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Short Calculus The Original Edition of “A First Course in Calculus”

This is a reprint of "A First Course in Calculus," which has gone through five editions since the early sixties. It covers all the topics traditionally taught in the first-year calculus sequence in a brief and elementary fashion. As sociological and educational conditions have evolved in v...

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Bibliographic Details
Main Author: Lang, Serge
Format: eBook
Language:English
Published: New York, NY Springer New York 2002, 2002
Edition:1st ed. 2002
Series:Undergraduate Texts in Mathematics
Subjects:
Functions Of Real Variables
Real Functions
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • I Numbers and Functions
  • 1. Integers, rational numbers and real numbers
  • 2. Inequalities
  • 3. Functions
  • 4. Powers
  • II Graphs and Curves
  • l. Coordinates
  • 2. Graphs
  • 3. The straight line
  • 4. Distance between two points
  • 5. Curves and equations
  • 6. The circle
  • 7. The parabola. Changes of coordinates
  • 8. The hyperbola
  • III The Derivative
  • l. The slope of a curve
  • 2. The derivative
  • 3. Limits
  • 4. Powers
  • 5. Sums, products, and quotients
  • 6. The chain rule
  • 7. Rate of change
  • IV Sine and Cosine
  • l. The sine and cosine functions
  • 2. The graphs
  • 3. Addition formula
  • 4. The derivatives
  • 5. Two basic limits
  • V The Mean Value Theorem
  • 1. The maximum and minimum theorem
  • 2. Existence of maxima and minima
  • 3. The mean value theorem
  • 4. Increasing and decreasing functions
  • VI Sketching Curves
  • 1. Behavior as x becomes very large
  • 2. Curve sketching
  • 3. Pol ar coordinates
  • 4. Parametric curves
  • VII Inverse Functions
  • XIV Taylor’s Formula
  • 1. Taylor’s formula
  • 2. Estimate for the remainder
  • 3. Trigonometric functions
  • 4. Exponential function
  • 5. Logarithm
  • 6. The arctangent
  • 7. The binomial expansion
  • XV Series
  • 1. Convergent series
  • 2. Series with positive terms
  • 3. The integral test
  • 4. Absolute convergence
  • 5. Power series
  • 6. Differentiation and integration of power series
  • Appendix 1. ? and ?
  • 1. Least upper bound
  • 2. Limits
  • 3. Points of accumulation
  • 4. Continuous functions
  • Appendix 2. Physics and Mathematics
  • Answers
  • Supplementary Exercises
  • 1. Definition of inverse functions
  • 2. Derivative of inverse functions
  • 3. The arcsine
  • 4. The arctangent
  • VIII Exponents and Logarithms
  • 1. The logarithm
  • 2. The exponential function
  • 3. The general exponential function
  • 4. Order of magnitude
  • 5. Some applications
  • IX Integration
  • 1. The indefinite integral
  • 2. Continuous functions
  • 3. Area
  • 4. Upper and lower sums
  • 5. The fundamental theorem
  • 6. The basic properties
  • X Properties of the Integral
  • 1. Further connection with the derivative
  • 2. Sums
  • 3. Inequalities
  • 4. Improper integrals
  • XI Techniques of Integration
  • 1. Substitution
  • 2. Integration by parts
  • 3. Trigonometric integrals
  • 4. Partial fractions
  • XII Some Substantial Exercises
  • 1. An estimate for (n!)1/n
  • 2. Stirling’s formula
  • 3. Wallis’ product
  • XIII Applications of Integration
  • 1.Length of curves
  • 2. Area in polar coordinates
  • 3. Volumes of revolution
  • 4. Work
  • 5. Moments

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