Author | Lang, Serge. author |
---|---|

Title | Short Calculus [electronic resource] : The Original Edition of "A First Course in Calculus" / by Serge Lang |

Imprint | New York, NY : Springer New York : Imprint: Springer, 2002 |

Connect to | http://dx.doi.org/10.1007/978-1-4613-0077-9 |

Descript | XII, 260 p. online resource |

SUMMARY

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 various ways over the past four decades, it has been found worthwhile to make the original edition available again. The audience consists of those taking the first calculus course, in high school or college. The approach is the one which was successful decades ago, involving clarity, and adjusted to a time when the students'background was not as substantial as it might be. We are now back to those times, so it's time to start over again. There are no epsilon-deltas, but this does not imply that the book is not rigorous. Lang learned this attitude from Emil Artin, around 1950

CONTENT

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 -- 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. Stirlingโ{128}{153}s formula -- 3. Wallisโ{128}{153} product -- XIII Applications of Integration -- 1. Length of curves -- 2. Area in polar coordinates -- 3. Volumes of revolution -- 4. Work -- 5. Moments -- XIV Taylorโ{128}{153}s Formula -- 1. Taylorโ{128}{153}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

Mathematics
Functions of real variables
Mathematics
Real Functions