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- The highest mathematics in exercises and tasks. Part 1
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Danko Pavel Efimovich, Popov Alexander Georgievich, Kozhevnikova Tatiana Yakovlevna. Content I parts covers the following sections of the program: analytical geometry, basics linear algebra...
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In 2 parts - Danko P.E., Popov A.G., Kozhevnikov T.Ya. download in PDF. The content of Part I covers the following sections ...
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(In 2 parts) (Danko P.E., Popov A.G., Kozhevnikov T.Ya.)
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Electronic library of books "Pavel Danko" book "Higher mathematics in exercises and problems.
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(In 2 parts) Books Mathematician P.E., Popov A.G., Kozhevnikov T.Ya. Year of publication: 1986 Format: djvu Publisher: High school Size: 21.8 Mb. Language: Russian0 (votes: 0) Assessment: The content of the I part covers the following sections of the program: analytical ...
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Danko PE D17 Higher mathematics in exercises and problems: Textbook. allowance. for universities / P. E. Danko, A. G. Popov, T. Ya. Kozhevnikova, S. P. Dan -ko.
The manual includes typical tasks, for clarity, accompanied by illustrations, and methods for them are discussed in detail ...
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Higher mathematics in exercises and problems in two parts Part.
ББК 22.lа73 Educational publication danko Pavel Efimovich, Popov Alexander Georgievich, Kozhevnikova Tatyana Yakovlevna HIGHER MATHEMATICS IN EXERCISES AND PROBLEMS In two parts Part.
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Danko P.E., Popov A.G., Kozhevnikov T. Ya. Thanks to its extensive coverage, this two-volume tutorial " The highest mathematics in exercises and tasks"for universities is an excellent assistant in mastering the exact discipline and preparing for exams.
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M .: Higher school, 1980 .-- 304 p. The content of Part I covers the following sections
Part 1 contains the program guidelinesrecommended educational literature and
- Ch. 2. - 68 p. Presented lecture notes and tasks for the course "Higher Mathematics", containing ...
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M .: Higher school, 1980 .-- 304 p. The content of Part I covers the following sections of the program
Mathematics in Examples and Problems (edited by L.I. Maisen).
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(In 2 parts) Danko P.E., Popov A.G., Kozhevnikov T.Ya. Tutorial for students of technical colleges.
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A.G. Popov, T. Ya. Kozhevnikova - Higher mathematics in exercises and tasks, part 1.
Typical tasks are given with detailed solutions. There are a large number of tasks for
Change of variable and integration by parts § 2. Integration of rational fractions ...
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Textbook. manual for students of technical colleges. At 2 o'clock, Part II. - 5th ed., Rev. and add. - M .: Higher. shk. , 1999. - 416 p., Ill. In the second parts multiple and curvilinear integrals, series, differential equations, probability theory are considered ...
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Higher mathematics in exercises and problems [part 1] (1986), A.G. Popov, T. Ya. Kozhevnikova Publishing House: MOSCOW "VYSHAYA
Typical tasks are given with detailed solutions. There are many tasks for independent work.
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Higher mathematics: solvers, problem solving guides. Not coping with tasks? More examples and explanations are needed on some topic of higher mathematics (from
Danko P., Popov A., Kozhevnikova T. "Higher mathematics in exercises and problems", volume 1, 1986.
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Higher mathematics in exercises and tasks, part 2 - P.E. Danko, A.G. Popov, T. Ya. Kozhevnikov. Technofile type: tutorial Format: RAR - djvu Size: 12MB Description: The content of the second part covers the following sections of the program: multiples and ...
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"Reshebniks" in higher mathematics. Danko P.E., Popov A.G., Kozhevnikova T. Ya. Higher mathematics in exercises and problems.
Kaplan I.A. Practical classes in higher mathematics, in 5 parts .. - Kharkov, Izd. Kharkov state University, 1967, 1971, 1972.
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Higher mathematics in exercises and problems Year of release: 1986, Popov, Kozhevnikova Genre: Textbook Publisher: "Higher school" Format: DjVu Quality: Scanned pages Number of pages: 719 Description: Contents of two volumes ...
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Danko, Danko, Popov: Higher mathematics in exercises and problems. In 2 parts.
In 2 parts. Textbook for universities. "The content of the first part covers the following sections of the program: analytical geometry, the basics of linear algebra ...
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Textbook for students of technical colleges.
6th ed. - M .: 2003. part 1 - 304s.; part 2 - 416p.
Each paragraph contains the necessary theoretical information. Typical tasks are given with detailed solutions. There are many tasks for independent work.
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Format: pdf (1986 , 4th ed., 304s.)
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Part 2.
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Part 1.
