Introduction Chapter 1. The policy of dispossession of peasants in the 1930s. in the Urals 1...
The history of the Kaliningrad Trade and Economic College is a page in the history of the region, which has been written since 1946. Over the past time, more than 25 thousand specialists have graduated from the college.
Since 2004, the college has become an experimental platform for the Moscow Institute for the Development of Secondary Vocational Education on the topic “Dissemination of European experience in the creation and organization of Adult Education Centers and Open Education Centers in the region.” For ten years he has been a member of the Russian Marketing Association and has the status of a social college. The latter was awarded to the college by the regional administration for its constant support of socially disadvantaged students, teachers, retirees, military personnel and members of their families, working teachers and staff.
Students are trained at the Kaliningrad Trade and Economic College in five faculties: technology and service, marketing management, law, economics and accounting, and non-traditional forms of education. The educational field of the college includes sixteen specialties. These include food preparation technology, food commerce, trade commerce, management, marketing, accountant-lawyer, banking, organization of services in a hotel complex, finance, tourism and much more.
The college has a Center for Career Guidance and Applicant Training. At the faculty of non-traditional forms of education, you can not only improve your qualifications, but also acquire a new specialty without interrupting your job. The current Open Education Center is focused on providing assistance in professional training in more than twenty specialties. Here you can improve your skills and undergo retraining. The methods are varied: business games, trainings, seminars, exercises, open meetings, conferences, project work. All this allows students to assimilate the proposed material as much as possible.
Cooperation with Kaliningrad State University, Kaliningrad State Technical University, and the Baltic State Academy allows the college to train specialists whose knowledge becomes capital and the main resource for the economic development of the region. Over the years of this interaction, more than two hundred graduates received higher education at a special faculty with a shortened period of study. All of them are in demand by the region’s economic complex, many have entered the elite of the region’s entrepreneurial corps.
The Kaliningrad Trade and Economic College has established communications and is actively interacting with Denmark, Sweden, Germany, Poland, and Finland. The team participates in international educational projects. Their topics are varied, it includes such important topics as “Assisting the Kaliningrad authorities in the development of small and medium-sized businesses”, “Assisting officers and unemployed members of their families in obtaining civilian specialties for subsequent employment”, “Training teachers in andragogy and developing entrepreneurial training programs activities in Kaliningrad" and the like.
In 1999, as part of an international project, thanks to the efforts of Lydia Ivanovna Motolyanets, Deputy Director for Academic Affairs, a simulation company was created - a model of an enterprise that reflects the activities of a real trade organization, an effective specialized form of advanced training for personnel at all levels working in the field of small business.
The mission of the team - to guarantee an education that will meet the needs of society and contribute to the formation of an integral personality - is fully fulfilled. Kaliningrad Trade and Economic College is professionalism, responsibility, prestige.
KTEK
PCC of Economics and Accounting
15 copies, 2006
Introduction. 4
The concept of derivative. 5
Partial derivatives. eleven
Inflection points. 16
Exercises to solve. 17
Test. 20
Answers to exercises.. 21
Literature. 23
Introduction
f(x x, then they call marginal product; If g(x) g(x) g′(x) called marginal cost.
For example, Let the function be known u=u(t) u while working t. ∆t=t 1 - t 0:
z avg. =
z avg. at ∆t→ 0: .
Production costs K x, so we can write K=K(x) ∆x K(x+∆x). ∆x ∆K=K(x+∆x)- K(x).
Limit 
called
Concept of derivative
Derivative of a function at point x 0 is called the limit of the ratio of the increment of a function to the increment of the argument, provided that the increment of the argument tends to zero.
Derivative function notation:
That. a-priory:
Algorithm for finding the derivative:
Let the function y=f(x) continuous on the segment , x
1. Find the increment of the argument:
x– new argument value
x 0- initial value
2. Find the increment of the function:
f(x)– new function value
f(x 0)- initial value of the function
3. Find the ratio of the increment of the function to the increment of the argument:
4. Find the limit of the found ratio at
Find the derivative of the function based on the definition of derivative.
Solution:
Let's give X increment Δх, then the new value of the function will be equal to:
Let's find the increment of the function as the difference between the new and initial values of the function:
We find the ratio of the function increment to the argument increment:
.
Let us find the limit of this ratio provided that:
Therefore, by definition of derivative: 
.
Finding the derivative of a function is called differentiation.
Function y=f(x) called differentiable on the interval (a;b), if it has a derivative at each point of the interval.
Theorem If the function is differentiable at a given point x 0, then it is continuous at this point.
The reverse statement is false, because There are functions that are continuous at some point but are not differentiable at that point. For example, the function at the point x 0 =0.
