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Solved tasks. Free Solutions to Problems in Higher Mathematics

Solved tasks. Free Solutions to Problems in Higher Mathematics

Soon the fairy tale is told, but not soon the deed is done

Unfortunately, new teaching materials appear on the site not as fast as many visitors would like. No wonder, because creating a quality lesson takes a lot of time. But what to do? Here you have a topic or a separate task that you did not find on my resource .... And you need to figure it out today!

Therefore, as the project developed, there was a need to create bank of ready solutions in various branches of higher mathematics. And this page should definitely please you, because the bank completely FREE! Right now you have the opportunity to download file archives with solved problems on various tower topics for free.

Quest source?

All tasks are performed personally by me, which guarantees high quality solutions.

A typical archive contains, as a rule, 200-300 tasks with ready-made solutions, and I tried to collect the most diverse materials, including rare tasks and examples of increased complexity. The hodgepodge consists of tests, individual assignments, completed at one time by me to order. In addition, I will please full-time students - the archives include typical calculations from the collection of Kuznetsov, IPD from the collection of Ryabushko, as well as probability theory from the collection of Chudesenko. For the reasons mentioned in the review Useful math websites I'm giving them away for free. Yes, of course, there are not very many calculations, but some students are very lucky!

Only, comrades, let's not complain or resent. Yes, the quality is high, all examples and tasks have been checked by teachers, but there may be typos, incorrect things and even errors in the solutions. Tasks are presented "as is". Doesn't make sense notify me about found jambs, because, firstly, they are technically difficult to fix, and secondly, there is simply no time (one repacking and re-uploading of the archive is worth something). The exception is pdf files that are located directly on the server, please email me if you find a SERIOUS oversight in them.

Perhaps some will have misunderstandings, and even misconceptions will arise. But still, almost all site visitors have a non-zero level of mathematical training, so sample solutions will do much more good than harm; ideal if you are already well versed in a particular topic and want to find additional examples.

Each collection of problems is accompanied by brief comments about the contents of the archive. Moreover, the bank is replenished with new materials from time to time.

In other words, come join us!

Ready-made tasks with solutions on the topic
Linear Algebra and Analytic Geometry

The archive contains about three hundred problems in linear algebra and analytic geometry, including several Kuznetsov's calculations and several Ryabushko's IPDs. Almost all typical problems of algebra have been solved: solving systems of equations, calculating determinants, studying systems for compatibility, problems on determining the rank of a matrix, matrix equations, problems on eigenvalues / eigenvectors. Analytical geometry is represented both by problems on the plane and in three-dimensional space.

Limit Solution Examples

The archive contains a couple of hundred split limits, including sequence limits. Examples of solving the limits with the help of remarkable equivalences are considered. The collection contains 3 calculations by Kuztsev and 3 variants of Ryabushko's IDZ. I won’t say what options exactly - enjoy the lottery game and entertainment! If you have read the existing lessons on the topic, then it will not be difficult to understand many of the limits of the collection.

Differentiation of functions of one variable

The archive is represented mainly by typical calculations from the collection of Kuznetsov (10 options), there are 3 options from Ryabushko. The examples will allow you to significantly improve your differentiation technique, since the derivatives from the Kuznetsov collection are quite heavy. Moreover, some derivatives have to be found in several stages! In addition to derivatives, there are typical tasks: finding derivatives of an arbitrary, “nth” order, examples using the Lagrange formula; tasks for constructing a tangent and a normal, and a few tasks for finding a derivative by defining a derivative.

Function Research

Perhaps the most time-consuming and painstaking selection. The solution contains 69 fully researched functions of one variable, including polynomial functions, fractional rational functions, functions with exponents, logarithms, and other rarer examples. It is quite likely that you will find your own example or a very similar function. The piggy bank acts as an application of the Functions and Graphs theme and is intended for self-examination / training.

Ready-made solutions of indefinite integrals

Mostly, indefinite integrals, but there are few definite integrals. The collection includes more than two hundred integrals, including very complex ones. Considered a lot of examples that did not fit in my lessons! The archive includes 6 standard calculations by Kuznetsov and 2 IDS Ryabushko. Kuznetsov's calculations contain tasks for finding the volumes of bodies, arc lengths and surface areas, moreover, not only in the Cartesian, but also in the polar coordinate system, as well as for parametrically specified functions.

Tasks for numerical and functional series

Another heavy brick. Of course, there are examples of research number series for convergence, moreover, there are creative series that require experience and a non-standard approach. My pride - there was no row that I could not crack. The problems of finding the sum of a series, various tasks for approximate calculations using a series are considered. Function series are represented by a common problem of finding the region of convergence (again with creative cases). At your service is the popular problem of expanding a particular solution of DE in a series. There are about a dozen examples of solutions to Fourier series.
And even the disclosure of uncertainties within with the help of series expansion. The archive contains 13 types of Kuznetsov and a couple of variants of Ryabushko. Bon Appetit!

