This article discusses solution algorithms typical tasks to find the maximum and minimum points, the largest and smallest values of the function. Examples of problem solving are provided. Presented training options, corresponding to task No. 12. Examples taken from open bank FIPI.
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VC. Kuznetsova,
mathematics teacher at State Budgetary Educational Institution “School No. 329”, Moscow,
Candidate of Pedagogical Sciences
Getting ready for the Unified State Exam
Student Guide
In this article we will talk about problems in which functions are considered and the conditions contain questions related to their study.
This is a whole group of problems included in the Unified State Examination in mathematics. Usually the question is about finding the maximum (minimum) points or determining the largest (smallest) value of a function on a given interval.
Considered:
Power and irrational functions.
Rational functions.
Research of works and private ones.
Logarithmic functions.
Trigonometric functions.
To successfully solve these problems, you need to know the theory of limits, the concept of a derivative, the properties of the derivative for studying graphs of functions and its geometric meaning. The properties of the derivative are necessary to study the behavior of a function as it increases and decreases.
What else do you need to know to solve problems on the study of functions: the table of derivatives and the rules of differentiation. This basic knowledge, on the topic of derivatives. Derivatives elementary functions you need to know perfectly well.
Properties of the derivative
1. The derivative at increasing intervals has a positive sign (when substituting a value from the interval into the derivative, a positive number is obtained).
This means that if the derivative at a certain point from a certain interval has positive value, then the graph of the function increases on this interval.
2. On decreasing intervals, the derivative has negative sign(when substituting a value from an interval into the derivative expression, a negative number is obtained).
This means that if the derivative at a certain point from a certain interval has negative meaning, then the graph of the function decreases on this interval.
Problems of finding maximum and minimum points
Algorithm for finding maximum (minimum) points of a function:
1. Find the derivative of the function f'(x).
2. Find the zeros of the derivative (by equating the derivative to zero f’(x)=0 and solve the resulting equation). We also find points at which the derivative does not exist (in particular, this applies to fractional rational functions).
3. We mark the obtained values on the number line and determine the signs of the derivative on these intervals by substituting the values from the intervals into the derivative expression.
The conclusion will be one of two:
1. The maximum point is the point at which the derivative changes value from positive to negative.
2. The minimum point is the point at which the derivative changes its value from negative to positive.
Problems to find the largest or smallest value
functions on an interval.
In another type of problem, you need to find the greatest or smallest value functions on a given interval.
Algorithm for finding the largest (smallest) value of a function:
1. Determine whether there are maximum (minimum) points. To do this, we find the derivative f’(x) , then solve f’(x)=0 .
2. We determine whether the obtained points belong to the given interval and write down those lying within its limits.
3. Substitute the boundaries into the original function (not into the derivative, but into the one given in the condition) given interval and points (maximum-minimum) lying within the interval.
4. Calculate the function values.
5. We select the largest (smallest) value from the obtained values, depending on what question was posed in the problem and then write down the answer.
Let's look at examples of solving problems involving the study of functions.
Example 1.
Find maximum and minimum points functions
Solution:
Let's find the zeros of the derivative:
Let us determine the signs of the derivative of the function and depict it in the figure
function behavior:
+ _ +
Y -4 4
Max min
The required maximum point is x= -4, the desired minimum point is x=4.
Answer: −4; 4.
Example 2.
Find smallest value functions on the interval.
Solution.
Let's find the derivative given function:
Let's find the zeros of the derivative:
_ +
Y 0 3 4
At point x=3 the given function has a minimum, which is its smallest value on given segment. Let's find this smallest value:
Answer: −54.
Example 3.
Find highest value functions on the interval.
Solution.
Let's find the derivative of the given function:
Let's find the zeros of the derivative:
Let's determine the signs of the derivative of the function and depict the behavior of the function in the figure:
_ +
Y -2 -1 0
At point x= -1, the given function has a maximum, which is its largest value on a given segment. Let's find this greatest value:
Answer: 6.
We invite you to solve training options for finding maximum and minimum points, the largest and smallest values of power and irrational functions. The tasks correspond to task No. 12 and are taken from the FIPI open bank.
Training on the topic
"Study of power and irrational functions"
Task No. 12
Option 1.
3. on the segment
on the segment
Option 2.
1. .
