An important concept in mathematics is function. With its help, one can visualize many processes occurring in nature, reflect using the formulas, tables and images on the graph the relationship between certain quantities. An example is the dependence of the pressure of a liquid layer on a body on immersion depth, acceleration - on the action of a certain force on an object, temperature increase - on transmitted energy, and many other processes. The study of the function involves the construction of a graph, the clarification of its properties, the domain of definition and values, the intervals of increase and decrease. An important point in this process is finding the extremum points. About how to do it right, and the conversation will go further.
About the concept on a concrete example
In medicine, the construction of a function graph can tell about the progress of the disease in the patient's body, clearly reflecting his condition. Suppose, on the OX axis, time is plotted in days, and on the OA axis, the temperature of the human body. The figure clearly shows how this figure rises sharply, and then falls. It is also easy to notice the singular points reflecting the moments when the function, increasing earlier, begins to decrease, and vice versa. These are extreme points, that is, critical values (maximum and minimum) in this case, the patient’s temperature, after which changes in his condition occur.
Tilt angle
It can be easily determined from the figure how the derivative of the function changes. If the straight lines of the chart go up over time, then it is positive. And the steeper they are, the more important is the derivative, since the angle of inclination is growing. During periods of decrease, this value takes negative values, turning to zero at the points of the extremum, and the graph of the derivative in the latter case is drawn parallel to the OX axis.
Any other process should be considered in a similar way. But best of all about this concept can tell the movement of various bodies, clearly shown in the graphs.
Traffic
Suppose a certain object moves in a straight line, evenly gaining speed. During this period, the change in the coordinate of the body graphically represents a certain curve, which the mathematician would call a parabola branch. At the same time, the function is constantly increasing, since the coordinates are changing faster every second. The velocity graph shows the behavior of the derivative, the value of which also increases. So, the movement has no critical points.
That would go on forever. But if the body suddenly decides to slow down, stop and start moving in a different direction? In this case, the coordinates will begin to decrease. And the function will go over a critical value and turn from increasing to decreasing.
In this example, we can again understand that the extremum points on the function graph appear at the moments when it ceases to be monotonic.
The physical meaning of the derivative
The previously described clearly showed that the derivative is essentially the rate of change of the function. In this specification, its physical meaning is concluded. Extreme points are critical areas on the chart. They can be clarified and discovered by calculating the value of the derivative, which turns out to be zero.
There is another sign, which is a sufficient condition for an extremum. The derivative in such places of the inflection changes its sign: from “+” to “-” in the region of the maximum and from “-” to “+” in the region of the minimum.
Gravity
Imagine another situation. Children, playing ball, threw him in such a way that he began to move at an angle to the horizon. At the initial moment, the speed of this object was the largest, but under the influence of gravity it began to decrease, and with each second by the same value, equal to approximately 9.8 m / s 2 . This is the value of the acceleration arising under the influence of gravity during free fall. On the moon, it would be about six times smaller.
The graph describing the movement of the body is a parabola with branches pointing down. How to find extremum points? In this case, this is the peak of the function, where the speed of the body (ball) takes zero value. The derivative of the function becomes zero. In this case, the direction, and hence the speed value, is reversed. The body flies down every second faster and faster, and accelerates by the same amount - 9.8 m / s 2 .
Second derivative
In the previous case, the graph of the velocity module is drawn as a straight line. This line is first directed downward, since the value of this quantity is constantly decreasing. Having reached zero at one point in time, then the indicators of this quantity begin to increase, and the direction of the graphic image of the speed module changes dramatically. Now the line is up.
Speed, being a derivative of the coordinate in time, also has a critical point. In this area, the function, initially decreasing, begins to increase. This is the place of the extremum point of the derivative function. In this case, the angle of inclination of the tangent becomes equal to zero. And the acceleration, being the second time derivative of the coordinate, changes sign from “-” to “+”. And the movement from equally slow becomes equally accelerated.
Acceleration graph
Now consider the four figures. Each of them displays a graph of the change over time of such a physical quantity as acceleration. In the case of "A", its value remains positive and constant. This means that the speed of the body, like its coordinate, is constantly increasing. If we imagine that the object will move in this way infinitely long, the function, reflecting the dependence of the coordinate on time, will be constantly increasing. It follows from this that it does not have critical areas. There are also no extremum points on the graph of the derivative, that is, a linearly varying velocity.
The same applies to case "B" with a positive and ever-increasing acceleration. True, the graphs for the coordinates and speeds here will be somewhat more complicated.
When acceleration tends to zero
Examining figure "B", one can observe a completely different picture characterizing the movement of the body. Its speed will be graphically represented by a parabola with branches pointing down. If we continue the line describing the change in acceleration until it intersects with the OX axis, and beyond, we can imagine that up to this critical value, where the acceleration turns out to be zero, the speed of the object will increase more slowly. The extremum point of the derivative of the coordinate function will be just at the top of the parabola, after which the body will radically change the nature of the movement and begin to move in the other direction.
In the latter case, “G”, the nature of the movement cannot be precisely determined. It is only known here that there is no acceleration for some period under consideration. This means that the object can remain in place or the movement occurs at a constant speed.