TABLE OF CONTENTS
Preface to the fourth edition 5
From the prefaces to the first, second and third editions 5
Chapter I. Analytical geometry on the plane
§ 1. Rectangular and polar coordinates 6
§ 2. Straight line. fifteen
§ 3. Curves of the second order 25
§ 4. Transformation of coordinates and simplification of equations of curves of the second order 32
§ 5. Determinants of the second and third orders and systems of linear equations in two and three unknowns 39
Chapter II. Elements of vector algebra
§ 1. Rectangular coordinates in space 44
§ 2. Vectors and the simplest actions on them. 45
§ 3. Scalar and vector products. Mixed work. 48
Chapter III. Analytical geometry in space
§ 1. Plane and line. 53
§ 2. Surfaces of the second order. 63
Chapter IV. Determinants and matrices
§ 1. The concept of an n-th order determinant. 70
§ 2. Linear transformations and matrices 74
§ 3. Reduction to the canonical form of the general equations of curves and surfaces of the second order 81
§ 4. Rank of the matrix. Equivalent Matrices 86
§ 5. Study of a system of m linear equations with n unknowns. 88
§ 6. Solution of a system of linear equations by the Gauss method 91
§ 7. Application of the Jordan-Gauss method to solving systems of linear equations 94
Chapter V. Fundamentals of Linear Algebra
§ 1. Linear spaces 103
§ 2. Transformation of coordinates in the transition to a new basis. 109
§ 3. Subspaces 111
§ 4. Linear transformations 115
§ 5. Euclidean space 124
§ 6. Orthogonal basis and orthogonal transformations 128
§ 7. Quadratic forms 131
Chapter VI. Introduction to Analysis
§ 1. Absolute and relative errors 136
§ 2. Function of one independent variable 137
§ 3. Plotting functions 140
§ 4. Limits 142
§ 5. Comparison of infinitesimal 147
§6. Continuity of function 149
Chapter VII. Differential calculus of functions of one independent variable
§ 1. Derivative and differential 151
§ 2. Research of functions 167
§ 3. Curvature of a plane line 183
§ 4. The order of tangency of plane curves 185
§ 5. Vector-function of scalar argument and its derivative. 185
§ 6. Accompanying trihedron of the space curve. Curvature and torsion 188
Chapter VIII. Differential calculus of functions of several independent variables
§ 1. Domain of function definition. Level 192 lines and surfaces
§ 2. Derivatives and differentials of functions of several variables. 193
§ 3. The tangent plane and the normal to the surface 203
§ 4. Extremum of a function of two independent variables 204
Chapter IX. Indefinite integral
§ 1. Direct integration. Variable Change and Integration by Parts 208
§ 2. Integration of rational fractions 218
§ 3. Integration of the simplest irrational functions 229
§ 4. Integration trigonometric functions 234
§ 5. Integration of different functions 242
Chapter X. Definite Integral
§ 1. Calculation of the definite integral 243
§ 2. Improper integrals 247
§ 3. Calculation of area flat figure 251
§ 4. Calculation of the arc length of a plane curve 254
§ 5. Calculation of body volume 255
§ 6. Calculation of the surface area of \u200b\u200brevolution 257
§ 7. Static moments and moments of inertia of plane arcs and figures. 258
§ 8. Finding the coordinates of the center of gravity. Gulden's theorems. 260
§ 9. Calculation of work and pressure 262
§ 10. Some information about hyperbolic functions 266
Chapter XI. Linear programming elements
§ 1. Linear inequalities and the domain of solutions of the system of linear inequalities 271
§ 2. The main problem of linear programming 274
§ 3. Simplex method 276
§ 4. Dual tasks 287
§ 5. Transport problem 288
294 responses
Part 2.
TABLE OF CONTENTS
Chapter 1. Double and triple integrals
§ 1. Double integral in rectangular coordinates b
§ 2. Change of variables in the double integral 10
§ 3. Calculation of the area of \u200b\u200ba flat figure 14
§ 4. Calculation of the volume of a body 16
§ 5. Calculation of surface area 17
§ 6. Physical applications of the double integral 20
§ 7. Triple integral 23
§ 8. Applications triple integral 28
§ 9. Integrals depending on a parameter. Differentiation and integration under the integral sign. thirty
§ 10. Gamma function. Beta Feature 35
Chapter II. Curvilinear and surface integrals
§ 1. Curvilinear integrals over arc length and coordinates. ... 42
§ 2. Independence curvilinear integral II kind from the contour of integration. Finding a function by its total differential 47
§ 3. Green's formula 50
§ 4. Calculation of area 51
§ 5. Surface integrals 52
§ 6. Stokes and Ostrogradsky - Gauss formulas. Elements of field theory 56
Chapter III. Ranks
§ 1. Number series 66
§ 2. Functional series 77
§ 3. Power series 81
§ 4. Expansion of functions in power series 86
§ 5. Approximate calculations of the values \u200b\u200bof functions using power series 91
§ 6. Application of power series to the calculation of limits and definite integrals 95
§ 7. Complex numbers and series with complex numbers 97
§ 8. Fourier series 106
§ 9. The Fourier integral 113
Chapter IV. Ordinary differential equations
§ 1. Differential equations of the first order 117
§ 2. Differential equations of higher orders 139
§ 3. Linear equations of higher orders 145
§ 4. Integration of differential equations by means of series 161
§ 5. Systems of differential equations 166
Chapter V. Elements of the theory of probability
§ 1. Random event, its frequency and probability. Geometric Probability 176
§ 2. Theorems of addition and multiplication of probabilities. Conditional probability 179
§ 3. Bernoulli's formula. Most likely event occurred 183
§ 4. Formula of total probability. Bayesian formula 186
§ 5. Random variable and the law of its distribution 188
§ 6. Mathematical expectation and variance of a random variable 192
§ 7. Fashion and median. 195
§ 8. Uniform distribution 196
§ 9. Binomial distribution law. Poisson's Law ... 197
§ 10. Exponential (exponential) distribution. Reliability function 200
§ 11. Normal distribution law. Laplace function .... 202
§ 12. Moments, asymmetry and kurtosis of a random variable .... 206
§ 13. Law of large numbers 210
§ 14. The Moivre-Laplace theorem 213
§ 15. Systems of random variables 214
§ 16. Lines of regression. Correlation 223
§ 17. Determination of characteristics of random variables on the basis of experimental data 228
§ 18. Finding the laws of distribution of random variables on the basis of experimental data 240
Chapter VI. Understanding Partial Differential Equations
§ 1. First-order partial differential equations 260
§ 2. Types of second order partial differential equations. Canonicalization 262
