Short Calculus

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. 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.- 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.