CORE 001: Single-Variable Calculus
Syllabus

The topic list for this project is: sequences and limits, the derivative and differentiation rules, applications of differentiation, the integral and methods of integration, applications of integration, differential geometry, infinite series and convergence, improper integrals and indeterminate forms, and power series and Fourier series.

Prerequisite: It is assumed that you are familiar with both techniques of proof and the principles of algebra, geometry, and trigonometry. If you are not completely comfortable with the concept of proof, then take MATH R002 (Geometry), if you are not familiar and comfortable with algbebra take MATH R003 (Algebra), and if you are not comfortable with trigonometry take MATH R004 (Trigonometry).

Instructor: George E. Hrabovsky, george@madscitech.org, 608-276-6832.

Task #1: Start and keep a notebook for your study. This should be bound and have at least 300 sheets. You may need more than one notebook of this size. Smaller notebooks than 300-sheets can be used, but the total number of sheets should be at least 300. Each set of 300 pages started and completed is worth a point towards your final total of 4. To begin your notebook you will need a list of topics. The one listed below is only one possible choice. This choice is the default. Any choice other than this one must be approved by your instructor.

Procedure for the Course

If a topic from the list below is underscored that means there is some resource material for it. If there is no resource material for it then you must develop that for yourself.

It is expected that you will develop one or more questions for each topic. Questions can be of the form who, what, when, where, why, and how.

Once you have written down a set of questions for a topic, you either answer each of these qurestions or you explain how you attempted to answer the question and failed. Don't be alarmed; even some elementary questions resist answering. You can learn a lot just by making the effort.

The next step is to ask a set of new questions based on your previous attempts at answering your first set of questions (this can include those questions you were unable to answer before). Answer each of those questions as best you can and create another set of questions for each answer. Answer each of those to the best of your ability and ask another set of questios for each, but do not answer them right away. If you are really interested in one or more of these questions attempt to answer them in a, "topic of personal interest," session; or you may answer them in a personal research project.

Wherever possible give at least three examples of any definition, principle, or procedure.

This course will require a little more than two pages of notes for each topic to fill a 300 page notebook.

1. The nature of single-variable calculus
2. Pure single-variable calculus
3. Applied single-variable calculus
4. Sequences
5. Limits of sequences
6. Limits of functions
7. One-sided limits
8. Arithmetic of limits
9. Calculating limits
10. Continuity
11. The intermediate value theorem
12. The extreme value theorem
13. Infinite limits
14. Limits at infinity
15. Topic of Personal Interest (including, but not limited to 5 practice problems, monotonic sequences, completeness, the formal definition of the limit, discontinuity, sufficient conditions for continuity, continuity of trigonometric functions, uniform continuity, infinite limits at infinity, asymptotes, subsequences, the Bolzano-Weierstrass theorem, the Cauchy criterion)
16. Topic of Personal Interest.
17. Topic of Personal Interest.
18. Review of topics to date
19. Derivatives
20. Computing derivatives
21. Higher order derivatives
22. Sum, difference, product, quotient, and power rules
23. The chain rule
24. Implicit differentiation
25. Trigonometric derivatives
26. Inverse trigonometric derivatives
27. The exponential function
28. The logarithmic function
29. Logarithmic differentiation
30. Hyperbolic functions
31. Inverse hyperbolic functions
32. Differentials
33. Topic of Personal Interest (including, but not limited to 5 practice problems, tangent and normal lines to curves, differentiation operators, differentiability and continuity, partial derivatives, affine approximations, differentiation of complex functions, convex and concave functions)
34. Topic of Personal Interest.
35. Topic of Personal Interest.
36. Review of topics 17-33
37. Review of topics to date
38. Related rates
39. Maxima and minima
40. Curve sketching
41. Newton's method
42. Taylor polynomials
43. Mean value theorem
44. The linear approximation
45. Optimization
46. Antiderivatives
47. Topic of Personal Interest (including, but not limited to 5 practice problems, direction fields, velocity and acceleration, Rolle's theorem, order of magnitude)
48. Topic of Personal Interest.
49. Topic of Personal Interest.
50. Review of topics 36-46
51. Review of topics to date
52. Indefinite integrals
53. Definite integrals
54. The fundamental theorem of calculus
55. Integration by substitution
56. Integration by parts
57. Integration by partial fractions
58. Numerical integration
59. Topic of Personal Interest (including, but not limited to 5 practice problems, construction of indefinite integrals, Riemann sums, integration of trigonometric functions, trigonometric substitutions, fractional power substitutions, Simpson's rule and error estimates, integrability, open and closed sets, compact sets)
60. Topic of Personal Interest.
61. Topic of Personal Interest.
62. Review of topics 49-57
63. Review of topics to date
64. Area
65. Arc length
66. Volume
67. Average values
68. Topic of Personal Interest (including, but not limited to 5 practice problems, the area between curves, volume by disks, volume by washers, volume by shells, volume by cross section, surfaces of revolution, probability distributions and densities, the natural logarithm, exponential functions, exponential growth and decay, moments and centroids)
69. Topic of Personal Interest.
70. Topic of Personal Interest.
71. Review of topics 60-66
72. Review of topics to date
73. Parametric curves
74. Arc length as a parameter
75. Curvature
76. Topic of Personal Interest (including, but not limited to 5 practice problems, plane curves, change of parameter, radius and circle of curvature, center of curvature, evolutes, curves in polar coordinates)
77. Topic of Personal Interest.
78. Topic of Personal Interest.
79. Review of topics to date
80. Infinite series
81. Convergence tests
82. Arithmetic of infinite series
83. Topic of Personal Interest (including, but not limited to 5 practice problems, Zeno's paradox, sums of sequences, difference equations, integral test, comparison test, limit comparison test, estimation of the sum of a series, alternating series, absolute convergence, ratio test, root test, infinite products, nonlinear difference equations)
84. Topic of Personal Interest.
85. Topic of Personal Interest.
86. Review of topics to date
87. Improper integrals
88. Indeterminate forms
89. Topic of Personal Interest (including, but not limited to 5 practice problems, convergence and divergence, indeterminate products, indeterminate differences, indeterminate powers)
90. Topic of Personal Interest.
91. Topic of Personal Interest.
92. Review of topics to date
93. Power series
94. Representing functions as power series
95. Taylor series
96. Fourier series
97. Topic of Personal Interest (including, but not limited to 5 practice problems, the expansion of the logarithm, estimate of the remainder, uniform convergence, differentiation and integration of power series, multiplication and division of power series, Abel's theorem, radius of convergence, binomial series, power series with complex terms, periodic functions, complex notation for trigonometric polynomials, convergence of Fourier series, Weierstrass approximation)
98. Topic of Personal Interest.
99. Topic of Personal Interest.
100. Review of topics to date

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