2. Find the minimum point of the function.
3. Find the smallest value of the function on the segment.
4. Find the largest value of the function on the segment.
Option 3.
1. Find the maximum point of the function.
2. Find the minimum point of the function.
3. Find the smallest value of the function
on the segment.
4,. Find the largest value of the function
on the segment.
Option 4.
1. Find the maximum point of the function
2. Find the minimum point of the function
3. Find the smallest value of the function
on the segment
4. Find the largest value of the function
on the segment
Option 5.
1. Find the maximum point of the function.
2. Find the minimum point of the function
3. Find the smallest value of the function
on the segment
4. Find the largest value of the function
on the segment
Option 6.
1. Find the maximum point of the function
2. Find the minimum point of the function
3. Find the smallest value of the function on the segment
4. Find the largest value of the function on the segment
Option 7.
1. Find the maximum point of the function
2. Find the minimum point of the function
3. Find the smallest value of the function
On the segment
4. Find the largest value of the function
On the segment
Option 8.
1. Find the maximum point of the function
2. Find the minimum point of the function
on the segment
4. Find the largest value of the function
On the segment
Option 9.
3. Find the smallest value of the function on the segment
4. Find the largest value of the function on the segment
Option 10.
on the segment
4. Find the largest value of the function
On the segment
Option 11.
1. Find the minimum point of the function
2. Find the smallest value of the function
on the segment
3. Find the minimum point of the function
4. Find the largest value of the function
On the segment
Option 12.
on the segment
4. Find the smallest value of the function
on the segment
Option 13.
1. Find the largest value of the function
On the segment
2. Find the largest value of the function
on the segment
Research of functions. In this article we will talk about problems in which functions are considered and the conditions contain questions related to their study. Let's consider the main theoretical points that need to be known and understood to solve them.
This is a whole group of problems included in the Unified State Examination in mathematics. Usually the question is about finding the maximum (minimum) points or determining the largest (smallest) value of a function on a given interval.Considered:
— Power and irrational functions.
— Rational functions.
— Study of works and private ones.
— Logarithmic functions.
— Trigonometric functions.
If you understand the theory of limits, the concept of a derivative, the properties of the derivative for studying graphs of functions and its , then such problems will not cause you any difficulty and you will solve them with ease.
The information below is theoretical points, the understanding of which will allow you to understand how to solve similar tasks. I will try to present them in such a way that even those who have missed this topic or have studied it poorly can solve such problems without much difficulty.
In the problems of this group, as already mentioned, it is required to find either the minimum (maximum) point of the function, or the largest (smallest) value of the function on the interval.
Minimum and maximum points.Properties of derivative.
Consider the graph of the function:
Point A is the maximum point; on the interval from O to A the function increases, and on the interval from A to B it decreases.
Point B is the minimum point; on the interval from A to B the function decreases, on the interval from B to C it increases.
At these points (A and B), the derivative becomes zero (equal to zero).
The tangents at these points are parallel to the axis ox.
I will add that the points at which the function changes its behavior from increasing to decreasing (and vice versa, from decreasing to increasing) are called extrema.
Important point:
1. The derivative at increasing intervals has a positive sign (nWhen you substitute a value from an interval into its derivative, you get a positive number).
This means that if the derivative at a certain point from a certain interval has a positive value, then the graph of the function on this interval increases.
2. At decreasing intervals, the derivative has a negative sign (when substituting a value from the interval into the derivative expression, a negative number is obtained).
This means that if the derivative at a certain point from a certain interval has a negative value, then the graph of the function decreases on this interval.
This needs to be clearly understood!!!
Thus, by calculating the derivative and equating it to zero, you can find points that divide the number line into intervals.At each of these intervals, you can determine the sign of the derivative and then draw a conclusion about its increase or decrease.
*Special mention should be made about the points at which the derivative does not exist. For example, we can obtain a derivative whose denominator vanishes at a certain x. It is clear that for such x the derivative does not exist. So, this point must also be taken into account when determining the intervals of increase (decrease).
The function at points where the derivative is equal to zero does not always change its sign. There will be a separate article about this. There will be no such tasks on the Unified State Examination itself.
The above properties are necessary to study the behavior of a function for increasing and decreasing.
