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 |
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. Stirlingโs formula -- 3. Wallisโ product -- XIII Applications of Integration -- 1. Length of curves -- 2. Area in polar coordinates -- 3. Volumes of revolution -- 4. Work -- 5. Moments -- XIV Taylorโs Formula -- 1. Taylorโs 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