The purpose of this course is to build rigorous mathematical theory for the fundamental calculus concepts, sequences and limits, continuous functions, and derivatives. We will rely on our intuition from calculus, but (unlike calculus) the emphasis will be not on calculations but on detailed understanding of concepts and on proofs of mathematical statements.
Final Exam is cumulative and covers all of the semester material, EXCEPT series. Series will not be on the test. Final exam checklist (with some extra practice questions) is here.
Solutions to midterm problems: Midterm I and Midterm II.
Exam II covers Bolzano-Weierstrass theorem, continuous functions and their properties, and limits of functions. A more detailed checklist is here. For practice questions, see supplementary notes on limits (below). Extra questions: 17.7, 17.9, 17.10, 17.13, 18.2, 18.4, and examples and exercise in section 20. The best preparation, as usual, is to go over past homeworks. (Don't look anywhere and see if you can solve an old hw question. Check solutions, notes, textbook etc. afterwards.) Also, go over your class notes/textbook and see if you understand proofs of all theorems. (Ask yourself in the middle of proof what the next step should be.)
Exam I covers all the material studied so far both lectures
and homework. A more detailed checklist, with
additional practice questions, is here.
On exam, you can refer to any results proved in the lectures or the homework.
(Please try to make clear references - eg "limit laws", "definition of supremum", or "monotone convergence
theorem"; if the result has no name, state or describe it clearly.) In some questions, you may be required to
argue from the definitions. (This means that you cannot use theorem but have to work from scratch.)
Any facts not proved in class or homework (for example, the Squeeze theorem) cannot be used.
The exam will be closed book, closed notes. No calculators, only pens, pencils allowed on desk.
A week-by-week schedule with homework and reading assignments will be posted here as the course progresses.
Week 13 (04/22 – 04/27). Theorem: if f is differentiable at x and
has a
local maximum or minimum at x, then f'(x)=0. The mean value theorem (to be continued next week).
Read section 29 (pp. 213-217 only).
Homework 11, due May 3 pdf . Solutions.
Week 12 (04/15 – 04/20). Derivatives.
Read section 28.
Homework 10, due Apr 26 pdf . Solutions
Week 11 (04/8 – 04/13). Limits of functions.
Read section 20 and supplementary notes . Make sure you can do the exercises in the notes and
fill in the gaps! It's a great review not only for limits, but for continuity as well.
Week 10 (03/26 – 03/30) . More on
continuous functions: important theorems about functions on closed interval.
Read section 18.
Homework 9, due Apr 12 pdf .
Solutions
Week 9 (03/19 – 03/23) . More on
continuous functions.
Finish reading section 17.
Homework 8, due Mar 29 pdf . (Updated Sat, Mar 24)
Solutions
Week 8 (03/12 – 03/16)
Bolzano-Weierstrass
thm (every bounded sequences contains a convergent subsequence). Continuous functions.
The sequences definition and the epsilon-delta definition.
Read section 17, pp 115-118, carefully. The rest of the section will be covered next week.
Also, review Bolzano-Weierstrass theorem in section 11.
Homework 7, due Mar 22 pdf .
Solutions
Week 6 (02/27 – 03/02) Increasing and
decresing sequences, convergence of monotone bounded sequence. Series. Idea of subsequences.
Read section 10 (p.55 only) and section 11 (up to 11.3) in the textbook, as well as
Class notes on series.
Homework 6, due Wednesday, Mar 7 pdf . Solutions
Week 5 (02/20 – 02/24) Axioms and properties of
natural, rational, and real numbers. Algebraic axioms, order axioms. Completeness of reals.
We covered roughly the following bits and pieces: section 1, pp.1-3; section 2 pp. 6-7 (plus a different proof of
irrationality of square roots of 2 and 3); section 3 pp.12-15 (we didn't prove all the parts in 3.1 -
please read them!) section 4 (we did a simpler version of the Archimedean property).
Section 6 was only briefly touched upon and is
optional. If you feel motivated, reading Chapter 1 in its entirety is a good idea.
Homework 5, due Mar 1 pdf
Solutions
Week 4 (02/13 – 02/17) Bounded and Unbounded sets and
sequences. Thm: a convergent sequence is bounded. Infinite limits.
Remaining parts of section 9, definition 4.2 in section 4.
Homework 4, due Feb 23 pdf Solutions
Week 3 (02/06 – 02/10) Arithmetics
of limits (limit laws). Proof of limit laws, their applications.
Read section 9 of the textbook.
Homework 3, due Feb 16 pdf Solutions
Week 2 (01/30 – 02/03) Sequences and their limits:
epsilon-definition, calculations and estimates.
Read section 8, begin reading section 9 of the textbook.
Homework 2, due Feb 9 pdf Solutions
Week 1 (01/23 – 01/27) Sequences and their limits:
intuition, neighborhoods and tails, epsilon-definition.
Class notes (discussion of limits).
Read them. Please also read sections 7 and 8 of the textbook.
Homework 1, due Feb 2 pdf Solutions
Important: Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. You are welcome to collaborate with others and to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed.
Students with Disabilities: If you have a physical,
psychological, medical, or learning disability that may impact on your
ability to carry out assigned course work, you are strongly urged to
contact the staff in the Disabled Student Services (DSS) office: Room
133 in the Humanities Building; 632-6748v/TDD. The DSS office will
review your concerns and determine, with you, what accommodations are
necessary and appropriate. A written DSS recommendation should be
brought to your lecturer who will make a decision on what special
arrangements will be made. All information and documentation of
disability is confidential. Arrangements should be made early in the
semester so that your needs can be accommodated.