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… Lue lisää 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.