A First Course in Calculus

✍ Scribed by Serge Lang

• Publisher: Springer Science & Business Media
• Year: 2012
• Tongue: English
• Leaves: 741
• Series: Undergraduate Texts in Mathematics
• Edition: 5
• Category: Library

✦ Synopsis

This fifth edition of Lang's book covers all the topics traditionally taught in the first-year calculus sequence. Divided into five parts, each section of A FIRST COURSE IN CALCULUS contains examples and applications relating to the topic covered. In addition, the rear of the book contains detailed solutions to a large number of the exercises, allowing them to be used as worked-out examples -- one of the main improvements over previous editions.

✦ Table of Contents

Cover
Title page
Copyright page
Foreword
Contents
Part One Review of Basic Material
CHAPTER1 Numbers and Functions
I, §1. INTEGERS, RATIONAL NUMBERS, AND REAL NUMBERS
I, §2. INEQUALITIES
I, §3. FUNCTIONS
I, §4. POWERS
CHAPTER II Graphs and Curves
II, §1. COORDINATES
II, §2. GRAPHS
II, §3. THE STRAIGHT LINE
II, §4. DISTANCE BETWEEN TWO POINTS
II, §5. CURVES AND EQUATIONS
II, §6. THE CIRCLE
II, §7. DILATIONS AND THE ELLIPSE
II, §8. THE PARABOLA
II, §9. THE HYPERBOLA
Part Two Differentiation and Elementary Functions
CHAPTER III The Derivative
III, §1. THE SLOPE OF A CURVE
III, §2. THE DERIVATIVE
III, §3. LIMITS
III, §4. POWERS
III, §5. SUMS, PRODUCTS, AND QUOTIENTS
III, §6. THE CHAIN RULE
III, §7. HIGHER DERIVATIVES
III, §8. IMPLICIT DIFFERENTIATION
III, §9. RATE OF CHANGE
CHAPTER IV Sine and Cosine
IV, §O. REVIEW OF RADIAN MEASURE
IV, §1. THE SINE AND COSINE FUNCTIONS
IV, §2. THE GRAPHS
IV, §3. ADDITION FORMULA
IV, §4. THE DERIVATIVES
IV, §5. TWO BASIC LIMITS
IV, §6. POLAR COORDINATES
CHAPTER V The Mean Value Theorem
V, §1. THE MAXIMUM AND MINIMUM THEOREM
V, §2. INCREASING AND DECREASING FUNCTIONS
V, §3. THE MEAN VALUE THEOREM
CHAPTER VI Sketching Curves
VI, §1. BEHAVIOR AS X BECOMES VERY LARGE
VI, §2. BENDING UP AND DOWN
VI, §3. CUBIC POLYNOMIALS
VI, §4. RATIONAL FUNCTIONS
VI, §5. APPLIED MAXIMA AND MINIMA
CHAPTER VII Inverse Functions
VII, §1. DEFINITION OF INVERSE FUNCTIONS
VII, §2. DERIVATIVE OF INVERSE FUNCTIONS
VII, §3. THE ARCSINE
VII, §4. THE ARCTANGENT
CHAPTER VIII Exponents and Logarithms
VIII, §1. THE EXPONENTIAL FUNCTION
VIII, §2. THE LOGARITHM
VIII, §3. THE GENERAL EXPONENTIAL FUNCTION
VIII, §4. SOME APPLICATIONS
VIII, §6. THE LOGARITHM AS THE AREAUNDER THE CURVE 1/ X
VIII, APPENDIX. SYSTEMATIC PROOF OF THE THEORY OFEXPONENTIALS AND LOGARITHMS
Part Three Integration
CHAPTER IX Integration
IX, §1. THE INDEFINITE INTEGRAL
IX, §2. CONTINUOUS FUNCTIONS
IX, §3. AREA
IX, §4. UPPER AND LOWER SUMS
IX, §5. THE FUNDAMENTAL THEOREM
CHAPTER X Properties of the Integral
X, §1. FURTHER CONNECTION WITH THE DERIVATIVE
X, §2. SUMS
X, §3. INEQUALITIES
X, §4. IMPROPER INTEGRALS
CHAPTER XI Techniques of Integration
XI, §1. SUBSTITUTION
XI, §2. INTEGRATION BY PARTS
XI, §3. TRIGONOMETRIC INTEGRALS
XI, §4. PARTIAL FRACTIONS
XI, §5. EXPONENTIAL SUBSTITUTIONS
CHAPTER XII Applications of Integration
XII, §1. VOLUMES OF REVOLUTION
XII, §2. AREA IN POLAR COORDINATES
XII, §3. LENGTH OF CURVES
XII, §4. PARAMETRIC CURVES
XII, §5. SURFACE OF REVOLUTION
XII, §6. WORK
XII, §7. MOMENTS AND CENTER OF GRAVITY
Part Four Taylor's Formula and Series
CHAPTER XIII Taylor's Formula
XIII, §1. TAYLOR'S FORMULA
XIII, §2. ESTIMATE FOR THE REMAINDER
XIII, §3. TRIGONOMETRIC FUNCTIONS
XIII, §4. EXPONENTIAL FUNCTION
XIII, §5. LOGARITHM
XIII, §6. THE ARCTANGENT
XIII, §7. THE BINOMIAL EXPANSION
XIII, §8. SOME LIMITS
CHAPTER XIV Series
XIV, §1. CONVERGENT SERIES
XIV, §2. SERIES WITH POSITIVE TERMS
XIV, §3. THE RATIO TEST
XIV, §4. THE INTEGRAL TEST
XIV, §5. ABSOLUTE AND ALTERNATING CONVERGENCE
XIV, §6. POWER SERIES
XIV, §7. DIFFERENTIATION AND INTEGRATION OFPOWER SERIES
APPENDIX To the First Four Parts
Ɛ and δ
APP., §1. LEAST UPPER BOUND
APP., §2. LIMITS
APP., §3. POINTS OF ACCUMULATION
APP., §4. CONTINUOUS FUNCTIONS
Part Five Functions of Several Variables
CHAPTER XV Vectors
XV, §1. DEFINITION OF POINTS IN SPACE
XV, §2. LOCATED VECTORS
XV, §3. SCALAR PRODUCT
XV, §4. THE NORM OF A VECTOR
XV, §5. PARAMETRIC LINES
XV, §6. PLANES
CHAPTER XVI Differentiation of Vectors
XVI, §1. DERIVATIVE
XVI, §2. LENGTH OF CURVES
CHAPTER XVII Functions of Several Variables
XVII, §1. GRAPHS AND LEVEL CURVES
XVII, §2. PARTIAL DERIVATIVES
XVII, §3. DIFFERENTIABILITY AND GRADIENT
CHAPTER XVIII The Chain Rule and the Gradient
XVIII, §1. THE CHAIN RULE
XVIII, §2. TANGENT PLANE
XVIII, §3. DIRECTIONAL DERIVATIVE
XVIII, §4. FUNCTIONS DEPENDING ONLY ON THE DISTANCE FROM THE ORIGIN
XVIII, §5. CONSERVATION LAW
Answers to Exercises
Index