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Lesson type: repetition and generalization.
Lesson format: lesson-consultation.
Lesson objectives:
- educational: repeat and summarize theoretical knowledge on the topics: “Geometric meaning of the derivative” and “Application of the derivative to the study of functions”; consider all types of B8 problems encountered on the Unified State Examination in mathematics; provide students with the opportunity to test their knowledge independent decision tasks; teach how to fill out exam form answers;
- developing: promote the development of communication as a method scientific knowledge, semantic memory and voluntary attention; formation of such key competencies as comparison, comparison, object classification, definition adequate ways solving an educational problem based on given algorithms, the ability to act independently in situations of uncertainty, control and evaluate one’s activities, find and eliminate the causes of difficulties;
- educational: to develop in students communication skills (communication culture, ability to work in groups); promote the development of the need for self-education.
Technologies: developmental education, ICT.
Teaching methods: verbal, visual, practical, problematic.
Forms of work: individual, frontal, group.
Educational and methodological support:
1. Algebra and the beginnings of mathematical analysis. 11th grade: textbook. For general education Institutions: basic and profile. levels / (Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin); edited by A. B. Zhizhchenko. – 4th ed. – M.: Education, 2011.
2. Unified State Exam: 3000 problems with answers in mathematics. All tasks of group B / A.L. Semenov, I.V. Yashchenko and others; edited by A.L. Semyonova, I.V. Yashchenko. – M.: Publishing house “Exam”, 2011.
3. Open bank tasks.
Equipment and materials for the lesson: projector, screen, PC for each student with a presentation installed on it, printout of a memo for all students (Appendix 1) And score sheet (Appendix 2) .
Preliminary preparation to the lesson: as homework students are asked to repeat theoretical material from the textbook on the topics: “Geometric meaning of the derivative”, “Application of the derivative to the study of functions”; The class is divided into groups (4 people each), each of which has students of different levels.
Lesson explanation: This lesson is taught in 11th grade at the stage of repetition and preparation for the Unified State Exam. The lesson is aimed at repetition and generalization theoretical material, to use it in solving exam problems. Lesson duration - 1.5 hours .
This lesson is not attached to the textbook, so it can be taught while working on any teaching materials. This lesson can also be divided into two separate ones and taught as final lessons on the topics covered.
Lesson progress
I. Organizational moment.
II. Setting goals lesson.
III. Repetition on the topic “Geometric meaning of derivatives.”
Oral frontal work using a projector (slides No. 3-7)
Work in groups: solving problems with hints, answers, with teacher consultation (slides No. 8-17)
IV. Independent work 1.
Students work individually on a PC (slides No. 18-26), and enter their answers into the evaluation sheet. If necessary, you can consult a teacher, but in this case the student will lose 0.5 points. If the student completes the work earlier, he can choose to solve additional tasks from the collection, pp. 242, 306-324 (additional tasks are assessed separately).
V. Mutual verification.
Students exchange assessment sheets, check a friend’s work, and assign points (slide No. 27)
VI. Correction of knowledge.
VII. Repetition on the topic “Application of the derivative to the study of functions”
Oral frontal work using a projector (slides No. 28-30)
Work in groups: solving problems with hints, answers, with teacher consultation (slides No. 31-33)
VIII. Independent work 2.
Students work individually on a PC (slides No. 34-46), and enter their answers on the answer form. If necessary, you can consult a teacher, but in this case the student will lose 0.5 points. If the student completes the work earlier, he can choose to solve additional tasks from the collection, pp. 243-305 (additional tasks are assessed separately).
IX. Peer review.
Students exchange assessment sheets, check a friend’s work, and assign points (slide No. 47).
X. Correction of knowledge.
Students work again in their groups, discuss the solution, and correct mistakes.
XI. Summing up.
Each student calculates their points and puts a grade on the score sheet.
Students submit to the teacher an assessment sheet and solutions to additional problems.
Each student receives a memo (slide No. 53-54).
XII. Reflection.
Students are asked to evaluate their knowledge by choosing one of the phrases:
- I succeeded!!!
- We need to solve a couple more examples.
- Well, who came up with this math!
XIII. Homework.
For homework Students are invited to choose to solve tasks from the collection, pp. 242-334, as well as from open bank tasks.
The derivative of a function $y = f(x)$ at a given point $x_0$ is the limit of the ratio of the increment of a function to the corresponding increment of its argument, provided that the latter tends to zero:
$f"(x_0)=(lim)↙(△x→0)(△f(x_0))/(△x)$
Differentiation is the operation of finding the derivative.
