Sergei Yakovenko’s Weblog

Monday, February 13, 2012

החזרת תרגיל בית מס’ 4

שלום לכולם,

התרגילים הבדוקים נמצאים בתוך מעטפה חומה שבתיבת דואר שלי בבניין זיסקינד.

בברכה,
דימה

לינק

בהצלחה לכולם!

The last effort

Here you can find the problems for the take-home exam. The rules of the game are outlined in the preamble, I copy them here for your convenience.

The following problems are suggested for the home exam, to be submitted no later than by March 8, 2012. Almost each problem consists of several subproblems, arranged in a specific order. This order is not accidental and should be considered as an implicit hint: solutions of subsequent problems are based on the preceding ones. Please take care to avoid the words “obvious”, “clearly” etc., use as few “plain” words as possible and instead write the intermediate assertions in a closed and precise form using the quantifiers and standard set theoretic notations.

The problems have different complexity: some are easier, some require additional ideas, but none of them is “computational”: if your solutions involves too many identical transformations and/or other computations, have a second look, whether you indeed answer the question that was asked, or something different.

To get the full score 100, it is not necessary to solve all problems and answer all questions: the grade will be awarded based on your demonstrated understanding of mathematics and not on your familiarity with some theorems.

Don’t forget to consult the lecture notes: sometimes you may find useful hints or examples there.

For your convenience Dima will soon post the Hebrew translation of these problems.

Don’t hesitate to ask questions in the comment field: we’ll try to answer them to the extent permissible for an independent home assignment

Good luck!

UPD (Feb 06, 2012, 8:30 am) A small correction of Problem 9 made (sign corrected + more accurate wording).

Integral: antiderivative and area

The last lecture (only partially exposed in the class) deals with the two seemingly unrelated problem: how to antidifferentiate functions (i.e., how to find a function when its derivative is known) and how to compute areas, in particular, under the graph of a given nonlinear function.

The answers turn out to be closely related by the famous Newton-Leibniz formula, which expresses the undergraph area through the antiderivative (primitive) of the function.

We discuss some tricks which allow to read the table of the derivatives from right to left (how to invert the Leibniz rule?) and find out that not all anterivatives can be “explicitly computed”. This “non-computability”, however, has its bright side: among “non-computable” antiderivatives we find functions which possess very special and useful properties, like the primitive of the power $x^{-1}=\frac1x$, which transforms multiplication into addition.

The lecture notes are available here.

Monday, January 23, 2012

Incomplete posting: differentiability of maps and functions

Filed under: Uncategorized — Sergei Yakovenko @ 8:33

Towards the next lecture on Jan 24, 2012, the notes for this and the previous lecture. It will be completed and uploaded once the subject is treated in full, then the link will be replaced.

Sunday, January 22, 2012

תרגיל בית מס’ 4

שלום,

מצורף קישור לתרגיל בית רביעי ואחרון.

תאריך הגשה: 31 בינואר. הפעם לא יתקבלו איחורים.

בהצלחה,

דימה

Tuesday, January 17, 2012

פתרון של תרגיל בית מס’ 2

בלינק זה תמצאו את הפתרון של תרגיל בית מס’ 2

בברכה,

דימה

Properties of continuous functions. Basic notions of topology

The standard list of properties of functions, continuous on intervals, includes theorems on intermediate value, on boundedness, on attainability of extremal values etc.

We explain that these results are manifestations of the following phenomenon. There are several properties of subsets of $\mathbb R^1$ (and, in general, arbitrary subsets of the Euclidean space), which can be defined using only the notions of limit and proximity. Such properties are called topological. Examples of such properties are openness/closeness, connectedness (arc-connectedness) and compactness.

The general principle then can be formulated (vaguely) as follows: the topological properties are preserved by continuous maps (or their inverses).

Wednesday, December 28, 2011

תרגיל בית מס’ 3

אין זמן למנוחה, ועל כן מוגש בפניכם תרגיל בית מס’ 3.

שימו לב כי יש לכם את כל הכלים לפתור את התרגיל, למרות שהוא עלול להיראות קשה ממבט ראשון.

תאריך אחרון להגשה: 10 בינואר.

בהצלחה!

Continuity and limits of functions of real variable

In the first lecture we introduce the notion of continuity of a function at a given point in its domain and a very close notion of a limit at a point outside of the “natural” domain.

This notion is closely related to the notion of sequential limit as introduced earlier. This paves a way to generalize immediately all arithmetic and order results from numeric sequences to functions.

The novel features involve one-sided limits, limits “at infinity” and continuity of composition of functions.

The (unfinished) notes, to be eventually replaced by a more polished text, are available here: follow the updates, this temporary link will eventually be erased.

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