§ 3. The equation of vibration of a string 265
§ 4. Equation of heat conduction 272
§ 5. The Dirichlet problem for a circle 278
Chapter VII. Elements of the theory of functions of a complex variable
§ 1. Functions of a complex variable. 282
§ 2. Derivative of a function of a complex variable 285
§ 3. The concept of a conformal mapping 287
§ 4. Integral of a function of a complex variable 291
§ 5. Taylor and Laurent series 295
§ 6. Calculation of the residues of functions. Application of residues to the calculation of integrals. 300
Chapter VIII. Elements of operational calculus
§ 1. Finding images of functions 305
§ 2. Finding the original from the image 307
§ 3. Convolution of functions. Image of derivatives and integral from the original 310
§ 4. Application of operational calculus to the solution of some differential and integral equations 312
§ 5. General formula of appeal 315
§ 6. Application of operational calculus to the solution of some equations of mathematical physics. 316
Chapter IX. Calculation methods
§ 1. Approximate solution of equations 321
Section 2. Interpolation 330
§ 3. Approximate calculation of definite integrals 334
§ 4. Approximate calculation of multiple integrals ... 338
§ 5. Application of the Monte Carlo method to the calculation of definite and multiple integrals 350
§ 6. Numerical integration of differential equations. 362
§ 7. Picard's method of successive approximations 368
§ 8. The simplest methods of processing experimental data 370
Chapter X. Foundations of the calculus of variations
§ 1. The concept of a functional 385
§ 2. The concept of variation of a functional 386
§ 3. The concept of an extremum of a functional. Particular cases of the integrability of the Euler equation 387
§ 4. Functionals depending on derivatives of higher orders 393
§ 5. Functionals depending on two functions of one independent variable 394
§ 6. Functionals depending on functions of two independent variables 395
§ 7. Parametric form of variational problems 396
§ 8. The concept of sufficient conditions for the extremum of a functional 397
398 replies
Appendix 409
2. Gnedenko B.V. Probability theory course. - M .: Nuka, 1988.
3. Chistyakov V.P. Probability theory course. - M .: Nauka, 1982.
4. Zubkov A.M., Sevastyanov B.A., Chistyakov V.P. Collection of problems in probability theory. - M .: Nauka, 1989.
5. Kolemaev VA, Staroverov OV, Turudaevsky VB Probability theory and math statistics... - M .: graduate School, 1991.
6. Collection of problems in mathematics for higher educational institutions. Part 3. Probability theory and mathematical statistics / Under. Ed. Efimova A.V. - M .: Nauka, 1990.
7. Feller V. Introduction to the theory of probability and its applications. T. 1, 2. - M .: Mir, 1984.
8. Shiryaev A.N. Probability. - M .: Nauka, 1980.
Directories
1. Handbook of Mathematics for Economists / Ed. IN AND. Ermakova. M .: Higher school, 1987.
Rules for choosing a test option, its design and offset
1. In the process of studying higher mathematics, a first-year student must complete two tests, the tasks of the second of which are contained in the section "Test options". Should not start control task until a sufficient number of tasks are solved on the educational material corresponding to this task. Experience shows that most often the inability to solve a particular problem of the control task is caused by the fact that the student has not fulfilled this requirement.
2. Control works must be executed in accordance with these rules. Works completed without following these rules will not be counted and returned to the student for revision.
3. Each test paper should be performed in a separate notebook, with ink of any color, except red, leaving fields for the reviewer's comments.
4. On the cover of the notebook, the surname, name, and patronymic of the student, the faculty (institute), the group number, the name of the discipline (higher mathematics), the number of the test, the number of the variant and the student's home address should be legibly written. At the end of the work, you should put the date of its completion and sign.
5. The number of the variant of the test, which the student performs, must match the last digit of the number of his grade book.
6. Problem solutions should be arranged in ascending order of numbers. Problem conditions should be rewritten in a notebook.
When solving problems, you need to justify each stage of the solution based on the theoretical provisions of the course.
The solution of problems and examples should be presented in detail, explaining all the actions performed and the formulas used. The solution to each problem must be brought to the final answer required by the condition. In intermediate calculations, you should not enter the approximate values \u200b\u200bof the roots, numbers i, e, etc.
The answer obtained should be checked in ways that follow from the essence of this problem. So, for example, having calculated the indefinite integral, it is necessary to check whether the integrand is equal to the derivative of the resulting antiderivative. It is also useful, if possible, to solve the problem in several ways and compare the results obtained.
8. Verification period control works - 10 working days. Students are required to submit written tests no later than 10 days before the start of the examination session. Otherwise, they will not be admitted to tests and exams.
9. After receiving the peer-reviewed work, the student must correct all the errors and shortcomings noted by the reviewer, make changes or additions recommended by the reviewer to the problem solutions, and send work for re-checking. In this regard, we recommend that when completing the test, leave several blank sheets at the end of the notebook for making corrections and additions later.
If the work is not accepted and there is no direct indication from the reviewer that the student can limit himself to presenting corrected solutions to individual problems, all the work must be done anew.
When corrections are submitted for re-checking, there must be a peer-reviewed work and a review of it. It is prohibited to make corrections to the text of the work after its review.
10. The peer-reviewed test papers, along with any corrections and additions made at the request of the reviewer, should be saved. The student must come to the exam with a review of the completed test. The student is not allowed to the exam without presenting the peer-reviewed test papers to the teacher.
