Практический смысл производной

Practical use of derivative

Table of contents

1.Introduction        2

2. The concept of the derivative        3

3. Using derivative in life situation        4

4. Using the derivative for solving tasks in economics        7

5. Conclusion        9

6. Bibliography        9

1.Introduction

The term of function is one of the basic definitions of mathematics. It did not appear, directly in this form, as we use it now, and like other fundamental definitions passed a long way of dialectical and historical development. The idea of functional dependence goes back to ancient Greek mathematics. For example, changing the area or volume of a shape depends on change of its size. However, the idea of functional dependence was understood intuitively by the ancient Greeks.

In the 16 - 17th centuries equipment, industry, navigation sat before the mathematics tasks that were impossible to solve the existing math methods of constant values. It was necessary to find out the new mathematical methods which different from the methods of elementary mathematics.

I have chosen the theme for my worknot accidentally. I think that using of the derivative allows solving many problems of high complexitymore effectively. Of course, using of the derivative for solving problems, requires us unconventional thinking. But knowledge of nonstandard methods and techniques of solving problems contributes to the development of my thinking and I’ll be able to apply my new knowledge in other spheres of human activity (computer science, Economics, physics, chemistry, etc.)It   proves the relevance of this work

For my work I set the aim – studying of the application of the derivative for solving practical tasks.

2. The concept of the derivative

There is a question about quickness of current changing during studying of varying quantities. So we're talking about the velocity of the plane, train, bus, rocket, the speed of falling stone, etc. We can talk about the velocity of execution of certain work, about the velocity of a chemical reaction, the velocity of population growth in the city; it is possible to speak about velocity to any magnitude that varies over time. For that, we can use the definition of derivative.

Consider the following task: let's some point moves in a straight line continuously and regularly. At some point of time t coordinate of a point x is equal to x(t). For definiteness, you may assume that we are talking about driving on the straight part of road.

The task is by a known dependence x(t) to find the speed  which the car moves in a time t, in other words - to find the instantaneous velocity at time t.

It is easy to solve this task, when the dependence x(t) is linear. At any time, the velocity will be equal to the ratio of distance traveled to the time during which this way passed. If the movement is uneven, it becomes more complicated.

It is obvious that at any time the car is moving with a certain speed. The average velocity during the interval will be calculated by the formula:

V =.

It is logical to assume that if this period of time to do very little, for this time interval the velocity value will not change. Then the average velocity for this interval will be different from the values of instantaneous velocity is very small.

That is, to solve the task we need to find average velocity on a very small time interval, i.e. ∆t is almost equal to zero.

Consider a specific example: the average velocity of a body thrown up

V =  – g –.

Now let's see what happens to this formula if we start to decrease the value of ∆t to zero, i.e. ∆t →0.

It is obvious that the tendency of ∆t to zero, the terms in which ∆t is involved will also tend to zero, i.e.,  will be very small and can be ignored.

Theformula V =  – g∙ remains. This will be the formula for instantaneous velocity at time  the velocity of change of the function f at the point . Instead, the mathematics says that we found the derivative of the function f at the point.Give the general definition of the derivative. The derivative of the function f at the point is the limit of the ratio of the increment function ∆f to the increment of the argument ∆x when the ∆x to zero.

Derivative of the function f at the point  is denoted in the following way: f’(). Thus, by definition, f’=. Using the formula of the increase can be written

fꞋ() =

Derivative is a powerful way for solving applied problems. With such problems now we have to deal with the representatives of various professions:

• Engineers, technologists try to organize production so that produced many products as possible;

• Designers try to develop a device for a space vehicle so that the weight of the device was the smallest;

• Economists try to plan the connection of the plant to sources of raw materials so transportation costs were minimal.

In the future I want to become an economist. I know that the use of derivative widely used for economic analysis as a mathematical tools. In Economics it is often necessary to find the optimal or the best value: highest productivity, maximum profit, maximum production, minimum costs, etc. Each indicator represents a function of one or more arguments. Thus, finding the optimal value of the indicator is reduced to finding the extremum of the function, which can be found through the derivative of a function

Examples of derivative formulas in the economic formulas:

• Productivity of the work P (t) = υꞋ (t), where υ (t) - volume of production or marginal cost productivity

• J(x) = yꞋ (x), where y(x) production costs depending on production volume x.

3. Using derivative in life situation

I cannot solve such serious problems, but I have one problem.

My friend decided to make his girlfriend a present for Valentine's Day. He ordered to the friend Denis to make the jewelry box. In the Studio he brought a piece of sheet metal of size 21cm x 30 cm. It needs to make an open top box of greatest volume, carved in the corners of the squares and counting the remaining edges.

I decide to help Denis to calculate the dimensions of the box.

