Computer-Supported Calculus

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.