
5-4 Differential Equations: Separation of Variables
5-5 Motion Under Gravity: Escape Velocity and Black Holes
CHAPTER 6: Definite Integrals
6-1 Introduction
6-2 The Problem of Areas
6-3 The Sigma Notation and Certain Special Sums
6-4 The Area Under a Curve: Definite Integrals
6-5 The Computation of Areas as Limits
6-6 The Fundamental Theorem of Calculus
6-7 Properties of Definite Integrals
CHAPTER 7: Applications of Integration
7-1 Introduction: The Intuitive Meaning of Integration
7-2 The Area between Two Curves
7-3 Volumes: The Disk Method
7-4 Volumes: The Method of Cylindrical Shells
7-5 Arc Length
7-6 The Area of a Surface of Revolution
7-7 Work and Energy
7-8 Hydrostatic Force
PART II
CHAPTER 8: Exponential and Logarithm Functions
8-1 Introduction
8-2 Review of Exponents and Logarithms
8-3 The Number e and the Function y = e <^>x
8-4 The Natural Logarithm Function y = ln x
8-5 Applications
Population Growth and Radioactive Decay
8-6 More Applications
CHAPTER 9: Trigonometric Functions
9-1 Review of Trigonometry
9-2 The Derivatives of the Sine and Cosine
9-3 The Integrals of the Sine and Cosine
9-4 The Derivatives of the Other Four Functions
9-5 The Inverse Trigonometric Functions
9-6 Simple Harmonic Motion
9-7 Hyperbolic Functions
CHAPTER 10 : Methods of Integration
10-1 Introduction
10-2 The Method of Substitution
10-3 Certain Trigonometric Integrals
10-4 Trigonometric Substitutions
10-5 Completing the Square
10-6 The Method of Partial Fractions
10-7 Integration by Parts
10-8 A Mixed Bag
10-9 Numerical Integration
CHAPTER 11: Further Applications of Integration
11-1 The Center of Mass of a Discrete System
11-2 Centroids
11-3 The Theorems of Pappus
11-4 Moment of Inertia
CHAPTER 12: Indeterminate Forms and Improper Integrals
12-1 Introduction. The Mean Value Theorem Revisited
12-2 The Interminate Form 0/0. L'Hospital's Rule
12-3 Other Interminate Forms
12-4 Improper Integrals
12-5 The Normal Distribution
CHAPTER 13: Infinite Series of Constants
13-1 What is an Infinite Series ?
13-2 Convergent Sequences
13-3 Convergent and Divergent Series
13-4 General Properties of Convergent Series
13-5 Series on Non-negative Terms: Comparison Tests
13-6 The Integral Test
13-7 The Ratio Test and Root Test
13-8 The Alternating Series Test
CHAPTER 14: Power Series
14-1 Introduction
14-2 The Interval of Convergence
14-3 Differentiation and Integration of Power Series
14-4 Taylor Series and Taylor's Formula
14-5 Computations Using Taylor's Formula
14-6 Applications to Differential Equations
14. 7 (optional) Operations on Power Series
14. 8 (optional) Complex Numbers and Euler's Formula
PART III
CHAPTER 15: Conic Sections
15-1 Introduction
15-2 Another Look at Circles and Parabolas
15-3 Ellipses
15-4 Hyperbolas
15-5 The Focus-Directrix-Eccentricity Definitions
15-6 (optional) Second Degree Equations
CHAPTER 16: Polar Coordinates
16-1 The Polar Coordinate System
16-2 More Graphs of Polar Equations
16-3 Polar Equations of Circles, Conics, and Spirals
16-4 Arc Length and Tangent Lines
16-5 Areas in Polar Coordinates
CHAPTER 17: Parametric Equations
17-1 Parametric Equations of Curves
17-2 The Cycloid and Other Similar Curves
17-3 Vector Algebra
17-4 Derivatives of Vector Function
17-5 Curvature and the Unit Normal Vector
17-6 Tangential and Normal Components of Acceleration
17-7 Kepler's Laws and Newton's Laws of Gravitation
CHAPTER 18: Vectors in Three-Dimensional Space
18-1 Coordinates and Vectors in Three-Dimensional Space
18-2 The Dot Product of Two Vectors
18-3 The Cross Product of Two Vectors
18-4 Lines and Planes
18-5 Cylinders and Surfaces of Revolution
18-6 Quadric Surfaces
18-7 Cylindrical and Spherical Coordinates
CHAPTER 19: Partial Derivatives
19-1 Functions of Several Variables
19-2 Partial Derivatives
19-3 The Tangent Plane to a Surface
19-4 Increments and Differentials
19-5 Directional Derivatives and the Gradient
19-6 The Chain Rule for Partial Derivatives
19-7 Maximum and Minimum Problems
19-8 Constrained Maxima and Minima
19-9 Laplace's Equation, the Heat Equation, and the Wave Equation
19-10 (optional) Implicit Functions
CHAPTER 20: Multiple Integrals
20-1 Volumes as Iterated Integrals
20-2 Double Integrals and Iterated Integrals
20-3 Physical Applications of Double Integrals
20-4 Double Integrals in Polar Coordinates
20-5 Triple Integrals
20-6 Cylindrical Coordinates
20-7 Spherical Coordinates
20-8 Areas of curved Surfaces
CHAPTER 21: Line and Surface Integrals
21-1 Green's Theorem, Gauss's Theorem, and Stokes' Theorem
21-2 Line Integrals in the Plane
21-3 Independence of Path
21-4 Green's Theorem
21-5 Surface Integrals and Gauss's Theorem
21-6 Maxwell's Equations : A Final Thought
Appendices
A: The Theory of Calculus
A-1 The Real Number System
A-2 Theorems About Limits
A-3 Some Deeper Properties of Continuous Functions
A-4 The Mean Value theorem
A-5 The Integrability of Continuous Functions
A-6 Another Proof of the Fundamental Theorem of Calculus
A-7 Continuous Curves With No Length
A-8 The Existence of e = lim h->0 (1 + h) <^>1/h
A-9 Functions That Cannot Be Integrated
A-10 The Validity of Integration by Inverse Substitution
A-11 Proof of the Partial fractions Theorem
A-12 The Extended Ratio Tests of Raabe and Gauss
A-13 Absolute vs Conditional Convergence
A-14 Dirichlet's Test
A-15 Uniform Convergence for Power Series
A-16 Division of Power Series
A-17 The Equality of Mixed Partial Derivatives
A-18 Differentiation Under the Integral Sign
A-19 A Proof of the Fundamental Lemma
A-20 A Proof of the Implicit Function Theorem
A-21 Change of Variables in Multiple Integrals
B: A Few Review Topics
B-1 The Binomial Theorem
B-2 Mathematical Induction
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New Features |
- Revision highlights include the early introduction of trigonometry, extensive reworking of the infinite series chapters, and the addition of new exercises at varying levels of difficulty.
- New topics include first-order nonlinear differential equations, elementary probability, and hyperbolic functions.
- Two long appendices (Variety of additional topics, Biographical notes) have been removed from the text (will be available in the text, CALCULUS GEMS).
- The text offers full coverage for the full majors on engineering calculus, but, remains shorter than most competition.
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