Table of derivatives of some elementary functions
| Function | Derivative |
| $c$ | $0$ |
| $x$ | $1$ |
| $x^n$ | $nx^(n-1)$ |
| $(1)/(x)$ | $-(1)/(x^2)$ |
| $√x$ | $(1)/(2√x)$ |
| $e^x$ | $e^x$ |
| $lnx$ | $(1)/(x)$ |
| $sinx$ | $cosx$ |
| $cosx$ | $-sinx$ |
| $tgx$ | $(1)/(cos^2x)$ |
| $ctgx$ | $-(1)/(sin^2x)$ |
Basic rules of differentiation
1. The derivative of the sum (difference) is equal to the sum (difference) of the derivatives
$(f(x) ± g(x))"= f"(x)±g"(x)$
Find the derivative of the function $f(x)=3x^5-cosx+(1)/(x)$
The derivative of a sum (difference) is equal to the sum (difference) of derivatives.
$f"(x) = (3x^5)"-(cos x)" + ((1)/(x))" = 15x^4 + sinx - (1)/(x^2)$
2. Derivative of the product
$(f(x) g(x)"= f"(x) g(x)+ f(x) g(x)"$
Find the derivative $f(x)=4x cosx$
$f"(x)=(4x)"·cosx+4x·(cosx)"=4·cosx-4x·sinx$
3. Derivative of the quotient
$((f(x))/(g(x)))"=(f"(x) g(x)-f(x) g(x)")/(g^2(x)) $
Find the derivative $f(x)=(5x^5)/(e^x)$
$f"(x)=((5x^5)"·e^x-5x^5·(e^x)")/((e^x)^2)=(25x^4·e^x- 5x^5 e^x)/((e^x)^2)$
4. Derivative complex function is equal to the product of the derivative of the external function and the derivative of the internal function
$f(g(x))"=f"(g(x)) g"(x)$
$f"(x)=cos"(5x)·(5x)"=-sin(5x)·5= -5sin(5x)$
Physical meaning of the derivative
If a material point moves rectilinearly and its coordinate changes depending on time according to the law $x(t)$, then instantaneous speed of a given point is equal to the derivative of the function.
The point moves along the coordinate line according to the law $x(t)= 1.5t^2-3t + 7$, where $x(t)$ is the coordinate at time $t$. At what point in time will the speed of the point be equal to $12$?
1. Speed is the derivative of $x(t)$, so let’s find the derivative of the given function
$v(t) = x"(t) = 1.5 2t -3 = 3t -3$
2. To find at what point in time $t$ the speed was equal to $12$, we create and solve the equation:
Geometric meaning of derivative
Recall that the equation of a line that is not parallel to the coordinate axes can be written in the form $y = kx + b$, where $k$ is the slope of the line. Coefficient $k$ equal to tangent the angle of inclination between the straight line and the positive direction of the $Ox$ axis.
The derivative of the function $f(x)$ at the point $x_0$ is equal to slope$k$ tangent to the graph at a given point:
Therefore, we can create a general equality:
$f"(x_0) = k = tanα$
In the figure, the tangent to the function $f(x)$ increases, therefore the coefficient $k > 0$. Since $k > 0$, then $f"(x_0) = tanα > 0$. The angle $α$ between the tangent and the positive direction $Ox$ is acute.
In the figure, the tangent to the function $f(x)$ decreases, therefore, the coefficient $k< 0$, следовательно, $f"(x_0) = tgα < 0$. Угол $α$ между касательной и положительным направлением оси $Ох$ тупой.
In the figure, the tangent to the function $f(x)$ is parallel to the $Ox$ axis, therefore, the coefficient $k = 0$, therefore, $f"(x_0) = tan α = 0$. The point $x_0$ at which $f "(x_0) = 0$, called extremum.
The figure shows a graph of the function $y=f(x)$ and a tangent to this graph drawn at the point with the abscissa $x_0$. Find the value of the derivative of the function $f(x)$ at point $x_0$.
The tangent to the graph increases, therefore, $f"(x_0) = tan α > 0$
In order to find $f"(x_0)$, we find the tangent of the angle of inclination between the tangent and the positive direction of the $Ox$ axis. To do this, we build the tangent to the triangle $ABC$.
Let's find the tangent of the angle $BAC$. (Tangential acute angle V right triangle called relation opposite leg to the adjacent leg.)
$tg BAC = (BC)/(AC) = (3)/(12)= (1)/(4)=$0.25
$f"(x_0) = tg BAC = 0.25$
Answer: $0.25$
The derivative is also used to find the intervals of increase and decrease of a function:
If $f"(x) > 0$ on an interval, then the function $f(x)$ is increasing on this interval.
If $f"(x)< 0$ на промежутке, то функция $f(x)$ убывает на этом промежутке.
The figure shows the graph of the function $y = f(x)$. Find among the points $х_1,х_2,х_3…х_7$ those points at which the derivative of the function is negative.
In response, write down the number of these points.
\(\DeclareMathOperator(\tg)(tg)\)\(\DeclareMathOperator(\ctg)(ctg)\)\(\DeclareMathOperator(\arctg)(arctg)\)\(\DeclareMathOperator(\arcctg)(arcctg) \)
ContentContent elements
Derivative, tangent, antiderivative, graphs of functions and derivatives.