TEST OPTIONS
Option O.
1 A 1, B 1, C 1, D 1... Find:
a) rib length;
c) the equation of the edge;
d) the equation of the face C 1;
2
and) Cramer's method;
b) Gauss method;
3 ... There are 12 items in the same packaging. It is known that four of them have first-class goods. 3 items are randomly selected. Calculate the probability that among them:
and) only packages with first class goods;
b) exactly one package with a first-class product.
4 ... The store has received shoes from two suppliers. The number of shoes received from the first supplier is 2 times more than from the second. It is known that, on average, 20% of shoes from the first supplier and 35% of shoes from the second supplier have "various defects in finishing. From the total mass, one package of shoes is taken at random. It turned out that it does not have a defect in finishing. What is the probability that it was made by the first provider?
5 .
| X | -2 | -1 | |||||
| R | 0,01 | R | 0,23 | 0,28 | 0,19 | 0,11 | 0,06 |
and) unknown probability r;
b) expected value M, variance D
in)
d)
6. It is known that, on average, 64% of students in the stream complete their tests on time. What is the probability that out of 100 students in the flow will delay the submission of tests:
and) 30 students ; b) 30 to 40 students?
Option 1.
1 . A 1, B 1, C 1, D 1... Find:
and) rib length;
b) cosine of the angle between vectors and;
in) edge equation;
d) face equation С 1;
2 .
and) Cramer's method;
b) Gauss method;
3 ... In a box of 25 identical in shape chocolates... It is known that 15 of them are of the "Bear in the North" variety, and the rest are of the "Red Riding Hood" variety. 3 candies are randomly selected. Calculate the probability that among them: a). All candies of the "Bear in the North" variety; b). Only one candy of this type.
4 ... The store has received a product of the same name, made by two enterprises. The first enterprise received 150 units, of which 30 were of the first grade, and 200 units from the second, of which 50 were of the first grade. From the total mass of the goods, one unit of the 1st grade is taken at random. What is. the likelihood that it was manufactured at the first plant?
5 X:
| X | -2 | -1 | |||||
| P | 0,2 | 0,31 | 0,24 | p | 0,07 | 0,04 | 0,01 |
and) unknown probability r;
b) expected value M, variance D <т данной случайной величины;
in) distribution function and build its graph;
d) distribution law of a random variable, if its values \u200b\u200bare specified by functional dependence
6. It is known that, on average, 14% of the glasses manufactured at this enterprise have a defect. What is the probability that out of 300 glasses of a given batch:
and) have a defect 45 ;
b) do not have a defect from 230 to 250?
Option 2.
1 ... In a Cartesian rectangular coordinate system, the vertices of the pyramid are given A 1, B 1, C 1, D 1... Find:
and) rib length;
b) cosine of the angle between vectors and;
in) edge equation;
d) face equation C 1;
2 ... Solve a system of linear equations
and) the Cramer method,
b) Gaussian method:
3 ... There are 15 people in the tourist group, among whom only 5 people speak English well. In London, the group was randomly accommodated in two hotels (3 people and 12 people, respectively). Calculate the probability that of the group members in the first hotel: and) all tourists speak good English;
b). Only one tourist speaks good English.
4 ... The buyer can purchase the product he needs in two stores. The probabilities of contacting each of the two stores depend on their location and are 0.3 and 0.7, respectively. The probability that by the time the buyer arrives, the product he needs will not be sold out is 0.8 for the first store and 0.4 for the second. What is the likelihood that the buyer will purchase the product he needs.
5 ... The law of distribution of a discrete random variable is given X:
| X | -2 | -1 | |||||
| R | 0,04 | 0,08 | 0,3 | 0,3 | 0,1 | 0,08 | R |
and) unknown probability r;
b) expected value M, variance D and the standard deviation sgiven random variable.
in) distribution function and build its graph;
d) distribution law of a random variable, if its values \u200b\u200bare specified by functional dependence
6. It was found that the consumer services company fulfills on average 60% of orders on time. What is the likelihood that out of 150 orders received over time, they will be completed on time:
and) 90 orders ; b) from 93 to 107 orders?
Option 3.
1 ... In a Cartesian rectangular coordinate system, the vertices of the pyramid are given A 1, B 1, C 1, D 1... Find:
and) rib length;
b) cosine of the angle between vectors and;
in) edge equation;
d) face equation C 1;
2 ... Solve a system of linear equations
and) Cramer's method;
b) Gauss method;
3 ... The package contains 12 identical books. It is known that every third book has a cover defect. 3 books are randomly selected. Calculate the probability that among them: and) all books have a defect in the cover; b). Only one book has this defect.
4 ... Two inspectors assess the quality of the manufactured products. The probability that the next item gets to the first inspector is 0.55; to the second controller - 0.45
The first inspector detects a defect with a probability of 0.8, and the second - with a probability of 0.9. Calculate the likelihood that a defective product will be considered serviceable.
5 The distribution law of a discrete random variable is given Xexploitation.
| X | -2 | -1 | |||||
| R | 0,42 | 0,23 | R | 0,10 | 0,06 | 0,03 | 0,01 |
and) unknown probability r;
b) expected value M,variance D
in) distribution function and build its graph;
d) distribution law of a random variable, if its values \u200b\u200bare specified by functional dependence
6. It is known that in this technological process 10% of products have a defect. What is the probability that in a batch of 400 items:
and) 342 products will not be defective;
b) 30 to 52 items will be defective.
