The figure shows a graph of the derivative of the function f(x) defined on the interval [–5; 6]. Find the number of points of the graph f (x), in each of which the tangent drawn to the graph of the function coincides or is parallel to the x-axis

The figure shows a graph of the derivative of a differentiable function y = f(x).

Find the number of points in the graph of the function, belonging to the segment[–7; 7], in which the tangent to the graph of the function is parallel to the straight line given by the equation y = –3x.

Material point M starts from point A and moves in a straight line for 12 seconds. The graph shows how the distance from point A to point M changed over time. The abscissa shows the time t in seconds, the ordinate shows the distance s in meters. Determine how many times during the movement the speed of point M went to zero (ignore the beginning and end of the movement).

The figure shows sections of the graph of the function y \u003d f (x) and the tangent to it at the point with the abscissa x \u003d 0. It is known that this tangent is parallel to the straight line passing through the points of the graph with the abscissas x \u003d -2 and x \u003d 3. Using this, find the value of the derivative f "(o).

The figure shows a graph y = f'(x) - the derivative of the function f(x), defined on the segment (−11; 2). Find the abscissa of the point at which the tangent to the graph of the function y = f(x) is parallel to the x-axis or coincides with it.

The material point moves rectilinearly according to the law x(t)=(1/3)t^3-3t^2-5t+3, where x is the distance from the reference point in meters, t is the time in seconds measured from the beginning of the movement. At what point in time (in seconds) was her speed equal to 2 m/s?

The material point moves along a straight line from the initial to the final position. The figure shows a graph of its movement. The abscissa shows the time in seconds, the ordinate shows the distance from initial position points (in meters). Find average speed dot movement. Give your answer in meters per second.

The function y \u003d f (x) is defined on the interval [-4; four]. The figure shows a graph of its derivative. Find the number of points in the graph of the function y \u003d f (x), the tangent in which forms an angle of 45 ° with the positive direction of the Ox axis.

The function y \u003d f (x) is defined on the segment [-2; four]. The figure shows a graph of its derivative. Find the abscissa of the graph point of the function y \u003d f (x), at which it takes smallest value on the segment [-2; -0.001].

The figure shows the graph of the function y \u003d f (x) and the tangent to this graph, drawn at the point x0. The tangent is given by the equation y = -2x + 15. Find the value of the derivative of the function y = -(1/4)f(x) + 5 at the point x0.

Seven points are marked on the graph of the differentiable function y = f(x): x1,..,x7. Find all marked points where the derivative of the function f(x) is greater than zero. Enter the number of these points in your answer.

The figure shows a graph y \u003d f "(x) of the derivative of the function f (x), defined on the interval (-10; 2). Find the number of points at which the tangent to the graph of the function f (x) is parallel to the line y \u003d -2x-11 or matches it.

The figure shows a graph of y \u003d f "(x) - the derivative of the function f (x). Nine points are marked on the x-axis: x1, x2, x3, x4, x5, x6, x6, x7, x8, x9.
How many of these points belong to the intervals of decreasing function f(x) ?

The figure shows the graph of the function y \u003d f (x) and the tangent to this graph, drawn at the point x0. The tangent is given by the equation y = 1.5x + 3.5. Find the value of the derivative of the function y \u003d 2f (x) - 1 at the point x0.

The figure shows a graph y=F(x) of one of antiderivative functions f(x). Six points with abscissas x1, x2, ..., x6 are marked on the graph. At how many of these points does the function y=f(x) take negative values?

The figure shows the schedule of the car along the route. Time is plotted on the abscissa axis (in hours), on the ordinate axis - the distance traveled (in kilometers). Find the average speed of the car on this route. Give your answer in km/h

The material point moves rectilinearly according to the law x(t)=(-1/6)t^3+7t^2+6t+1, where x is the distance from the reference point (in meters), t is the time of movement (in seconds). Find its speed (in meters per second) at time t=6 s

The figure shows a graph of the antiderivative y \u003d F (x) of some function y \u003d f (x), defined on the interval (-6; 7). Using the figure, determine the number of zeros of the function f(x) in a given interval.

The figure shows a graph y = F(x) of one of the antiderivatives of some function f(x) defined on the interval (-7; 5). Using the figure, determine the number of solutions to the equation f(x) = 0 on the segment [- 5; 2].

The figure shows a graph of a differentiable function y=f(x). Nine points are marked on the x-axis: x1, x2, ... x9. Find all marked points where the derivative of f(x) is negative. Enter the number of these points in your answer.

The material point moves rectilinearly according to the law x(t)=12t^3−3t^2+2t, where x is the distance from the reference point in meters, t is the time in seconds measured from the beginning of the movement. Find its speed (in meters per second) at time t=6 s.

The figure shows the graph of the function y=f(x) and the tangent to this graph drawn at the point x0. The tangent equation is shown in the figure. find the value of the derivative of the function y=4*f(x)-3 at the point x0.

(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 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 negative meaning, then the graph of the function decreases on this interval.
3. The derivative at the point x is angular coefficient 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 highest value located 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 given segment-- 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.

B8. USE

1. The figure shows a graph of the function y=f(x) and a tangent to this graph, drawn at a point with the abscissa x0. Find the value of the derivative of the function f(x) at the point x0. Answer: 2

2.

Answer: -5

3.

On the interval (–9; 4).

Answer: 2

4.

