# Sergei Yakovenko's blog: on Math and Teaching

## Monday, February 13, 2012

### החזרת תרגיל בית מס’ 4

שלום לכולם,

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

בברכה,
דימה

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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.

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