What else you need to know to solve the specified problems: the table of derivatives and the rules of differentiation. There is no way without this. This is basic knowledge on the topic of derivatives. You should know the derivatives of elementary functions perfectly well.
Calculating the derivative of a complex functionf(g(x)), imagine the functiong(x) this is a variable and then calculate the derivativef’(g(x)) By tabular formulas as an ordinary derivative of a variable. Then multiply the result by the derivative of the functiong(x) .
Watch Maxim Semenikhin's video tutorial on complex functions:
Problems of finding maximum and minimum points
Algorithm for finding maximum (minimum) points of a function:
1. Find the derivative of the function f’(x).
2. Find the zeros of the derivative (by equating the derivative to zero f’(x)=0 and solve the resulting equation). We also find points at which the derivative does not exist(in particular this applies to fractional rational functions).
3. We mark the obtained values on the number line and determine the signs of the derivative on these intervals by substituting the values from the intervals into the derivative expression.
The conclusion will be one of two:
1. The maximum point is the pointin which the derivative changes value from positive to negative.
2. The minimum point is the pointin which the derivative changes its value from negative to positive.
Problems to find the largest or smallest value
functions on an interval.
In another type of problem, you need to find the largest or smallest value of a function on a given interval.
Algorithm for finding the largest (smallest) value of a function:
1. Determine whether there are maximum (minimum) points. To do this, we find the derivative f’(x) , then we decide f’(x)=0 (points 1 and 2 from the previous algorithm).
2. We determine whether the obtained points belong to the given interval and write down those lying within its limits.
3. We substitute into the original function (not into the derivative, but into the one given in the condition) the boundaries of the given interval and the points (maximum-minimum) lying within the interval (step 2).
4. Calculate the function values.
5. We select the largest (smallest) value from those obtained, depending on what question was posed in the problem and then write down the answer.
Question: why is it necessary to look for maximum (minimum) points in problems of finding the largest (smallest) value of a function?
The best way to illustrate this is to look at the schematic representation of the graphs of the specified functions:
In cases 1 and 2, it is enough to substitute the boundaries of the interval to determine the largest or smallest value of the function. In cases 3 and 4, it is necessary to find the zeros of the function (maximum-minimum points). If we substitute the boundaries of the interval (without finding the zeros of the function), we will get the wrong answer, this can be seen from the graphs.
And the whole point is that, given a given function, we cannot see what the graph looks like on the interval (whether it has a maximum or minimum within the interval). Therefore, be sure to find the zeros of the function!!!
If the equation f'(x)=0 will not have a solution, this means that there are no maximum-minimum points (Figure 1,2), and to find the problem in question in this function We substitute only the boundaries of the interval.
Another important point. Remember that the answer must be an integer or finite number decimal. When you calculate the largest and smallest value of a function, you will get expressions with e and pi, as well as expressions with the root. Remember that you do not need to calculate them completely, and it is clear that the result of such expressions will not be the answer. If you want to calculate such a value, then do it (numbers: e ≈ 2.71 Pi ≈ 3.14).
I wrote a lot, maybe I got confused? By specific examples you will see that everything is simple.
Next I want to tell you little secret. The fact is that many problems can be solved without knowledge of the properties of the derivative and even without the rules of differentiation. I will definitely tell you about these nuances and show you how it’s done? do not miss!
But then why did I present the theory at all and also say that it is necessary to know it. That's right - you need to know. If you understand it, then no problem in this topic will baffle you.
The “tricks” you will learn about will help you when solving specific (some) prototype problems. TOIt is, of course, convenient to use these techniques as an additional tool. The problem can be solved 2-3 times faster and save time on solving part C.
All the best!
Sincerely, Alexander Krutitskikh.
P.S: I would be grateful if you tell me about the site on social networks.
Getting ready for the Unified State Exam
“The devil is not as scary as he is painted” - there is a saying. Exams are coming up soon. Single State exam- this is just an exam and summing up the results of schooling. Not the easiest. Your task is one thing - to make the most of your efforts in the time remaining before this test and try to prepare better. Try to solve it as much as possible more tasks- this will allow you to feel both your knowledge and the time it takes to complete tasks. Since the exam is allotted certain time- this will also be important for you - to clearly control your time in order to get as much done as possible.
For a bad student, the crocodile flies and the exam bites! Don't make the crocodile laugh and be on top!