First method (practical).
First I tried to find suitable sizes in a practical way. I took a suitable sized sheet of paper and began to cut square corners. In depend on length cutting out square boxes are having different volumes. So it is necessary to calculate the x side of the squares cut out, which box has the greatest volume. I made 4 boxes (Fig. 2), the resulting dimensions I recorded in the table. The volume of the resulting boxes I found on формулеV=a∙b∙c 
where is length, b - width, C - height of boxes, x is the side of the cut out squares.

Conclusion: a clear pattern is not observed, when increasing the cut side of a square x the amount initially increased, and then began to decrease. Method is clearly ineffective.

Second method (theoretical).

Then I decided to find a suitable size using the derivative.
Given: a rectangular sheet with dimensions 21cm× 30 cm
So, x cm length of side of square cut out. It is easy to see that

The size of the box is calculated by the formula:

, that is a function of the variable x. Let’s find the derivative of the resulting function

VꞋ = 12-204х+630.

=

- does not satisfy the conditions of the task.

=  – is the only solution,

So,x≈4cm;   a=30 – 2∙4 = 22cm; b=21 – 2∙4 = 13cm

V = 4 2213 = 1144(cm³).

The results comply with the created table.
Thus, I spent much less time by calculating the largest volume using the derivative and the result was the most accurate.

4. Using the derivative for solving tasks in economics

I understand that it is not always possible in the economy to usethe limit value due to the indivisibility of many economic calculations as well as the discreteness of some of the economic time indicators (for example, annual, quarterly, monthly, etc.). At the same time you can effectively use a limit value in many cases. I considered the examples of problems which solution involves the derivative and explains the economic meaning of the results.

Task 1.

The volume of production and produced by the workers in a working day, expressed by the function

, where t is time, h; and.

It is necessary to calculate productivity and its velocity of change after 1 hour after start and 1 hour before the end of the day.
Solution:
Labour productivity is expressed by the formula .

Then,

Productivity in 1 hour after the start is

(conventional units)

Productivity for 1 hour before finishing work

(conventional units)

The velocity of change in labour productivity means to the end of the working

So, ,

Figure 3

The figure 3 displaysthe productivity of labor during eight hours. You can see that the highest productivitywill be on the third hourand then it will gradually decrease.

So the labor productivity at the beginning of the working day is growing, and by the end is reduced, so the it is necessary to make job scheduling in agreement with this fact.

Task 2
Cement plant produces X tons of cement per day. By the contract it must daily put to build company not least 20 tons of cement.  Production capacity of the plant is such that the production of cement cannot exceed 90 tons per day.
To determine at what volume of production the unit cost will be greatest (lowest) if the cost function has the form: 

К = - х³+98х²+200х.

Solution.

Unit costs will be

К/х= - х²+98х+200

Our task is to find the largest and smallest values of the function.

Figure 4

y= - х²+98х+200 in the interval [20;90].

y’ = - 2x+98

y’ = 0 or   - 2x+98 = 0, x = 49

x = 49, critical point of the function.  Calculate the value of the function at the ends of the intervals and at the critical point.

You can see everything on theFigure 4.

f(20) = 1760, f(49) = 2601, f(90) = 920.

So, with the release of 49 tons of cement per day unit costs  is maximum, it is not economically viable but with the release of 90 tons per day is minimum, therefore, it may advice  to operate the plant at maximum capacity and  to find ways to improve the technology, as will continue to operate the law of diminishing returns. It will not increase output without reconstruction.

The language of mathematics is universal, that is an effective reflection of the universality of the laws surrounding diversity of the world.

The term of the derivative in Economics answers many important questions:

-limit values in microeconomics help to define the measure of response of quantity demanded for a product or service

- The optimal level of taxation

- Maximization of production, where it needs to make the conditions: marginal cost should equal marginal revenue

5. Conclusion

In my opinion, a derivative is an essential tool of economic analysis that allows to deepen the economic meaning of mathematical concepts and to express economic laws by means of mathematical formulas. The economic meaning of the derivative is the velocity of change of some economic process over time or with respect to the other studied factor. Many of the laws the theory of production and consumption, supply and demand are direct consequences of mathematical theorems.

This work helped to form the vision of a holistic picture of the information world, contributed to the development of imagination, creativity, had to the independent research activity, expanded the range of interests, fascinated, and helped to form the ability to make parallels and conclusions. The result of this work is the fact that now I can apply this knowledge in other subjects.

6. Bibliography

• Mathematics for the IB Diploma SL; Cambridge University Press, 2007-2010.
• F.F. Nagibin «Extremes»/- М. «Prosveshenie» 1966. P.30-35.
• V. G. boltyansky, «What is differentiation?», "Popular lectures on mathematics", Issue 17, Gostekhizdat, 1955, p. 64
• Stefan Banach "Differential and integral calculus". // - M., 1991
• Rybnikov K. A. "History of mathematics"
•  Advanced mathematics for economists. / under the editorship of N. W. Kremer M., 1999.