When the derivative is positive. Graphs of functions, derivatives of functions

The line y=3x+2 is tangent to the graph of the function y=-12x^2+bx-10. Find b , given that the abscissa of the touch point is less than zero.

Solution

Let x_0 be the abscissa of the point on the graph of the function y=-12x^2+bx-10 through which the tangent to this graph passes.

The value of the derivative at the point x_0 is equal to the slope of the tangent, i.e. y"(x_0)=-24x_0+b=3. On the other hand, the tangent point belongs to both the graph of the function and the tangent, i.e. -12x_0^2+bx_0-10= 3x_0 + 2. We get a system of equations \begin(cases) -24x_0+b=3,\\-12x_0^2+bx_0-10=3x_0+2. \end(cases)

Solving this system, we get x_0^2=1, which means either x_0=-1 or x_0=1. According to the condition of the abscissa, the touch points are less than zero, therefore x_0=-1, then b=3+24x_0=-21.

Answer

Condition

The figure shows a graph of the function y=f(x) (which is a broken line made up of three straight line segments). Using the figure, compute F(9)-F(5), where F(x) is one of the antiderivatives of f(x).

Solution

According to the Newton-Leibniz formula, the difference F(9)-F(5), where F(x) is one of the antiderivatives of the function f(x), is equal to the area of ​​the curvilinear trapezoid bounded by the graph of the function y=f(x), straight lines y=0 , x=9 and x=5. According to the graph, we determine that the specified curvilinear trapezoid is a trapezoid with bases equal to 4 and 3 and a height of 3.

Its area is equal to \frac(4+3)(2)\cdot 3=10.5.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Condition

The figure shows a graph of y \u003d f "(x) - the derivative of the function f (x), defined on the interval (-4; 10). Find the intervals of decreasing function f (x). In your answer, indicate the length of the largest of them.

Solution

As you know, the function f (x) decreases on those intervals, at each point of which the derivative f "(x) is less than zero. Considering that it is necessary to find the length of the largest of them, three such intervals are naturally distinguished from the figure: (-4; -2) ;(0;3);(5;9).

The length of the largest of them - (5; 9) is equal to 4.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Condition

The figure shows a graph of y \u003d f "(x) - the derivative of the function f (x), defined on the interval (-8; 7). Find the number of maximum points of the function f (x) belonging to the interval [-6; -2].

Solution

The graph shows that the derivative f "(x) of the function f (x) changes sign from plus to minus (there will be a maximum at such points) at exactly one point (between -5 and -4) from the interval [-6; -2 Therefore, there is exactly one maximum point on the interval [-6;-2].

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Condition

The figure shows a graph of the function y=f(x) defined on the interval (-2; 8). Determine the number of points where the derivative of the function f(x) is equal to 0 .

Solution

If the derivative at a point is equal to zero, then the tangent to the graph of the function drawn at this point is parallel to the Ox axis. Therefore, we find such points at which the tangent to the function graph is parallel to the Ox axis. On this chart, such points are extremum points (maximum or minimum points). As you can see, there are 5 extremum points.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Condition

The line y=-3x+4 is parallel to the tangent to the graph of the function y=-x^2+5x-7. Find the abscissa of the point of contact.

Solution

The slope of the line to the graph of the function y=-x^2+5x-7 at an arbitrary point x_0 is y"(x_0). But y"=-2x+5, so y"(x_0)=-2x_0+5. Angular the coefficient of the line y=-3x+4 specified in the condition is -3.Parallel lines have the same slopes.Therefore, we find such a value x_0 that =-2x_0 +5=-3.

We get: x_0 = 4.

Answer

Source: "Mathematics. Preparation for the exam-2017. profile level. Ed. F. F. Lysenko, S. Yu. Kulabukhova.

Condition

The figure shows a graph of the function y=f(x) and marked points -6, -1, 1, 4 on the x-axis. At which of these points is the value of the derivative the smallest? Please indicate this point in your answer.

(fig.1)

Figure 1. Graph of the derivative

Derivative Plot Properties

1. On increasing intervals, the derivative is positive. If the derivative at a certain point from some interval has a positive value, then the graph of the function on this interval increases.
2. On decreasing intervals, the derivative is negative (with a minus sign). If the derivative at a certain point from some interval has a negative value, then the graph of the function on this interval decreases.
3. The derivative at the point x is equal to the slope of the tangent drawn to the graph of the function at the same point.
4. At the maximum-minimum points of the function, the derivative is equal to zero. The tangent to the function graph at this point is parallel to the OX axis.

Example 1

According to the graph (Fig. 2) of the derivative, determine at what point on the segment [-3; 5] the function is maximum.

Figure 2. Graph of the derivative

Solution: On this segment, the derivative is negative, which means that the function decreases from left to right, and the largest value is on the left side at point -3.

Example 2

According to the graph (Fig. 3) of the derivative, determine the number of maximum points on the segment [-11; 3].

Figure 3. Graph of the derivative

Solution: The maximum points correspond to the points where the sign of the derivative changes from positive to negative. On this interval, the function changes sign twice from plus to minus - at point -10 and at point -1. So the number of maximum points is two.

Example 3

According to the graph (Fig. 3) of the derivative, determine the number of minimum points in the segment [-11; -one].

Solution: The minimum points correspond to the points where the sign of the derivative changes from negative to positive. On this segment, only -7 is such a point. This means that the number of minimum points on a given segment is one.