Find derivatives of functions
1) 
.
2) 
.
Let us perform identical transformations of the function:
Higher order derivatives
Second order derivative is called the derivative of the first derivative. Designated
Derivative of n-order is called the derivative of the (n-1)th order derivative.
For example, 
Partial derivatives
Partial derivative a function of several variables with respect to one of these variables is called the derivative taken with respect to this variable, provided that all other variables remain constant.
For example, for function 
the first order partial derivatives will be equal to:
Maximum and minimum function
The argument value at which the function has the greatest value is called maximum point.
The argument value at which the function has the smallest value is called minimum point.
The maximum point of a function is the boundary point of transition of the function from increasing to decreasing, the minimum point of the function is the boundary point of transition from decreasing to increasing.
Function y=f(x) has (local) maximum at point if for all x
Function y=f(x) has (local) minimum at point if for all X, sufficiently close to the inequality
The maximum and minimum values of a function are collectively called extremes, and the points at which they are reached are called extremum points.
Theorem (a necessary condition for the existence of an extremum) Let the function be defined on an interval and have the greatest (smallest) value at the point . Then, if at a point there is a derivative of this function, then it is equal to zero, i.e. .
Proof:
Let the function have the greatest value at point x 0, then for any the following inequality holds: .
For any point 
If x > x 0, then, i.e. 
If x< x 0 , то , т.е. 
Because there is , something that is possible only if they are equal to zero, therefore, .
Consequence:
If at a point the differentiable function takes the greatest (smallest) value, then at the point the tangent to the graph of this function is parallel to the Ox axis.
The points at which the first derivative is zero or does not exist are called critical - these are possible extremum points.
Note that, since the equality of the first derivative to zero is only a necessary condition for an extremum, it is necessary to further investigate the question of the presence of an extremum at each point of a possible extremum.
Theorem(sufficient condition for the existence of an extremum)
Let the function y = f(x) is continuous and differentiable in some neighborhood of the point x 0. If, when passing through a point x 0 from left to right, the first derivative changes sign from plus to minus (from minus to plus), then at the point x 0 function y = f(x) has a maximum (minimum). If the first derivative does not change sign, then this function does not have an extremum at the point x 0 .
Algorithm for studying a function for an extremum:
1. Find the first derivative of the function.
2.Equate the first derivative to zero.
3.Solve the equation. The found roots of the equation are critical points.
4.Plot the found critical points on the numerical axis. We get a series of intervals.
5. Determine the sign of the first derivative in each of the intervals and indicate the extrema of the function.
6.To plot a graph:
Ø determine the values of the function at the extremum points
Ø find the points of intersection with the coordinate axes
Ø find additional points
The tin can has the shape of a round cylinder of radius r and heights h. Assuming that a clearly fixed amount of tin is used to make a can, determine at what ratio between r And h the jar will have the largest volume.
The amount of tin used will be equal to the total surface area of the can, i.e. . (1)
From this equality we find:
Then the volume can be calculated using the formula: 
. The problem will be reduced to finding the maximum of the function V(r). Let's find the first derivative of this function: 
. Let us equate the first derivative to zero:
. We find: . (2)
This point is the maximum point, because the first derivative is positive at and negative at .
Let us now establish at what ratio between the radius and height of the bank the largest volume will occur. To do this, divide equality (1) by r 2 and use relation (2) for S. We get: . Thus, a jar whose height is equal to its diameter will have the greatest volume.
Sometimes it is quite difficult to study the sign of the first derivative to the left and right of a possible extremum point, then you can use second sufficient condition for extremum:
Theorem Let the function y = f(x) has at the point x 0 possible extremum finite second derivative. Then the function y = f(x) has at the point x 0 maximum if , and minimum if .
Note This theorem does not solve the question of the extremum of a function at a point if the second derivative of the function at a given point is equal to zero or does not exist.
Inflection points
The points of the curve at which the convexity is separated from the concavity are called inflection points.
Theorem (necessary condition for the inflection point): Let the graph of a function have an inflection point and the function has a continuous second derivative at the point x 0, then
Theorem (sufficient condition for the inflection point): Let the function have a second derivative in some neighborhood of the point x 0, which has different signs to the left and right of x 0. then the graph of the function has an inflection at the point .
Algorithm for finding inflection points:
1. Find the second derivative of the function.
2. Equate the second derivative to zero and solve the equation: . Plot the resulting roots on the number line. We get a series of intervals.
3. Find the sign of the second derivative in each of the intervals. If the signs of the second derivative in two adjacent intervals are different, then we have an inflection point for a given value of the root; if the signs are the same, then there are no inflection points.
