Computer-Supported Calculus

Das Symbolische Rechnen zur Losung mathematischer Probleme erlebt einen neuen Aufschwung durch den Einsatz speziell dafur entwikckelter Softwaresysteme. Nach dem Erfolg von Mathematica und Maple erobert nun MACSYMA den Markt. MACSYMA wurde seit den 60er Jahren am MIT entwikckelt und immer weiter verbessert.

Functions, limits, and continuity.- 1 Functions.- 1.1 Introduction.- 1.2 Functions and their graphs.- 1.3 Polynomials.- 1.4 Rational functions.- 1.5 Inverse functions.- 2 Elementary functions used in calculus.- 2.1 Exponential and logarithmic functions.- 2.2 Trigonometric functions.- 2.3 Inverse trigonometric functions.- 2.4 Hyperbolic functions.- 2.5 Inverse hyperbolic functions.- 3 Limits and continuity.- 3.1 Limits.- 3.2 One-sided limits.- 3.3 Infinite limits.- 3.4 Continuous functions.- 3.5 Continuous functions on closed intervals.- 3.6 Proofs.- Derivatives.- 4 Differentiation.- 4.1 Tangency.- 4.2 Differentiability.- 4.3 Derivative function.- 4.4 Special derivatives.- 4.5 Rectilinear motion and velocity.- 4.6 Approximations.- 4.7 Higher derivatives.- 4.8 Acceleration.- 5 Differentiation rules.- 5.1 Product and quotient rules.- 5.2 Chain rule and implicit differentiation.- 5.3 Rates of change.- 5.4 Derivatives of inverse functions.- 6 Extremum problems.- 6.1 Terminology.- 6.2 Necessary condition for a local extremum.- 6.3 First-derivative test.- 6.4 Second-derivative test.- 6.5 Optimal inventory.- 6.6 Convexity.- 6.7 Analysis of graphs.- 6.8 Proofs.- 7 Mean value theorem.- 7.1 Mean value theorem.- 7.2 Rule of l''Hospital.- 7.3 Taylor theorem.- 7.4 Antiderivatives.- 7.5 Iterative methods.- 7.6 Newton method.- 7.7 Fixed points.- 7.8 Proofs.- Integrals.- 8 Definite integrals.- 8.1 Introduction.- 8.2 Riemann sums.- 8.3 Definite integral.- 8.4 Numerical integration: trapezoid method.- 8.5 Numerical integration: Simpson method.- 8.6 Proofs.- 9 Fundamental theorem of calculus.- 9.1 Indefinite integral.- 9.2 Position and distance from velocity.- 9.3 Fundamental theorem of calculus.- 9.4 List of integrals.- 9.5 Proofs.- 10 Integration techniques.- 10.1 Changing variables.- 10.2 Integration by parts.- 10.3 Rational functions.- 10.4 Improper integrals: infinite intervals.- 10.5 Improper integrals: unbounded integrands.- 11 Applications of integrals.- 11.1 Area.- 11.2 Area by polar coordinates.- 11.3 Arc length.- 11.4 Volume.- 11.5 Solids of revolution: volume.- 11.6 Solids of revolution: surface area.- 11.7 Moments and centroids.- 11.8 Centroids of three-dimensional bodies.- 11.9 Work.- 11.10 Hydrostatic force.- Series and approximations.- 12 Sequences and series.- 12.1 Sequences and convergence.- 12.2 Series.- 12.3 Convergence criteria for series.- 12.4 Proofs.- 13 Series expansions and approximations.- 13.1 Series of functions.- 13.2 Power series.- 13.3 Differentiation of power series.- 13.4 Taylor series.- 13.5 Lagrange interpolation.- 13.6 Proofs.- Appendixes.- A Introduction to MACSYMA.- A.1 MACSYMA inputs and outputs.- A.2 Getting on-line help.- A.3 Expressions.- A.4 Constants.- A.5 Numbers.- A.6 Assignments.- A.7 Equations.- A.8 Functions.- A.9 Lists.- A.10 Expanding expressions.- A.11 Simplifying expressions.- A.12 Factoring expressions.- A.13 Making substitutions.- A.14 Extracting parts of an expression.- A.15 Trigonometric functions.- A.16 A simple program.- A.17 Plotting.- B Numbers.- B.1 Arithmetic operations.- B.2 Real numbers.- B.3 Absolute value.- B.4 Equations and inequalities.- B.5 Two fundamental properties of real numbers.- B.6 Complex numbers.- C Analytical geometry.- C.2 Lines.- C.3 Circles.- C.4 Sine, cosine, and tangent.- C.5 Polar coordinates.- D Conic sections.- D.l Conic sections.- D.2 Circle.- D.3 Parabola.- D.4 Ellipse.- D.5 Hyperbola.