Option 4.
1 ... In a Cartesian rectangular coordinate system, the vertices of the pyramid are given A 1, B 1, C 1, D 1... Find:
and) rib length;
b) cosine of the angle between vectors and;
in) edge equation;
d) face equation С 1;
2 ... Solve a system of linear equations
and) Cramer's method;
b) Gauss method;
3. 24 identical pens are prepared for the exam .. It is known that a third of them have a purple core, the rest have a blue core. Three pens are randomly selected. Calculate
the likelihood that;
and) all pens have a purple shaft; b) only one pen has a purple shaft.
4 ... A passenger can purchase a ticket at one of two ticket offices. The probability of going to the first cashier is 0.4, and to the second - 0.6. The probability that by the time the passenger arrives, the tickets he needs will be sold out is 0.35 for the first ticket office and 0.7 for the second. The passenger visited one of the ticket offices and purchased a ticket. What is the likelihood that he acquired it at the second box office?
5
| X | -2 | -1 | |||||
| R | R | 0,29 | 0,12 | 0,15 | 0,21 | 0,16 | 0,04 |
and) unknown probability r;
b) expected value M, variance D and standard deviation s of a given random variable.
in) distribution function and build its graph;
d) distribution law of a random variable, if its values \u200b\u200bare specified by functional dependence
6 ... According to the TV studio, it was found that on average 20% of color TVs fail during the warranty period. What is the likelihood that out of 225 sold TVs will work properly during the warranty period:
and) 164 TVs;
b) from 172 to 184 TVs.
Option 5.
1. In a Cartesian rectangular coordinate system, the vertices of the pyramid are given A 1, B 1, C 1, D 1... Find:
and) rib length;
b) cosine of the angle between vectors and;
in) edge equation;
d) face equation C 1;
2. Solve a system of linear equations
and) Cramer's method;
b) Gauss method;
3. There are 40 deputies in the lower house of parliament, among whom the first party has 20 representatives, the second - 12 representatives, the third - 5 representatives, and the rest consider themselves independent. Three deputies are selected at random. Calculate the probability that among them:
and) ... Only representatives of the first party, b) ... Only one deputy from the first party.
4. Two QCD specialists check the quality of the manufactured products, and each product can be checked with the same probability by any of them. The probability of detecting a defect by the first specialist is 0.8, and the second is 0.9. From the mass of tested products, one was chosen at random, it turned out to be defective. What is the likelihood that the second controller made the mistake?
5. The distribution law of a discrete random variable X is given:
| X | -2 | -1 | |||||
| R | 0,05 | 0,12 | 0,18 | 0,30 | R | 0,12 | 0,05 |
and) unknown probability r;
b) expected value M, variance D and the standard deviation s of the given random variable;
in) distribution function F (x) and build her a schedule;
d) distribution law of a random variable, if its values \u200b\u200bare specified by functional dependence
6. When assessing the quality of the products, it was found that, on average, one third of the shoes produced by the factory have various defects in finishing. What is the likelihood that a batch of 200 pairs arrives at the store:
and) will have defects in finishing 60 pairs;
b) will be free from defects in finishing from 120 to 148 pairs.
Option 6.
1. In a Cartesian rectangular coordinate system, the vertices of the pyramid are given A 1, B 1, C 1, D 1... Find:
and)rib length;
b) cosine of the angle between vectors and;
in) edge equation;
d) face equation C 1;
2. Solve a system of linear equations
and) Cramer's method;
b) Gauss method;
3. There are 18 identical bottles of beer in a box without labels. It is known that a third of them are Zhigulevskoye. 3 bottles are randomly selected. Calculate the probability that among them: a) only beer of the Zhigulevskoye variety; b) exactly one bottle of this variety.
4. Two identical boxes contain Constructor pencils. It is known that a third of the pencils in the first box and 0,25 in the second, they have hardness TM. A box is chosen at random, and one pencil is taken out of it at random. It turns out to be hardness TM. What is the likelihood that it is taken out of the first box?
5. The distribution law of a discrete random variable X is given:
| x | -2 | -1 -1 | |||||
| R | 0,16 | 0,25 | 0,25 | 0,16 | 0,10 | R | 0,03 |
and) unknown probability r;
b) expected value M, variance D and the standard deviation s of the given random variable.
in) distribution function F (x) and build her a schedule;
d) distribution law of a random variable, if its values \u200b\u200bare specified by functional dependence
6. It is known that the probability of having a boy is 0.51, and that of a girl is 0.49. What is the likelihood that 300 newborns will be:
and) 150 boys;
b) 150 to 200 boys?
Option 7.
1. In a Cartesian rectangular coordinate system, the vertices of the pyramid A 1, B 1, C 1, D 1 are given. Find:
and) rib length;
b) cosine of the angle between vectors and;
in) edge equation;
d) face equation С 1;
2. Solve a system of linear equations
and) Cramer's method;
b) Gauss method;
3. There are 20 girls in the student group. It is known that 5 of them do not like to read detective stories. Three girls are randomly chosen and presented to them according to the detective story. Calculate the probability that: a). All girls will appreciate this gift; b). Only one girl will appreciate this gift.
4. The commodity expert of the fruit and vegetable base determines the grade of the consignment of apples received from the permanent supplier. It is known that on average 40% of the crop grown by the supplier is made up of first grade apples. The probability that the merchant will recognize the first-class batch as the first grade is 0.85. In addition, he has a 0.2 probability of making the mistake of judging a non-premium batch as first-rate. What is the likelihood that he will misidentify the apple batch?