The task of adding coordinates
Let's move on to the tasks that are often encountered when studying algebra at school and are offered to prepare for the exam. The figure below shows a graph of the function. It is required to calculate the sum of extremum points.
We do this for the ordinate axis, determining the coordinates of the critical areas where the change in the characteristics of the function is observed. Simply put, we find the values along the OX axis for the inflection points, and then proceed to the addition of the obtained terms. According to the graph, it is obvious that they take the following values: -8; -7; -5; -3; -2; 1; 3. In total, this is -21, which is the answer.
Optimal solution
It is not worth explaining how important it may be in the implementation of practical tasks to select the optimal solution. After all, there are many ways to achieve the goal, and the best way out, as a rule, is only one. This is extremely necessary, for example, in the design of ships, spaceships and aircraft, architectural structures to find the optimal shape of these man-made objects.
The speed of vehicles largely depends on the competent minimization of the resistance that they experience when moving through water and air, on overloads arising under the influence of gravitational forces and many other indicators. A ship at sea needs qualities such as stability during a storm; minimal draft is important for a river ship. When calculating the optimal design, the extremum points on the graph can clearly give an idea of the best solution to a complex problem. The tasks of such a plan are often solved in economics, in economic fields, in many other life situations.
From ancient history
Tasks for extremum even occupied the ancient sages. Greek scholars have successfully solved the mystery of squares and volumes through mathematical calculations. They were the first to realize that on a plane of various shapes that have the same perimeter, the circle always has the largest area. Similarly, a ball is endowed with a maximum volume among other objects in space with the same surface size. Such famous personalities as Archimedes, Euclid, Aristotle, Apollonius devoted themselves to solving such problems. Heron was able to find the extremum points very well, who, having resorted to calculations, built ingenious devices. These included machines moving through steam, working on the same principle as pumps and turbines.
The construction of Carthage
There is a legend, the plot of which is built on solving one of the extreme problems. The result of the business approach, which was shown by the Phoenician princess, who turned to the sages for help, was the construction of Carthage. The land for this ancient and illustrious city was presented to Didone (the so-called ruler) by the leader of one of the African tribes. The area of the allotment did not seem to him at first very large, since under the contract it was to be covered with cowhide. But the princess ordered her soldiers to cut it into thin strips and make a belt out of them. It turned out so long that it covered the site where the whole city fit.
The origins of mathematical analysis
And now we are transported from ancient times to a later era. Interestingly, in the 17th century, Kepler prompted a meeting with a wine seller to understand the basics of mathematical analysis. The merchant was so versed in his profession that he could easily determine the volume of the beverage in the barrel by simply lowering the iron rope there. Reflecting on such a curiosity, the famous scientist managed to solve this dilemma for himself. It turns out that the skilled barrels of those times got the hang of making vessels so that at a certain height and radius of the circumference of the fastening rings, they had maximum capacity.
This was a reason for Kepler to further thought. Bochars came to the optimal solution by the method of long search, mistakes and new attempts, passing on their experience from generation to generation. But Kepler wanted to speed up the process and learn to do the same in a short time by mathematical calculations. All his achievements, picked up by his colleagues, turned into the now known theorems of Fermat and Newton - Leibniz.
The task of finding the maximum area
Imagine that we have a wire whose length is 50 cm. How to make a rectangle with the largest area from it?
When starting a solution, one should proceed from simple and well-known truths. It is clear that the perimeter of our figure will be 50 cm. It also consists of doubled lengths on both sides. This means that by designating one of them as “X”, the other can be expressed as (25 - X).
From here we get the area equal to X (25 - X). This expression can be represented as a function that takes many values. The solution to the problem requires finding the maximum of them, which means that you need to know the extremum points.
To do this, find the first derivative and equate it to zero. The result is a simple equation: 25 - 2X = 0.
From it we learn that one of the sides is X = 12.5.
Therefore, the other: 25 - 12.5 = 12.5.
It turns out that the solution to the problem will be a square with a side of 12.5 cm.
How to find the maximum speed
Consider another example. Imagine that there exists a body whose rectilinear motion is described by the equation S = - t 3 + 9t 2 - 24t - 8, where the distance traveled is expressed in meters and time in seconds. It is required to find the maximum speed. How to do it? Downloaded find the speed, that is, the first derivative.
We get the equation: V = - 3t 2 + 18t - 24. Now, to solve the problem, again we need to find the extremum points. This must be done in the same way as in the previous task. We find the first derivative of speed and equate it to zero.
We get: - 6t + 18 = 0. Hence t = 3 s. This is the time when the speed of the body is critical. Substitute the obtained data into the velocity equation and get: V = 3 m / s.
But how to understand that this is precisely the maximum speed, because the critical or critical points of a function can be its largest or smallest values? To check, it is necessary to find the second derivative of speed. It is expressed by the number 6 with a minus sign. This means that the point found is the maximum. And in the case of a positive value of the second derivative would be a minimum. So, the solution found was the right one.
The problems given as an example are only part of those that can be solved by being able to find the extremum points of the function. In fact, there are many more. And such knowledge opens up unlimited possibilities for human civilization.