Ready-made solutions of differential equations

A good hundred diffusers. Various methods of solving are applied in the tasks differential equations of the first order, including those that I did not consider in the lessons. There are diffuses of increased complexity - DE of the second order and higher orders, examples using the method of variation of arbitrary constants. Systems of differential equations are attached. The archive contains 8 standard calculations by Kuznetsov and 2 variants by Ryabushko. In addition, I packed a collection of exam papers, in addition to differential equations, they contain problems for series and multiple integrals.

Examples of solutions to multiple and curvilinear integrals

Double integral, change the order of bypassing the area, calculate the area using a double integral. Triple integral, calculating the volume of a body using a triple integral. Examples of solutions not only in Cartesian, but also in polar, cylindrical coordinate systems. The main applications of multiple integrals: finding the center of gravity of a flat plate, the center of mass of a body, and other problems. Curvilinear integrals along a curve, along a contour. A hit task for finding the work of a force along a contour and using Green's formula.

Do you really believe that it is difficult?!

Examples of solving problems in field theory

Among other things, the collection includes the most popular problems with solutions of the following types: finding the directional derivative and gradient, calculating the flow of a vector field through a part of the plane, calculating the flow of a vector field through a closed surface using the Ostrogradsky-Gauss formula, calculating the circulation of a vector field along a contour. The archive contains several calculations from Kuznetsov's collection.

Examples of problem solving in combinatorics and probability theory

In the following pdf-kah you can find additional tasks for the lessons on combinatorics And probability theory. At the same time, I did not have a goal to collect hundreds of examples, on the contrary - I tried to filter out obvious duplicates. In addition, in most files, tasks are ordered in order of increasing complexity:

Tasks with economic content

Many people know how difficult it is to “swing out” after a long winter vacation, and one day after the holidays I found a great way to get back to my working rhythm - to create collections of ready-made problems with a pronounced economic content, which are mothballed in my collection and which do not benefit anyone. These are some problems of economic and mathematical modeling, these are problems of financial mathematics (interest, rent, etc.), and, finally, these are tasks that are more related to the economy. So let's fix the gloomy January weather with the following useful materials:

Tasks of financial mathematics

In fact, it is too loud - in the next file you will find urgent and practically useful tasks for interest and calculation of payments on deposits / loans.
I retouched the few solutions I found and made them as clear as possible! Moreover, along the way, I figured out something myself =)

And, of course, a calculator for tasks! Yes, at least in order to instantly find the percentage of interest from any number:

Intersectoral balance problem (Leontiev model)

One of the most famous economic and mathematical models that describes many sectors of the economy, part of the products of which are mutually spent as a result of production, and the other part is the final product. Having trouble with direct/total cost matrices, painstaking inverses, and fractional matrix multiplication? The following program will instantly carry out calculations that you just get tired of doing on a regular microcalculator:

This program was created in MS Excel and is available to active users of the library. Or for a moderate fee.

Demo version can be seen. The calculations were carried out for the most common educational case - three industries; Moreover, the decision can be printed and handed over!

And what is especially nice, here is an example of practical use matrix algebras.

Typical tasks for economic indices

The following modest pdf, of course, does not cover the whole variety of economic indices, but I have thoroughly analyzed and systematized, perhaps, all varieties of the most popular task on price indices / physical volume of sales / turnover+ similar task to cost and total cost indices. Even a child will understand:

... and if he also uses with this program, then never make mistakes!

Tasks of economic statistics

As in the previous file, all higher mathematics here is limited to the simplest arithmetic operations =) ... And it would be funny if it were not so sad (Economics is not my profile subject, but, nevertheless, your humble servant once had to torture several dozen tasks on socio-economic and production statistics, which were "appendages" to the control work on mathematical statistics. In this regard, I do not guarantee super quality, although, on the other hand, there is no hack either:

And finally, to the delight of full-time students, I open two more distributions:

Typical calculations on the theory of probability from the collection of Chudesenko for free!

The first 22 tasks were solved, some more, some less. The first option for demonstration is posted directly on the site: The rest of the options (2, 5, 7, 8, 9, 10, 11, 12, 15, 20, 21, 24, 25) due to their light weight, I moved from the file exchanger to the newly created library site. Also, the archive can be useful in that Chudesenko's collection contains quite difficult problems in probability theory, and perhaps you will find what you were looking for.

IDZ from Ryabushko's problem book - for free!