5. The distribution law of a discrete random variable X is given:
| x | -2 | -1 -1 | |||||
| R | 0,06 | R | 0,12 | 0,2 | 0,3 | 0 ,1 | 0,03 |
and) unknown probability r;
b) expected value M, variance D to the standard deviation s of a given random variable.
in) distribution function F (x) and build her a schedule;
d) distribution law of a random variable, if its values \u200b\u200bare specified by functional dependence
6. The probability of normal electricity consumption per day for the consumer services enterprise is 0.7. What is the probability that, out of 90 days, the enterprise normally consumes electricity:
and) within 60 days;
b) 60 to 90 days?
Option 8.
1.
and) rib length;
b) cosine of the angle between vectors and;
in) edge equation;
d) face equation С 1;
2. Solve a system of linear equations
and) Cramer's method;
b) Gauss method;
5x + 2y + Зz \u003d 3,
7x + 3 at + 5z \u003d 6.
3. The box contains 30 identical commemorative coins. It is known that 5 of them have a non-standard percentage of gold. Three coins are selected at random. Calculate the probability that: a). All coins have a non-standard percentage of gold; b). Only one coin has a non-standard gold percentage.
4. The store received two batches of the same name, equal in quantity. It is known that 25% of the first batch and 40% of the second batch are first class goods. What is the likelihood that the randomly selected item will not be first grade?
5. The distribution law of a discrete random variable is given X:
| X | -2 | -1 | |||||
| R | 0,02 | 0,38 | 0,30 | r | 0,08 | 0,04 | 0,02 |
and) unknown probability r;
6) expected value M, variance D and the standard deviation s of the given random variable.
in) distribution function F (x) and build her a schedule;
d) distribution law of a random variable, if its values \u200b\u200bare specified by functional dependence
6. It is known that the probability of a daily train arriving at a station is 0.2. What is the probability that the train will be late at the station within 200 days:
and) 50 times;
b) 100 to 150 times?
Option 9.
1. In the Cartesian rectangular coordinate system, the vertices of the pyramid are given, C 1, D 1. Find:
and) rib length;
b) cosine of the angle between vectors and;
in) edge equation;
d) face equation С 1;
2. Solve a system of linear equations
and) Cramer's method;
b) Gauss method;
3. There are 32 identical buns on display. It is known that among them a quarter of buns with raisins, the rest with cinnamon. Three rolls are randomly selected. Calculate the probability that: a). All selected raisin buns; b). Just one raisin bun.
4. The cans are sealed by two machines with the same capacity. The share of cans with a defective closure for the first machine is 1%, and for the second 0.5%. What is the likelihood that a jar taken at random will have a closure defect?
5. The distribution law of a discrete random variable X is given:
| X | -2 | -1 | |||||
| r | 0,08 | 0,1 | 0,14 | 0,1 | 0,1 | 0,1 | r |
and) unknown probability r;
b) expected value M, variance D and the standard deviation s of this random variable;
in) distribution function F (x) and build her a schedule;
d) distribution law of a random variable, if its values \u200b\u200bare specified by functional dependence
6. It has been established that a third of buyers, when visiting a fashion store, buys clothes for themselves. What is the probability that out of 150 store visitors:
and) exactly 50 people will purchase the product;
b) from 100 to 120 people will purchase the product?
Textbook for students of technical colleges.
4th ed., Rev. and additional - M .: Higher. shk., 1986. part 1 - 304s.; part 2 - 416p.
Each paragraph contains the necessary theoretical information. Typical tasks are given with detailed solutions. There are many tasks for independent work.
(Note: Modern, 6th ed., 2006-2007, as I understand it, stereotyped - the same 304 and 416 pages)
Part 1.
Format: djvu / zip
The size: 8, 7 Mb
Download: ifolder.ru
Part 2.
Format: djvu / zip
The size: 1 2.1 Mb
Download: ifolder.ru
Part 1.