ELECTRONIC EDUCATIONAL AND METHODOLOGICAL MANUAL: “We are preparing for the State Examination and the Unified State Examination. PROBLEMS WITH INTEREST.” (Gilmieva G.G.)
Presentation for a lesson. Content: theoretical part solving problems with percentages, methods for solving problems. There is a button “open solution”, “close solution” to check the correctness of the solution to the problem. Addition: tasks for independent work. The content is structured in such a way that you can jump to any block of tasks. Made in a form convenient for teachers and students to regulate viewing. Great way presenting material on a smart board and on any type of screen
Methods for solving problems on simple and complex percentage growth. (authors: Gilmieva G.G., Amanullina Z.A.)
Abstract: “Students often have difficulty solving problems with percentages. One reason is that commonly used mathematics textbooks tend to give standard tasks on interest. Word problems, including problems involving percentages, are found in Unified State Exam tests in mathematics, both in the 9th and 11th grades. The article outlines a methodology for solving problems involving simple and complex interest growth (the so-called “banking problems”). this work can be used by teachers to develop elective course dedicated to word problems with percentages, and will also be useful to students educational institutions For self-study for final tests." (Download Word file)
Several ways to solve one geometric problem. (authors: Gilmieva G.G., Khusnutdinova L.G.)
Abstract: The article discusses three ways to solve the stereometric problem C2 from Unified State Exam test, including coordinate method. (Download Word file)
Article “Learning to pass the State Exam and the Unified State Exam” . (author Gilmieva G.G.) The task of a mathematics teacher is to psychologically and methodically prepare schoolchildren for the Unified State Exam in such a way that he independently manages to score the maximum number of points possible for him. The article provides recommendations. Download DOS file.
Article. “We are preparing for the Unified State Exam in mathematics. Using the domain of functions and the set of function values to solve equations."(author Gilmieva G.G.)
The article presents a method for solving equations of the form f(x)=g(x), based on the use of the domain of definition of the function. Another property of a function - boundedness - can help find the roots of an equation (or inequality), or refute the statement about their existence. The article shows a method for solving equations based on this property. It is often called the “mini-max” method or the “majorant method”. Solutions of examples are given.
Mood! This is what is the main assistant! Man - it sounds! A Clever man- this sounds even cooler! Begin!
I bring to your attention material (prototypes of problems) for preparing for the Unified State Exam in mathematics (problems are presented with answers).
1. Simple text fordachi 1/Rounding down (6 examples), Rounding up (13 examples), Miscellaneous tasks(12 examples)/ Download Word file
2. Simple word problems 2/Rounding with excess (6 examples), Rounding with deficiency (1 example), Percentage, rounding (31 examples), Various tasks (12 examples)/ Download Word file
3. Graphs and diagrams/Determination of quantities from a graph (22 examples), Determination of quantities from a diagram (18 examples), Calculation of quantities from a graph or diagram (5 examples)/ Download Word file
4. Choosing the best option/Choice of va-ri-an-ta from two possible (5 examples), Choice of va-ri-an-ta from three possible (24 examples), Choice of va-ri-an-ta from four possible (5 examples)/ Download Word file
5. Calculation of lengths and areas/Triangle (58 examples), Rectangle (33 examples), Parallelogram (12 examples), Rhombus (10 examples), Trapezoid (26 examples), Arbitrary quadrilateral(28 examples), Polygon (3 examples), Problems on a square lattice (15 examples), Circle and its elements (24 examples), Inscribed and circumscribed circles (13 examples), Vectors (24 examples ), Coordinate plane(59 examples)/ Download Word file
6. Probability theory/Classical definition of probability (39 examples), Theorems on the probabilities of events (29 examples)/ Download Word file
7. The simplest equations/Linear, quadratic, cubic equations (9 examples), Rational equations(8 examples), Irrational equations(9 examples), Exponential equations(10 examples), Logarithmic equations(14 examples), Trigonometric equations(3 examples)/ Download Word file