Example 4

According to the graph (Fig. 3) of the derivative, determine the number of extremum points.

Solution: The extremum is the point of both minimum and maximum. Find the number of points at which the derivative changes sign.

Further in the class, it is advisable to consider the key task: according to the graph of the derivative, the students must come up with (of course, with the help of the teacher) various questions related to the properties of the function itself. Naturally, these issues are discussed, if necessary, corrected, summarized, recorded in a notebook, after which the stage of solving these tasks begins. Here it is necessary to ensure that students not only give the correct answer, but are able to argue (prove) it, using the appropriate definitions, properties, rules.
Let's give an example of such a task: on the board (for example, using a projector), students are offered a graph of the derivative, 10 tasks were formulated on it (not quite correct or duplicate questions were rejected).
The function y = f(x) is defined and continuous on the interval [–6; 6].
From the graph of the derivative y \u003d f "(x), determine:

1) the number of intervals of increasing function y = f(x);
2) the length of the interval of decreasing function y = f(x);
3) the number of extremum points of the function y = f(x);
4) the maximum point of the function y = f(x);
5) the critical (stationary) point of the function y = f(x), which is not an extremum point;
6) the abscissa of the graph point at which the function y = f(x) takes the largest value on the segment ;
7) the abscissa of the graph point at which the function y = f(x) takes the smallest value on the segment [–2; 2];
8) the number of points of the graph of the function y = f(x), in which the tangent is perpendicular to the axis Oy;
9) the number of points in the graph of the function y = f(x), in which the tangent forms an angle of 60° with the positive direction of the Ox axis;
10) the abscissa of the point of the graph of the function y = f (x), in which the slope of the tangent takes the smallest value.
Answer: 1) 2; 2) 2; 3) 2; 4) –3; 5) –5; 6) 4; 7) –1; 8) 3; 9) 4; 10) –2.
To consolidate the skills of studying the properties of a function at home, students can be offered a task related to reading the same graph, but in one case it is a graph of a function, and in the other it is a graph of its derivative.

The article was published with the support of the forum of system administrators and programmers. On "CyberForum.ru" you will find forums about topics such as programming, computers, software discussion, web programming, science, electronics and home appliances, career and business, recreation, people and society, culture and art, home and economy, cars, motorcycles and more. On the forum you can get free help. You will learn more on the website, which is located at: http://www.cyberforum.ru/differential-equations/ .

The function y = f(x) is defined and continuous on the interval [–6; 5]. The figure shows:
a) graph of the function y = f(x);
b) graph of the derivative y \u003d f "(x).
Determine from the schedule:
1) minimum points of the function y = f(x);
2) the number of intervals of decreasing function y = f(x);
3) the abscissa of the point of the graph of the function y = f(x), in which it takes the largest value on the segment;
4) the number of points in the graph of the function y = f(x) in which the tangent is parallel to the Ox axis (or coincides with it).
Answers:
a) 1) -3; 2; four; 2) 3; 3) 3; 4) 4;
b) 1) –2; 4.6;2) 2; 3) 2; 4) 5.
For control, work can be organized in pairs: each student prepares a graph of the derivative on a card for his partner in advance and below offers 4-5 questions to determine the properties of the function. At the lessons they exchange cards, perform the proposed tasks, after which each checks and evaluates the work of the partner.

The final work in the form of the Unified State Examination for 11-graders necessarily contains tasks for calculating the limits, intervals of decreasing and increasing the derivative of a function, finding extremum points and plotting graphs. A good knowledge of this topic allows you to correctly answer several questions of the exam and not experience difficulties in further professional training.

Fundamentals of differential calculus is one of the main topics of mathematics of the modern school. She studies the use of the derivative to study the dependences of variables - it is through the derivative that you can analyze the increase and decrease of a function without referring to the drawing.

Comprehensive preparation of graduates for passing the exam on the educational portal "Shkolkovo" will help to deeply understand the principles of differentiation - understand the theory in detail, study examples of solving typical problems and try your hand at independent work. We will help you to eliminate gaps in knowledge - to clarify your understanding of the lexical concepts of the topic and the dependencies of quantities. Students will be able to repeat how to find intervals of monotonicity, which means the rise or fall of the derivative of a function on a certain interval, when the boundary points are included and not included in the intervals found.

Before starting the direct solution of thematic problems, we recommend that you first go to the "Theoretical Reference" section and repeat the definitions of concepts, rules and tabular formulas. Here you can also read how to find and record each interval of increasing and decreasing functions on the derivative graph.

All the information offered is presented in the most accessible form for understanding practically from scratch. The site provides materials for perception and assimilation in several different forms - reading, video viewing and direct training under the guidance of experienced teachers. Professional educators will tell you in detail how to find the intervals of increase and decrease of the derivative of a function using analytical and graphical methods. During the webinars, it will be possible to ask any question of interest both in theory and in solving specific problems.

Remembering the main points of the topic, look at the examples of increasing the derivative of a function, similar to the tasks of the exam options. To consolidate what you have learned, look in the "Catalogue" - here you will find practical exercises for independent work. The tasks in the section are selected at different levels of complexity, taking into account the development of skills. For each of them, for example, solution algorithms and correct answers are attached.

By choosing the "Constructor" section, students will be able to practice studying the increase and decrease of the derivative of a function on real versions of the USE, which are constantly updated with the latest changes and innovations.