Derivative Let the function \(f(x)\) be defined in some neighborhood of the point \(x_0\).
Derivative of the function \(f\) at the point \(x_0\) called limit
\(f"(x_0)=\lim_(x\rightarrow x_0)\dfrac(f(x)-f(x_0))(x-x_0),\)
if this limit exists.
The derivative of a function at a point characterizes the rate of change of this function at a given point.
| Function | Derivative |
| \(const\) | \(0\) |
| \(x\) | \(1\) |
| \(x^n\) | \(n\cdot x^(n-1)\) |
| \(\dfrac(1)(x)\) | \(-\dfrac(1)(x^2)\) |
| \(\sqrt(x)\) | \(\dfrac(1)(2\sqrt(x))\) |
| \(e^x\) | \(e^x\) |
| \(a^x\) | \(a^x\cdot \ln(a)\) |
| \(\ln(x)\) | \(\dfrac(1)(x)\) |
| \(\log_a(x)\) | \(\dfrac(1)(x\ln(a))\) |
| \(\sin x\) | \(\cos x\) |
| \(\cos x\) | \(-\sin x\) |
| \(\tg x\) | \(\dfrac(1)(\cos^2 x)\) |
| \(\ctg x\) | \(-\dfrac(1)(\sin^2x)\) |
Rules of differentiation\(f\) and \(g\) are functions depending on the variable \(x\); \(c\) is a number.
2) \((c\cdot f)"=c\cdot f"\)
3) \((f+g)"= f"+g"\)
4) \((f\cdot g)"=f"g+g"f\)
5) \(\left(\dfrac(f)(g)\right)"=\dfrac(f"g-g"f)(g^2)\)
6) \(\left(f\left(g(x)\right)\right)"=f"\left(g(x)\right)\cdot g"(x)\) - derivative of a complex function
Geometric meaning of derivative Equation of a line- not parallel to the axis \(Oy\) can be written in the form \(y=kx+b\). The coefficient \(k\) in this equation is called slope of a straight line. It is equal to tangent inclination angle this straight line.
Straight angle- the angle between the positive direction of the \(Ox\) axis and this straight line, measured in the direction positive angles(that is, in the direction of least rotation from the \(Ox\) axis to the \(Oy\) axis).
The derivative of the function \(f(x)\) at the point \(x_0\) is equal to the slope of the tangent to the graph of the function at this point: \(f"(x_0)=\tg\alpha.\)
If \(f"(x_0)=0\), then the tangent to the graph of the function \(f(x)\) at the point \(x_0\) is parallel to the axis \(Ox\).
Tangent equation
Equation of the tangent to the graph of the function \(f(x)\) at the point \(x_0\):
\(y=f(x_0)+f"(x_0)(x-x_0)\)
Monotonicity of the function If the derivative of a function is positive at all points of the interval, then the function increases on this interval.
If the derivative of a function is negative at all points of the interval, then the function decreases on this interval.
Minimum, maximum and inflection points positive on negative at this point, then \(x_0\) is the maximum point of the function \(f\).
If the function \(f\) is continuous at the point \(x_0\), and the value of the derivative of this function \(f"\) changes with negative on positive at this point, then \(x_0\) is the minimum point of the function \(f\).
The points at which the derivative \(f"\) is equal to zero or does not exist are called critical points functions \(f\).
Internal points of the domain of definition of the function \(f(x)\), in which \(f"(x)=0\) can be minimum, maximum or inflection points.
Physical meaning of the derivative If a material point moves rectilinearly and its coordinate changes depending on time according to the law \(x=x(t)\), then the speed of this point is equal to the derivative of the coordinate with respect to time:
Acceleration material point in equal to the derivative of the speed of this point with respect to time:
\(a(t)=v"(t).\)
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The figure shows a graph of the derivative of the function f(x), defined on the interval (-1;17). Find the intervals of decrease of the function f(x). In your answer, indicate the length of the largest of them. f(x) 
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The figure shows a graph of the derivative of the function f(x), defined on the interval (-9; 2). At what point on the segment -8; -4 function f(x) takes highest value? On the segment -8; -4 f(x) 
The function y = f(x) is defined on the interval (-5; 6). The figure shows a graph of the function y = f(x). Find among the points x 1, x 2, ..., x 7 those points at which the derivative of the function f(x) is equal to zero. In response, write down the number of points found. Answer: 3 Points x 1, x 4, x 6 and x 7 are extremum points. At point x 4 there is no f (x)
Literature 4 Algebra and beginning analysis class. Textbook for educational institutions basic level/ Sh. A. Alimov and others, - M.: Prosveshchenie, Semenov A. L. Unified State Examination: 3000 problems in mathematics. – M.: Publishing House “Exam”, Gendenshtein L. E., Ershova A. P., Ershova A. S. A visual guide to algebra and the beginnings of analysis with examples for grades 7-11. – M.: Ilexa, Electronic resource Open Unified State Exam task bank.