A solid appendage to all archives + tasks that have not yet been proposed on this page, in particular, examples with functions of several variables. Of course, far from all 4 volumes are closed, but in some sections there will be a good half of the options, or even more! Solved individual tasks can be used both "for their intended purpose" and as additional self-training / training. long awaited problem solver Ryabushko received a separate web page.

And a number of works of various universities of the country:

4) MPEI tests(Moscow Economic Institute).

On the this moment that's all for now, check back from time to time - I'll come up with something else!

In this section, we are preparing for the USE in mathematics as a basic, specialized level - we present analysis of problems, tests, a description of the exam and useful recommendations. Using our resource, you will at least understand how to solve problems and be able to successfully pass the exam in mathematics in 2019. Getting started!

The USE in mathematics is a mandatory exam for any student in the 11th grade, so the information presented in this section is relevant for everyone. Mathematics exam is divided into two types - basic and profile. In this section, I provide an analysis of each type of task with a detailed explanation for the two options. The tasks of the exam are strictly thematic, therefore, for each number, you can give precise recommendations and give the theory necessary specifically for solving this type of task. Below you will find links to tasks, by clicking on which you can study the theory and analyze examples. Examples are constantly updated and updated.

The structure of the basic level of the exam in mathematics

The Foundation Level Mathematics Exam consists of one part , including 20 tasks with a short answer. All tasks are aimed at testing the development of basic skills and practical skills in applying mathematical knowledge in everyday situations.

The answer to each of tasks 1–20 is integer, final decimal , or digit sequence .

A task with a short answer is considered completed if the correct answer is recorded in the answer sheet No. 1 in the form provided for in the instructions for completing the task.

Secondary general education

Line UMK G.K. Muravina. Algebra and the beginnings of mathematical analysis (10-11) (deep)

Line UMK Merzlyak. Algebra and the Beginnings of Analysis (10-11) (U)

Maths

Preparation for the exam in mathematics (profile level): tasks, solutions and explanations

We analyze tasks and solve examples with the teacher

The profile-level examination paper lasts 3 hours 55 minutes (235 minutes).

Minimum Threshold- 27 points.

The examination paper consists of two parts, which differ in content, complexity and number of tasks.

The defining feature of each part of the work is the form of tasks:

part 1 contains 8 tasks (tasks 1-8) with a short answer in the form of an integer or a final decimal fraction;
part 2 contains 4 tasks (tasks 9-12) with a short answer in the form of an integer or a final decimal fraction and 7 tasks (tasks 13-19) with a detailed answer (full record of the decision with the rationale for the actions performed).

Panova Svetlana Anatolievna, mathematic teacher the highest category schools, 20 years of work experience:

“In order to receive a school certificate, a graduate must pass two mandatory exams in the form of the Unified State Examination, one of which is mathematics. In accordance with the Concept for the Development of Mathematical Education in Russian Federation The USE in mathematics is divided into two levels: basic and specialized. Today we will consider options for the profile level.

Task number 1- checks the ability of USE participants to apply the skills acquired in the course of 5-9 grades in elementary mathematics in practical activities. The participant must have computational skills, be able to work with rational numbers, be able to round decimal fractions, be able to convert one unit of measurement to another.

Example 1 An expense meter was installed in the apartment where Petr lives cold water(counter). On the first of May, the meter showed an consumption of 172 cubic meters. m of water, and on the first of June - 177 cubic meters. m. What amount should Peter pay for cold water for May, if the price of 1 cu. m of cold water is 34 rubles 17 kopecks? Give your answer in rubles.

Solution:

1) Find the amount of water spent per month:

177 - 172 = 5 (cu m)

2) Find how much money will be paid for the spent water:

34.17 5 = 170.85 (rub)

Answer: 170,85.

Task number 2- is one of the simplest tasks of the exam. The majority of graduates successfully cope with it, which indicates the possession of the definition of the concept of function. Task type No. 2 according to the requirements codifier is a task for using acquired knowledge and skills in practical activities and Everyday life. Task No. 2 consists of describing, using functions, various real relationships between quantities and interpreting their graphs. Task number 2 tests the ability to extract information presented in tables, diagrams, graphs. Graduates need to be able to determine the value of a function by the value of the argument with various ways of specifying the function and describe the behavior and properties of the function according to its graph. It is also necessary to be able to find the largest or smallest value from the function graph and build graphs of the studied functions. The mistakes made are of a random nature in reading the conditions of the problem, reading the diagram.

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Example 2 The figure shows the change in the exchange value of one share of a mining company in the first half of April 2017. On April 7, the businessman purchased 1,000 shares of this company. On April 10, he sold three-quarters of the purchased shares, and on April 13 he sold all the remaining ones. How much did the businessman lose as a result of these operations?