TABLE OF CONTENTS
Preface to the fourth edition 5
From the prefaces to the first, second and third editions 5
Chapter I. Analytical geometry on the plane
§ 1. Rectangular and polar coordinates 6
§ 2. Straight line. fifteen
§ 3. Curves of the second order 25
§ 4. Transformation of coordinates and simplification of equations of curves of the second order 32
§ 5. Determinants of the second and third orders and systems of linear equations in two and three unknowns 39
Chapter II. Elements of vector algebra
§ 1. Rectangular coordinates in space 44
§ 2. Vectors and the simplest actions on them. 45
§ 3. Scalar and vector products. Mixed work. 48
Chapter III. Analytical geometry in space
§ 1. Plane and line. 53
§ 2. Surfaces of the second order. 63
Chapter IV. Determinants and matrices
§ 1. The concept of an n-th order determinant. 70
§ 2. Linear transformations and matrices 74
§ 3. Reduction to the canonical form of the general equations of curves and surfaces of the second order 81
§ 4. Rank of the matrix. Equivalent Matrices 86
§ 5. Study of a system of m linear equations with n unknowns. 88
§ 6. Solution of a system of linear equations by the Gauss method 91
§ 7. Application of the Jordan-Gauss method to solving systems of linear equations 94
Chapter V. Fundamentals of Linear Algebra
§ 1. Linear spaces 103
§ 2. Transformation of coordinates in the transition to a new basis. 109
§ 3. Subspaces 111
§ 4. Linear transformations 115
§ 5. Euclidean space 124
§ 6. Orthogonal basis and orthogonal transformations 128
§ 7. Quadratic forms 131
Chapter VI. Introduction to Analysis
§ 1. Absolute and relative errors 136
§ 2. Function of one independent variable 137
§ 3. Plotting functions 140
§ 4. Limits 142
§ 5. Comparison of infinitesimal 147
§6. Continuity of function 149
Chapter VII. Differential calculus of functions of one independent variable
§ 1. Derivative and differential 151
§ 2. Research of functions 167
§ 3. Curvature of a plane line 183
§ 4. The order of tangency of plane curves 185
§ 5. Vector-function of scalar argument and its derivative. 185
§ 6. Accompanying trihedron of the space curve. Curvature and torsion 188
Chapter VIII. Differential calculus of functions of several independent variables
§ 1. Domain of function definition. Level 192 lines and surfaces
§ 2. Derivatives and differentials of functions of several variables. 193
§ 3. The tangent plane and the normal to the surface 203
§ 4. Extremum of a function of two independent variables 204
Chapter IX. Indefinite integral
§ 1. Direct integration. Variable Change and Integration by Parts 208
§ 2. Integration of rational fractions 218
§ 3. Integration of the simplest irrational functions 229
§ 4. Integration of trigonometric functions 234
§ 5. Integration of different functions 242
Chapter X. Definite Integral
§ 1. Calculation of the definite integral 243
§ 2. Improper integrals 247
§ 3. Calculation of the area of \u200b\u200ba flat figure 251
§ 4. Calculation of the arc length of a plane curve 254
§ 5. Calculation of body volume 255
§ 6. Calculation of the surface area of \u200b\u200brevolution 257
§ 7. Static moments and moments of inertia of plane arcs and figures. 258
§ 8. Finding the coordinates of the center of gravity. Gulden's theorems. 260
§ 9. Calculation of work and pressure 262
§ 10. Some information about hyperbolic functions 266
Chapter XI. Linear programming elements
§ 1. Linear inequalities and the domain of solutions of a system of linear inequalities 271
§ 2. The main problem of linear programming 274
§ 3. Simplex method 276
§ 4. Dual tasks 287
§ 5. Transport problem 288
294 responses
Part 2.
TABLE OF CONTENTS
Chapter 1. Double and triple integrals
§ 1. Double integral in rectangular coordinates b
§ 2. Change of variables in the double integral 10
§ 3. Calculation of the area of \u200b\u200ba flat figure 14
§ 4. Calculation of the volume of a body 16
§ 5. Calculation of surface area 17
§ 6. Physical applications of the double integral 20
§ 7. Triple integral 23
§ 8. Applications of the triple integral 28
§ 9. Integrals depending on a parameter. Differentiation and integration under the integral sign. thirty
§ 10. Gamma function. Beta Feature 35
Chapter II. Curvilinear and surface integrals
§ 1. Curvilinear integrals over arc length and coordinates. ... 42
§ 2. Independence of a curvilinear integral of the second kind from the contour of integration. Finding a function by its total differential 47
§ 3. Green's formula 50
§ 4. Calculation of area 51
§ 5. Surface integrals 52
§ 6. Stokes and Ostrogradsky - Gauss formulas. Elements of field theory 56
Chapter III. Ranks
§ 1. Number series 66
§ 2. Functional series 77
§ 3. Power series 81
§ 4. Expansion of functions in power series 86
§ 5. Approximate calculations of the values \u200b\u200bof functions using power series 91
§ 6. Application of power series to the calculation of limits and definite integrals 95
§ 7. Complex numbers and series with complex numbers 97
§ 8. Fourier series 106
§ 9. The Fourier integral 113
Chapter IV. Ordinary differential equations
§ 1. Differential equations of the first order 117
§ 2. Differential equations of higher orders 139
§ 3. Linear equations of higher orders 145
§ 4. Integration of differential equations by means of series 161
§ 5. Systems of differential equations 166
Chapter V. Elements of the theory of probability
§ 1. Random event, its frequency and probability. Geometric Probability 176
§ 2. Theorems of addition and multiplication of probabilities. Conditional probability 179
§ 3. Bernoulli's formula. Most likely event occurred 183
§ 4. Formula of total probability. Bayesian formula 186
§ 5. Random variable and the law of its distribution 188
§ 6. Mathematical expectation and variance of a random variable 192
§ 7. Fashion and median. 195
§ 8. Uniform distribution 196
§ 9. Binomial distribution law. Poisson's Law ... 197
§ 10. Exponential (exponential) distribution. Reliability function 200
§ 11. Normal distribution law. Laplace function .... 202
§ 12. Moments, asymmetry and kurtosis of a random variable .... 206
§ 13. Law of large numbers 210
§ 14. The Moivre-Laplace theorem 213
§ 15. Systems of random variables 214
§ 16. Lines of regression. Correlation 223
§ 17. Determination of characteristics of random variables on the basis of experimental data 228
§ 18. Finding the laws of distribution of random variables on the basis of experimental data 240
Chapter VI. Understanding Partial Differential Equations
§ 1. First-order partial differential equations 260
§ 2. Types of second order partial differential equations. Canonicalization 262
§ 3. The equation of vibration of a string 265
§ 4. Equation of heat conduction 272
§ 5. The Dirichlet problem for a circle 278
Chapter VII. Elements of the theory of functions of a complex variable
§ 1. Functions of a complex variable. 282
§ 2. Derivative of a function of a complex variable 285
§ 3. The concept of a conformal mapping 287
§ 4. Integral of a function of a complex variable 291
§ 5. Taylor and Laurent series 295
§ 6. Calculation of the residues of functions. Application of residues to the calculation of integrals. 300
Chapter VIII. Elements of operational calculus
§ 1. Finding images of functions 305
§ 2. Finding the original from the image 307
§ 3. Convolution of functions. Image of derivatives and integral from the original 310
§ 4. Application of operational calculus to the solution of some differential and integral equations 312
§ 5. General formula of appeal 315
§ 6. Application of operational calculus to the solution of some equations of mathematical physics. 316
Chapter IX. Calculation methods
§ 1. Approximate solution of equations 321
Section 2. Interpolation 330
§ 3. Approximate calculation of definite integrals 334
§ 4. Approximate calculation of multiple integrals ... 338
§ 5. Application of the Monte Carlo method to the calculation of definite and multiple integrals 350
§ 6. Numerical integration of differential equations. 362
§ 7. Picard's method of successive approximations 368
§ 8. The simplest methods of processing experimental data 370
Chapter X. Foundations of the calculus of variations
§ 1. The concept of a functional 385
§ 2. The concept of variation of a functional 386
§ 3. The concept of an extremum of a functional. Particular cases of the integrability of the Euler equation 387
§ 4. Functionals depending on derivatives of higher orders 393
§ 5. Functionals depending on two functions of one independent variable 394
§ 6. Functionals depending on functions of two independent variables 395
§ 7. Parametric form of variational problems 396
§ 8. The concept of sufficient conditions for the extremum of a functional 397
398 replies
Appendix 409
Name: Higher mathematics in exercises and problems - Part 1. 1986.