Find the value of the derivative of the function f(x) at the point x0 Answer: 0.5

5. Find the point of contact between the line y = 3x + 8 and the graph of the function y = x3+x2-5x-4. Indicate the abscissa of this point in your answer. Answer: -2

6.

Determine the number of integer values ​​of the argument for which the derivative of the function f(x) is negative. Answer: 4

7.

Answer: 2

8.

Find the number of points where the tangent to the graph of the function f(x) is parallel to or coincides with the line y=5–x. Answer: 3

9.

Interval (-8; 3).

Direct y = -20. Answer: 2

10.

Answer: -0.5

11

Answer: 1

12. The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa x0.

Find the value of the derivative of the function f(x) at the point x0. Answer: 0.5

13. The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa x0.

Find the value of the derivative of the function f(x) at the point x0. Answer: -0.25

14.

Find the number of points where the tangent to the graph of the function f(x) is parallel to or coincides with the line y = x+7. Answer: 4

15

Find the value of the derivative of the function f(x) at the point x0. Answer: -2

16.

interval (-14;9).

Find the number of maximum points of the function f(x) on the interval [-12;7]. Answer: 3

17

on the interval (-10; 8).

Find the number of extremum points of the function f(x) on the interval [-9;7]. Answer: 4

18. The line y = 5x-7 touches the graph of the function y = 6x2 + bx-1 at a point with an abscissa less than 0. Find b. Answer: 17

19

Answer:-0,25

20

Answer: 6

21. Find the tangent to the graph of the function y=x2+6x-7, parallel to the line y=5x+11. In your answer, indicate the abscissa of the point of contact. Answer: -0,5

22.

Answer: 4

23. f "(x) on the interval (-16; 4).

On the segment [-11; 0] find the number of maximum points of the function. Answer: 1

B8 Graphs of functions, derivatives of functions. Function research . USE

1. The figure shows a graph of the function y=f(x) and a tangent to this graph, drawn at a point with the abscissa x0. Find the value of the derivative of the function f(x) at the point x0.

2. The figure shows a graph of the derivative of the function f(x) defined on the interval (-6; 5).

At what point of the segment [-5; -1] f(x) takes the smallest value?

3. The figure shows a graph of the derivative of the function y = f(x), defined

On the interval (–9; 4).

Find the number of points where the tangent to the graph of the function f(x) is parallel to the line

y = 2x-17 or the same.

4. The figure shows the graph of the function y = f(x) and the tangent to it at the point with the abscissa x0.

Find the value of the derivative of the function f(x) at the point x0

5. Find the point of contact between the line y = 3x + 8 and the graph of the function y = x3+x2-5x-4. Indicate the abscissa of this point in your answer.

6. The figure shows a graph of the function y = f(x), defined on the interval (-7; 5).

Determine the number of integer values ​​of the argument for which the derivative of the function f(x) is negative.

7. The figure shows a graph of the function y \u003d f "(x), defined on the interval (-8; 8).

Find the number of extremum points of the function f(x) belonging to the segment [-4; 6].

8. The figure shows a graph of the function y \u003d f "(x), defined on the interval (-8; 4).

Find the number of points where the tangent to the graph of the function f(x) is parallel to or coincides with the line y=5–x.

9. The figure shows a graph of the derivative of the function y = f(x) defined on

Interval (-8; 3).

Find the number of points where the tangent to the graph of a function is parallel

Direct y = -20.

10. The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa x0.

Find the value of the derivative of the function f(x) at the point x0.

11 . The figure shows a graph of the derivative of the function f (x), defined on the interval (-9; 9).

Find the number of minimum points of the function $f(x)$ on the segment [-6;8]. 1

12. The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa x0.

Find the value of the derivative of the function f(x) at the point x0.

13. The figure shows the graph of the function y=f(x) and the tangent to it at the point with the abscissa x0.

Find the value of the derivative of the function f(x) at the point x0.

14. The figure shows a graph of the derivative of the function f (x), defined on the interval (-6; 8).

Find the number of points where the tangent to the graph of the function f(x) is parallel to or coincides with the line y = x+7.

15 . The figure shows the graph of the function y = f(x) and the tangent to it at the point with the abscissa x0.

Find the value of the derivative of the function f(x) at the point x0.

16. The figure shows a graph of the derivative of the function f(x) defined on

interval (-14;9).

Find the number of maximum points of the function f(x) on the interval [-12;7].

17 . The figure shows a graph of the derivative of the function f(x) defined

on the interval (-10; 8).

Find the number of extremum points of the function f(x) on the interval [-9;7].

18. The line y = 5x-7 touches the graph of the function y = 6x2 + bx-1 at a point with an abscissa less than 0. Find b.

19 . The figure shows the graph of the derivative of the function f(x) and the tangent to it at the point with the abscissa x0.

Find the value of the derivative of the function f(x) at the point x0.

20 . Find the number of points in the interval (-1;12) where the derivative of the function y = f(x) shown on the graph is equal to 0.

21. Find the tangent to the graph of the function y=x2+6x-7, parallel to the line y=5x+11. In your answer, indicate the abscissa of the point of contact.

22. The figure shows the graph of the function y=f(x). Find the number of integer points in the interval (-2;11) where the derivative of the function f(x) is positive.

23. The figure shows the graph of the function y= f "(x) on the interval (-16; 4).

On the segment [-11; 0] find the number of maximum points of the function.