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| Date | Sections | What is covered. Comments | Homework assignments |
| 1/24 | s. 1.1 | Logic, quantifiers. | Read s.1.1, 1.2, p. 7 #1 a, c, #2 a, c, d, # 3 b |
| 1/26 | s. 1.2 | Functions. Comparison of infinite sets. Countable and uncountable sets | p. 13 #1, 2, 4, |
| 1/29 | s. 1.2 | Properties of countable sets | |
| 1/31 | Ch. 2 | Construction of real numbers: infinite decimal (binary)
fractions and Dedekind cuts.
Read carefully s. 2.1.1 and 2.4.2 Read over the rest of Ch. 2, but do not bother much with Cauchy sequences, we will return to this topic later. |
p. 68 # 1, 2 (not collected now) To be collected Fri. 2/2: p 13 # 4 (find a one line proof), p. 7 # 1 a, 2d |
| 2/2 | s. 3.1.1 | Dedekind cuts (continuation). Completeness of reals. Supremum, infimum. | p. 84 # 1 (sup, inf only), # 4 Also read the handout about Dedekind cuts |
| 2/5 | s. 3.1.1- 3.1.2 | Definition of limit. Main properties | Prove that the sequence (-1)-n does not have a limit. (will be collected Fri,. 2/9) Read handout #4 about arithmetic operations with limits. |
| 2/7 | See handout # 4 | Properties of limits. Completeness and Cauchy sequences. Limsup and liminf | Prove Pinching Principle. p. 84 #
2, 5 |
| 2/9 | Cauchy sequences and completeness (continued) | ||
| 2/12 | Complete metric spaces. Completion. Equivalence relations, quotient spaces. | p. 37 # 3, 5, 6, p. 55 # 3 (use completeness of reals and Cauchy sequences) | |
| 2/14 | See Handout 4 | Working with limits. | Click here to get the assignment (PDF, 36K). This assignment will be collected Fri. 2/16 |
| 2/16 | S. 3.2 | Subsequences, limit points. Open sets | Click here to get the assignment (PDF, 36K). This assignment will be collected Fri 2/23 |
| 2/19 | President's day. | ||
| 2/21 | s 3.2 | Closed sets, accumulation points. | p. 98 # 8, 10, 12, 12, 13 |
| 2/23 | s. 4.1 | Continuous functions | Click here to get the assignment (PDF, 29K). |
| 2/26 | s. 4.1, 9.2 | Inverse images of open sets. Relative topology. | p 126 # 3, 14, p. 139 # 12, p. 384 # 1 |
| 2/28 | s. 4.1 | Limit of a function. Connection with limit of sequences | Click here to get the assignment (PDF, 30K). |
| 3/2 | s. 4.1 | Composition of functions. Putting limit inside the function. | Click here to get the assignment (PDF, 33K). |
| 3/5 | s. 4.2 | Intermediate value theorem. | p. 138 # 3 (ignore the question about open intervals), 4, 7, 8, 17. Will be collected Fri 3/9 |
| 3/7 | s. 4.2 | Compactness | p. 106 # 1, 6, p 126 # 8, p. 138 # 11, 15 |
| 3/9 | s. 4.2 | Compactness, continued. Uniform continuity. | Work out examples of continuous but not uniformly continuous functions. p. 107 # 7, 8 |
| 3/12 | |||
| 3/14 | Midterm | ||
| 3/16 | s. 5.1 | Derivatives. Definitions and main properties. O and o notation. | p. 152 # 1-4, 7. Compute (first take limit as k goes to infinity, and then as n goes to infinity) |
| 3/19 | s. 5.1, 5.2 | Maximum and minimum. Mean value theorem | p. 164 # 2, 6, 10, 11, 12 This assignment will be collected Fri 3.23 |
| 3/21 | s. 5.2 | Mean value theorem (continued) | Read s. 5.3 (calculus of
derivatives). p. 164 # 3, 13, p. 176 # 1, 2, 3. |
| 3/23 | s. 5.4 | Taylor's formula | p. 176 # 3, 4, 5, 15, p. 192 # 2, 3, 8, 16 |
| 3/26 |
Spring recess |
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| 3/28 | |||
| 3/30 | |||
| 4/2 | s. 5.4 | Lagrange's remainder theorem | Click here to get the assignment (PDF, 41K). This assignment (except problem #1) will be collected Fri 4/6 |
| 4/4 | s. 5.4 | Applications of the Taylor's formula and Lagrange remainder theorem | Click here to get the assignment. This assignment will be collected Fri 4/6 |
| 4/6 | s. 5.4 | Prove second mean value theorem,
and get the L'Hopitalle's rule (case 0/0) from it. p. 194 # 19, 20 |
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| 4/9 | s. 6.1, 6.2 | Riemann Integral, Definition | p. 217 # 2, p. 231 # 1, 3, 4, 6 Collected Fri 4/13 |
| 4/11 | s. 6.1 | Fundamental theorem of calculus, properties of integral | |
| 4/13 | s. 8.1 | Logarithm defined as integral | p. 232 # 9, p 235# 3, p. 335 # 6, 7, p. 218 # 5 |
| 4/16 | s 7.1, 7.2 | Series, definitions, convergence, Cauchy criterion | p. 262 # 1, 2, 3, 9, 13, 14 This assignment except # 9 will be collected Fri 4/20 |
| 4/18 | s. 7.1,7.2 | Absolutely convergent series, reordering. Tests of convergence | Estimate the minimal number N such
that  |
| 4/20 | Take home part of the test 2 will be given | ||
| 4/23 | Test 2 in class part | p. 274 # 2, 3, 5, 7, 11, 13 a Will be collected Fri 4/27 | |
| 4/25 | s. 7.3 | Integration and differentiation of sequences and series. | |
| 4/27 | s. 7.4 | Power Series | |
| 4/30 | |||
| 5/2 | |||
| 5/4 | |||
| 5/7 | |||
| 5/9 | |||