The content of the first part covers the following sections of the program: analytical geometry, the basics of linear algebra, differential calculus of functions of one and several variables, integral calculus of functions of one independent variable, elements of linear programming.
Each paragraph contains the necessary theoretical information. Typical tasks are given with detailed solutions. There are many tasks for independent work.
TABLE OF CONTENTS
Preface to the fourth edition 5
From the prefaces to the first, second and third editions 5
Chapter I. Analytical geometry on a plane
§ 1. Rectangular and polar coordinates 6
§ 2. Straight line. fifteen
§ 3. Curves of the second order 25
§ 4. Transformation of coordinates and simplification of equations of curves of the second order 32
§ 5. Determinants of the second and third orders and systems of linear equations in two and three unknowns 39
Chapter II. Elements of vector algebra
§ 1. Rectangular coordinates in space 44
§ 2. Vectors and the simplest actions on them. 45
§ 3. Scalar and vector products. Mixed work. 48
Chapter III. Analytical geometry in space
§ 1. Plane and line. 53
§ 2. Surfaces of the second order. 63
Chapter IV. Determinants and matrices
§ 1. The concept of a determinant of order n. 70
§ 2. Linear transformations and matrices 74
§ 3. Reduction of general equations of curves and surfaces of the second order to canonical form 81
§ 4. Rank of the matrix. Equivalent Matrices 86
§ 5. Study of a system of m linear equations with n unknowns. 88
§ 6. Solution of a system of linear equations by the Gauss method 91
§ 7. Application of the Jordan-Gauss method to solving systems of linear equations 94
Chapter V. Fundamentals of Linear Algebra
§ 1. Linear spaces 103
§ 2. Transformation of coordinates in the transition to a new basis. 109
§ 3. Subspaces 111
§ 4. Linear transformations 115
§ 5. Euclidean space 124
§ 6. Orthogonal basis and orthogonal transformations 128
§ 7. Quadratic forms 131
Chapter VI. Introduction to Analysis
§ 1. Absolute and relative errors 136
§ 2. Function of one independent variable 137
§ 3. Plotting functions 140
§ 4. Limits 142
§ 5. Comparison of infinitesimal 147
§6. Continuity of function 149
Chapter VII. Differential calculus of functions of one independent variable
§ 1. Derivative and differential 151
§ 2. Research of functions 167
§ 3. Curvature of a plane line 183
§ 4. The order of tangency of plane curves 185
§ 5. Vector-function of scalar argument and its derivative. 185
§ 6. Accompanying trihedron of the space curve. Curvature and torsion 188
Chapter VIII. Differential calculus of functions of several independent variables
§ 1. Domain of function definition. Level 192 lines and surfaces
§ 2. Derivatives and differentials of functions of several variables. 193
§ 3. The tangent plane and the normal to the surface 203
§ 4. Extremum of a function of two independent variables 204
Chapter IX. Indefinite integral
§ 1. Direct integration. Variable Change and Integration by Parts 208
§ 2. Integration of rational fractions 218
§ 3. Integration of the simplest irrational functions 229
§ 4. Integration of trigonometric functions 234
§ 5. Integration of different functions 242
Chapter X. Definite integral
§ 1. Calculation of the definite integral 243
§ 2. Improper integrals 247
§ 3. Calculation of the area of \u200b\u200ba flat figure 251
§ 4. Calculation of the arc length of a plane curve 254
§ 5. Calculation of body volume 255
§ 6. Calculation of the surface area of \u200b\u200brevolution 257
§ 7. Static moments and moments of inertia of plane arcs and figures. 258
§ 8. Finding the coordinates of the center of gravity. Gulden's theorems. 260
§ 9. Calculation of work and pressure 262
§ 10. Some information about hyperbolic functions 266
Chapter XI. Linear programming elements
§ 1. Linear inequalities and the domain of solutions of a system of linear inequalities 271
§ 2. The main problem of linear programming 274
§ 3. Simplex method 276
§ 4. Dual tasks 287
§ 5. Transport problem 288
294 responses
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