Math 401/501 - Advanced Calculus I

Time:  Tuesday & Thursday 11 a.m. - 12:15 p.m. (Lectures), Wednesday 9 - 9:50 a.m. (Recitations),   MITCH 221
 
Lecturer:  Anna Skripka,  skripka [at] math [dot] unm [dot] edu
Recitations TA:  Chamsol Park,  parkcs [at] math [dot] unm [dot] edu
 
Topics:  Real numbers, sequences, series, limits, functions of one variable, continuity, differentiation, integration. Chapters 1-6 of Wade's book.
Prerequisites:  Math 264 and two math courses 300-level or above are required; in particular, Math 327, 321, 313 are recommended
Goals:  Students will learn how to construct rigorous proofs, which show comprehension of formal definitions and stated hypotheses; form logical conclusions; construct counterexamples; provide theoretical justifications for all calculations.
 
Textbook:  W.R. Wade, An Introduction to Analysis, Prentice Hall; 4th edition, 2009, ISBN-10: 0132296381.
Supplementary books:  T. Tao, Analysis I, Hindustan Book Agency, 2nd edition, 2009, ISBN-10: 8185931941, related website;
J.L. Taylor, Foundations of Analysis, AMS, 2012, ISBN-10: 0821889842.
 
Office hours:
Monday 1:30 - 2:30 p.m., SMLC 346
Tuesday 1:15 - 2:30 p.m., SMLC 212
Wednesday 1:30 - 2:30 p.m., SMLC 346
Thursday 1:15 - 2:30 p.m., SMLC 212
Friday 11 a.m. - noon, SMLC 346
 
Grades:  25% homework, 25% each of two midterm exams, 25% final exam.
 
TENTATIVE Schedule (will be updated at the end of each week)
 
Week
Important Dates
Sections
Topics
Homework
1
Jan 17, 18, 19
1.1, 1.2, 1.3
Real number system. Ordered field axioms. Supremum, infimum.
#1.2.0-1.2.3; 1.2.4(a,b,d,e); (a,c) of 1.2.5, 1.2.7, 1.2.8
2
Jan 24, 25, 26
1.3, 1.4
Properties of sup, inf. Density of rationals. Mathematical induction.
#1.3.0-1.3.8, 1.3.10, 1.3.11; 1.4.2-1.4.5, 1.4.8
submit for grade on Feb 2: #1.3.0(c), 1.3.6(a), 1.3.7(b), 1.3.10, 1.4.4(b)
3
Jan 31, Feb 1, 2
Drop with no grade: Feb 3
1.4, 1.5
Binomial formula. Inverse functions. Images and preimages. DeMorgan's Laws.
#1.5.1, 1.5.2(b,d), 1.5.3(a,b,c,d), 1.5.4, 1.5.5, 1.5.6(b), 1.5.7
submit for grade on Feb 9: #1.4.3(c), 1.4.8, 1.5.1(a,b) for the function in ε), 1.5.5(iii), 1.5.7 (a)=>(b)
4
Feb 7, 8, 9
1.6
Countable and uncountable sets.
#1.6.0, 1.6.1, 1.6.2, 1.6.4, 1.6.5
submit for grade on Feb 16: #1.6.0(c,d), 1.6.1, 1.6.3, 1.6.5(a)
5
Feb 14, 15, 16
2.1, 2.2
Limits of sequences and their properties.
#2.1.0-2.1.8, 2.2.0-2.2.9
submit for grade on Feb 23: #2.1.0(a,c), 2.1.2(c), 2.2.2(b), 2.2.3(d), 2.2.6
6
Feb 21, 22, 23
2.3, 2.4
Monotone convergence and Bolzano-Weierstrass theorems. Cauchy sequences.
#2.3.0, 2.3.1, 2.3.5-2.3.7, 2.4.0, 2.4.2-2.4.5, 2.4.7, 2.4.8
7
Feb 28 (Exam 1 on 1.1-2.2), Mar 1, 2
2.5, 3.1
lim sup, lim inf. Limits of functions.
#2.5.1, 2.5.7
submit for grade on Mar 9: #2.3.5, 2.3.6, 2.4.3(c), 2.4.5, 2.5.7
8
Mar 7, 8, 9
3.1, 3.2, 3.3
Properties of limits of functions. Continuity. Extreme Value Theorem.
#3.1.0, 3.1.1, 3.1.4-3.1.6, 3.2.1-3.2.3, 3.2.7, 3.3.1
submit for grade on Mar 23: #3.1.1(a), 3.1.5, 3.1.6, 3.2.1(b), 3.3.1(d)
Mar 12-19: Spring Break
9
Mar 21, 22, 23
3.3, 3.4, 4.1, 4.2
Intermediate Value Theorem. Uniform continuity. Differentiability.
#3.3.0(a,b,c), 3.3.2-3.3.5, 3.4.1, 3.4.2, 3.4.4, 3.4.5(a,b), 3.4.6, 4.1.0(a,b), 4.1.2, 4.1.4, 4.1.8, 4.2.2, 4.2.4-4.2.6
submit for grade on Mar 30: #3.3.0(a), 3.3.4, 3.4.2(c), 3.4.5(b), 4.1.4
10
Mar 28, 29, 30
4.1-4.4
Mean Value and Taylor's theorems. L'Hoptial's rule.
#4.3.1(a,b)-4.3.6, 4.4.1-4.4.3, 4.4.5(a-f)
submit for grade on Apr 13: #4.3.1(b), 4.3.4, 4.3.5, 4.4.2, 4.4.5(f)
11
Apr 4, 5, 6 (Exam 2 on 2.3-4.2)
5.1
Review. Riemann integrable functions.
12
Apr 11, 12, 13
Withdrawal deadline: Apr 14
5.1, 5.2
Riemann integral and its properties.
#5.1.0(a,b,c), 5.1.2(a,b,c(α)), 5.1.4, 5.2.0(a,b,c), 5.2.3
13
Apr 18, 19, 20
5.2, 5.3
Fundamental Theorem of Calculus. Integration by parts. Change of variables.
#5.3.1
submit for grade on Apr 25: #5.1.2(b,c(α)), 5.1.4, 5.2.3(b),5.3.1(b)
14
Apr 25, 26, 27
6.1-6.4
Series.
#6.1.1, 6.1.3, 6.4.1(c,d,e); (a-c) of 6.1.0, 6.1.2, 6.2.1, 6.2.2, 6.3.1, 6.4.2
submit for grade on May 4: #6.1.3(c), 6.4.1(c), 6.3.1(b), 5.2.7, 3.3.3
15
May 2, 3, 4
Series. Review.
Questions and exercises
16
May 9 (Tuesday), 12:30-2:30 p.m.: Final Exam
 
Make up policy: There will be no late homework accepted. A make-up exam will be given or the final exam grade will be used as a substitute grade for a missed midterm in case of a documented absence prescribed by the university (family emergency, serious medical problem, official UNM function).
 
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