8. Planimetry Problems related to angles /Right triangle: calculation of angles (55 examples), Right triangle: calculation of angles external corners(29 examples), Right triangle: calculation of elements (75 examples), Isosceles triangle: calculation of angles (38 examples), (Isosceles triangle: calculation of elements (49 examples), General triangles (15 examples), Parallelogram (17 examples), Rectangle (10 examples), Trapezoid (26 examples), Central and inscribed angles (21 examples), Tangent, chord, secant (9 examples), Circle inscribed in a triangle (10 examples), Circle, inscribed in a quadrilateral (9 examples), Circle inscribed in a polygon (3 examples), Circle described around a triangle (12 examples), Circle described around a quadrilateral ( 15 examples), Circle described around a polygon (3 examples)/ Download Word file
9. Derivative and antiderivative /Physical meaning derivative (5 examples), Geometric meaning derivative, tangent (16 examples), Application of derivative to the study of functions (22 examples), Antiderivative (4 examples)/ Download Word file
10. Stereometry 1/Cube (11 examples), Rectangular parallelepiped(8 examples), Prism (39 examples), Pyramid (34 examples), Elements of composite polyhedra (16 examples), Surface area of a composite polyhedron (16 examples), Volume of a composite polyhedron (13 examples), Combinations of bodies (18 examples), Cylinder (15 examples), Cone (17 examples), Sphere (6 examples)/ Download Word file
11. Calculations and transformations/Number conversions rational expressions(6 examples), Transformations of algebraic expressions and fractions (23 examples), Transformations of numbers irrational expressions(10 examples), Transformations of alphabetic irrational expressions (12 examples), Transformations of numeric expressions demonstrative expressions(17 examples), Conversions of alphabetic exponential expressions (30 examples), Conversions of numeric expressions logarithmic expressions(29 examples), Transforming literal logarithmic expressions (3 examples), Calculating values trigonometric expressions(20 examples), Conversions of numerical trigonometric expressions (27 examples), Conversions of alphabetic trigonometric expressions (2 examples)/ Download Word file
12. Problems with applied content/Linear equations and inequalities (2 examples), Quadratic and power equations and inequalities (17 examples), Rational equations and inequalities (14 examples), Irrational equations and inequalities (9 examples), Exponential equations and inequalities (4 examples), Logarithmic equations and inequalities (4 examples), Trigonometric equations and inequalities (16 examples), Miscellaneous problems (5 examples) ./ Download Word file
13. Stereometry 2/Stereometry 2 (1 example), Rectangular parallelepiped (20 examples), Prism (20 examples), Pyramid (22 examples), Cylinder (9 examples), Cone (21 example), Sphere (9 examples)/ Download Word file
14. Word problems/Problems on percentages, alloys and mixtures (17 examples), Problems on movement in a straight line (29 examples), Problems on movement in a circle (5 examples), Problems on movement on water (13 examples), Problems for joint work (24 examples), Problems on progression (9 examples)/ Download Word file
15. The largest and smallest value of a function/Study of power and ir-ra-ci-o-nal functions (51 examples), Study of ra-tsi-o-nal functions (10 examples), Study of pro-iz-ve-de- tions and private ones (28 examples), Study of ka-za-tel-nyh and log-ga-rif-mi-che-sky functions (18 examples), Study of tri-go-no-met-ri-che-sky functions (29 examples), Study of functions without the help of derivatives (12 examples)/ Download Word file
Source: http://reshuege.ru (if you wish, on this site you can test your knowledge and take testing to pass the Unified State Exam in mathematics).
These are no longer flowers! Those who work always have a rich harvest! To activate your brain, be sure to eat sweets in the form of fruits and berries. And tasty, and healthy, and “relaxing”!
Below are prototypes of Unified State Exam assignments - WITHOUT ANSWERS (from the site reshuege.rf). Assignments in doc format, each assignment contains one example. For those who want to polish the solution, go to the website resuege.rf. There are many options out there. Test form passing the Unified State Exam, which will allow you to try to rehearse for passing the exam. My advice is to decide to find your gaps and have time to learn how to solve problems that are difficult for you. And if you get stuck in making decisions, contact me on this website and together we’ll look at solutions. The main thing in successful passing the Unified State Exam- this is a serious attitude towards him. This means that it is better to work a little now so that you can sunbathe on the beach later - after successfully passing the exam! Good luck!
P rototypes of Unified State Exam assignments (source: reshuege.rf, Dmitry Gushchin’s website)
Determining a value from a graph or diagram
Planimetry. Calculation of lengths and areas
Selecting an option from 2 or 3 possible
Planimetry. problems with angles