4. Find the ordinates of the inflection points.
Examine the curve for convexity and concavity. Find inflection points.
1) find the second derivative:
2) Solve the inequality 2x<0 x<0 при x кривая выпуклая
3) Solve the inequality 2x>0 x>0 for x the curve is concave
4) Let’s find the inflection points, for which we equate the second derivative to zero: 2x=0 x=0. Because at the point x=0 the second derivative has different signs on the left and right, then x=0 is the abscissa of the inflection point. Let's find the ordinate of the inflection point:
(0;0) inflection point.
Exercises to solve
No. 1 Find the derivatives of these functions, calculate the value of the derivatives for a given value of the argument:
| 1. | 5.  | 9. | |||
| 2. | 6. | 10.  |
|||
3.  | 7. | 11.  |
|||
4.  | 8.  | 12.  |
|||
13.  | 14.  | ||||
| 15. | 16. | ||||
No. 2 Find derivatives of complex functions:
1.  | 2.  |
| 3. | 4. |
5.  | 6.  |
| 7. | 8. |
| 9. | 10. |
| 11. | 12. |
| 13. | 14.  |
15.  | 16. |
| 17. | 18. |
No. 3 Solve problems:
1. Find the angular coefficient of the tangent drawn to the parabola at point x=3.
2. A tangent and a normal are drawn to the parabola y=3x 2 -x at the point x=1. Make up their equations.
3. Find the coordinates of the point at which the tangent to the parabola y=x 2 +3x-10 forms an angle of 135 0 with the OX axis.
4. Create an equation for the tangent to the graph of the function y=4xx2 at the point of intersection with the OX axis.
5. For what values of x is the tangent to the graph of the function y=x 3 -x parallel to the straight line y=x.
6. The point moves rectilinearly according to the law S=2t 3 -3t 2 +4. find the acceleration and speed of the point at the end of the 3rd second. At what point in time will the acceleration be zero?
7. When is the speed of a point moving according to the law S=t 2 -4t+5 equal to zero?
#4 Explore functions using derivatives:
1. Examine the monotonicity of the function y = x 2
2. Find the intervals of increasing and decreasing functions 
.
3. Find the intervals of increase and decrease of the function.
4. Explore the maximum and minimum function 
.
5. Examine the function for extremum 
.
6. Investigate the function y=x3 for extremum
7. Examine the function for extremum 
.
8. Divide the number 24 into two terms so that their product is greatest.
9. It is necessary to cut a rectangle with an area of 100 cm 2 from a sheet of paper so that the perimeter of this rectangle is the smallest. What should be the sides of this rectangle?
10. Examine the function y=2x 3 -9x 2 +12x-15 for an extremum and construct its graph.
11. Examine the curve for concavity and convexity.
12. Find the intervals of convexity and concavity of the curve 
.
13. Find the inflection points of the functions: a) ; b) .
14. Explore the function and build its graph.
15. Investigate the function and build its graph.
16. Explore the function 
and plot it.
17. Find the largest and smallest value of the function y=x 2 -4x+3 on the segment
Test questions and examples
1. Define derivative.
2. What is called argument increment? function increment?
3. What is the geometric meaning of the derivative?
4. What is called differentiation?
5. List the main properties of the derivative.
6. Which function is called complex? reverse?
7. Give the concept of a second-order derivative.
8. Formulate a rule for differentiating a complex function?
9. The body moves rectilinearly according to the law S=S(t). What can you say about movement if:
5. The function increases over a certain interval. Does it follow from this that its derivative is positive on this interval?
6. What are called extrema of a function?
7. Does the largest value of a function on a certain interval necessarily coincide with the value of the function at the maximum point?
8. The function is defined on . Could the point x=a be the extremum point of this function?
10. The derivative of the function at the point x 0 is zero. Does it follow from this that x 0 is the extremum point of this function?
Test
1. Find derivatives of these functions:
A)  | e) |
| b) | and)  |
With)  | h)  |
| d) | And) |
2. Write the equations of the tangents to the parabola y=x 2 -2x-15: a) at the point with the abscissa x=0; b) at the point of intersection of the parabola with the abscissa axis.
3. Determine the intervals of increasing and decreasing function
4. Explore the function and graph it
5. Find at time t=0 the speed and acceleration of a point moving according to the law s =2e 3 t
Answers to exercises
5. 
7. 
9. 
11. 
12. 
13. 
14. 
2. 
3. 
4. 
(the result was obtained by applying the quotient derivative formula). You can solve this example differently:
5. 
8. The product will be greatest if each term is equal to 12.
9. The perimeter of the rectangle will be smallest if the sides of the rectangle are 10 cm, i.e. you need to cut out a square.
17. On a segment, the function takes the greatest value equal to 3 when x=0 and the smallest value equal to –1 at x=2.
Literature
| 1. | Vlasov V.G. Lecture notes on higher mathematics, Moscow, Iris, 96. | |
| 2. | Tarasov N.P. Course of higher mathematics for technical schools, M., 87 | |
| 3. | I.I.Valuta, G.D. Diligul Mathematics for technical schools, M., Science, 90g | |
| 4. | I.P.Matskevich, G.P.Svirid Higher Mathematics, Minsk, Higher. School, 93 | |
| 5. | V.S. Shchipachev Fundamentals of Higher Mathematics, M. Higher School89 | |
| 6. | V.S. Shchipachev Higher Mathematics, M. Higher School 85 | |
| 7. | V.P.Minorsky Collection of problems in higher mathematics, M. Nauka 67g | |
| 8. | O.N.Afanasyeva Collection of problems in mathematics for technical schools, M.Nauka 87g | |
| 9. | V.T.Lisichkin, I.L.Soloveichik Mathematics, M.Higher school 91g | |
| 10. | N.V. Bogomolov Practical lessons in mathematics, M. Higher school 90 | |
| 11. | H.E. Krynsky Mathematics for economists, M. Statistics 70g | |
| 12. | L.G.Korsakova Higher mathematics for managers, Kaliningrad, KSU, 97. |
KALININGRAD TRADE AND ECONOMICS COLLEGE
on studying the topic
"function derivative"
for students of specialty 080110 “Economics and Accounting”, 080106 “Finance”,
080108 “Banking”, 230103 “Automated information processing and management systems”
Compiled by E.A. Fedorova
KALININGRAD
Reviewers: Natalya Vladimirovna Gorskaya, teacher, Kaliningrad Trade and Economic College
This manual examines the basic concepts of differential calculus: the concept of derivative, properties of derivatives, application in analytical geometry and mechanics, basic differentiation formulas are given, examples are given to illustrate theoretical material. The manual is supplemented with exercises for independent work, answers to them, questions and sample tasks for intermediate knowledge control. Intended for students studying the discipline “Mathematics” in secondary specialized educational institutions, studying full-time, part-time, evening, external, or having free attendance.
KTEK
PCC of Economics and Accounting
15 copies, 2006
Introduction. 4
Requirements for knowledge and skills... 5
The concept of derivative. 5
Geometric meaning of derivative. 7
Mechanical meaning of derivative. 7
Basic rules of differentiation. 8
Formulas for differentiating basic functions. 9
Derivative of the inverse function. 9
Differentiation of complex functions. 10
Derivatives of higher orders. eleven
Partial derivatives. eleven
Studying functions using derivatives. eleven
Increasing and decreasing function. eleven
Maximum and minimum functions. 13
Convexity and concavity of a curve. 15
Inflection points. 16
General scheme for studying functions and constructing graphs. 17
Exercises to solve. 17
Test questions and examples.. 20
Test. 20
Answers to exercises.. 21
Literature. 23
Introduction
Mathematical analysis provides a number of fundamental concepts with which an economist operates: function, limit, derivative, integral, differential equation. In economic research, specific terminology is often used to refer to derivatives. For example, if f(x) is a production function that expresses the dependence of the output of any product on the cost of the factor x, then they call marginal product; If g(x) there is a cost function, i.e. function g(x) expresses the dependence of total costs on the volume of production x, then g′(x) called marginal cost.
Marginal analysis in economics– a set of techniques for studying changing values of costs or results when volumes of production, consumption, etc. change. based on an analysis of their limit values.
For example, finding labor productivity. Let the function be known u=u(t), expressing the quantity of products produced u while working t. Let's calculate the amount of products produced over time ∆t=t 1 - t 0:
u=u(t 1)-u(t 0)=u(t 0 +∆t)-u(t 0).
Average labor productivity is called the ratio of the quantity of products produced to the time spent, i.e. z avg. =
Worker productivity at the moment t 0 the limit to which it tends is called z avg. at ∆t→ 0: . Calculating labor productivity thus comes down to calculating the derivative:
Production costs K homogeneous production is a function of the quantity of production x, so we can write K=K(x). Suppose the quantity of output increases by ∆x. The production quantity x+∆x corresponds to production costs K(x+∆x). Consequently, the increase in the quantity of products ∆x corresponds to an increase in production costs ∆K=K(x+∆x)- K(x).
The average increase in production costs is ∆K/∆x. This is an increase in production costs per unit increase in the quantity of production.
Limit called marginal production costs.
