Extreme value and derivative of function
First, the teaching objectives
1 knowledge and skills
〈 1〉 Combining with the function image, we can understand the necessary and sufficient conditions for the derivative function to obtain the extreme value at a certain point.
〈2〉 In order to understand the concept of extreme value of function, we will use derivative to find the maximum and minimum value of function.
2 processes and methods
Combined with examples, with the help of intuitive perception of function graphics, this paper explores the relationship between extreme value and derivative of function.
3 Emotion and value
The universality and effectiveness of sensory derivative in the study of function properties can help students realize that extreme value is the local property of function and enhance their thinking consciousness of combining numbers with shapes.
Second, the key point: using derivative to find the extreme value of function.
Difficulties: Necessary and sufficient conditions for a function to obtain an extreme value at a certain point.
Third, the basic process of teaching
The relationship between monotonicity and derivative of recall function and existing knowledge
Ask questions and arouse curiosity.
Organize students to explore independently and get the definition of extreme value of function.
Through examples and exercises, deepen and improve the understanding of the definition of function extreme value.
Fourth, the teaching process
(1) Create scenarios and introduce new lessons.
What is the relationship between the derivative of 1. function and monotonicity?
(Please ask students C to answer, and students A and B to supplement)
Extreme value and derivative of function II. Observe the graph 1.3.8, showing the extreme value and derivative t =-4.9t2+6.5t+ 10 of the function function of the height h of high platform divers with time, and answer the following questions.
Extreme value and derivative extreme value of function and derivative extreme value and derivative extreme value of lesson plan function.
Extreme value of function and derivative teaching plan
Extreme value and derivative of function teaching plan extreme value and derivative teaching plan of function
(1) When t=a, what are the height of the high platform diver from the water surface, the derivative of the extreme value of the function and the derivative teaching plan when t=a?
(2) What are the characteristics of the image near point t=a?
(3) What is the variation law of derivative sign near 3)t = a?
* * * Same induction: the function h(t) is at point A, h/(a)=0, and near t=a, when t < a, the extreme value and derivative lesson plans of the function increase monotonously, and the extreme value and derivative lesson plans of the function are > 0; When T > A, the extreme value and derivative lesson plans of the function decrease monotonously, and are less than 0, that is, when T passes through A from small to large near A, the extreme value and derivative lesson plans of the function are positive first and then negative, and the extreme value and derivative lesson plans of the function change continuously, so h/(a)=0.
3. This is true for this case, and does it also have this property for other continuous functions?
Second > exploration and discussion
The extreme value and derivative of function 1, observe the image of y=f(x) shown in figure 1.3.9, and answer the following questions:
Extreme value and derivative of function teaching plan (1) What is the relationship between the function values of point A and point B and the function values near these two points?
(2) What is the derivative value of the function y=f(x) at point A.B?
(3) What are the symbols of the derivative of y = f (x) near point a.b, and what does it matter?
2, the definition of extreme value:
We call point A the minimum point of function y=f(x), and f(a) the minimum value of function y=f(x);
Point b is called the maximum point of function y=f(x), and f(a) is called the maximum point of function y=f(x).
The maximum and minimum points are called extreme points, and the maximum and minimum points are called extreme points.
3. Through the above exploration, can we sum up the necessary and sufficient conditions for a differentiable function to obtain an extreme value at a certain point x0?
Necessary and sufficient conditions: f(x0)=0 and the derivative values around the point x0 are opposite in sign.
4. Guide students to observe the chart 1.3. 1 1 and answer the following questions:
(1) Find the poles in the graph and explain which points are the maximum points and which points are the minimum points.
(2) Is the maximum value necessarily greater than the minimum value?
5. Practice in class:
As shown in the figure, it is a function of y=f(x). Try to find the extreme point of the function y=f(x), and point out which are the maximum points and which are the minimum points. What if the visualization of function is changed to the visualization of teaching plan with derivative function y= extreme value and derivative?
Extreme Value and Derivative of Function Teaching Plan III: Examples
Example 4 teaching plan for finding the extreme value of function and derivative
Teacher's analysis: ① Find f/(x), f/(x)=0, and find the function pole; ② The sign of f/(x) near the pole x0 is determined by the monotonicity of the function, so as to determine which point is the maximum point and which point is the minimum point, and thus to find the extreme value of the function.
Students do it by hand and the teacher guides them.
Solution: ∵ Extreme value and derivative lesson plan of function ∴ Extreme value and derivative lesson plan of function =x2-4=(x-2)(x+2) Let the extreme value and derivative lesson plan of function =0, and the solution is x=2, or x=-2.
Extreme value of function and derivative teaching plan
Extreme value of function and derivative teaching plan
The following discussion is divided into two situations:
(1) When the extreme value and derivative of the function are > 0, that is, X is > 2 or X.
(2) When the extreme value and derivative of the function are less than 0, that is, -2 < x < 2.
When x changes, the extreme value and derivative of the function, and the change of f(x) are as follows:
x
(-∞,-2)
-2
(-2,2)
2
(2,+∞)
Extreme value of function and derivative teaching plan
+
_
+
f(x)
Monotonically increasing
Extreme value of function and derivative teaching plan
Extreme value of function and monotonic decrease of derivative teaching plan
Extreme value of function and derivative teaching plan
Monotonically increasing
Therefore, when x=-2, f(x) has a maximum value, and the maximum value is f(-2)= the extreme value and derivative of the function; When x=2, f(x) has a pole.
Small value, and the minimum value is f(2)= extreme value and derivative of function.
The extreme value and derivative of the function function are shown in the picture of the teaching plan:
The extreme value and derivative of the function are summarized in the teaching plan: the method to find the extreme value of the function y=f(x) is:
Extreme value and derivative lesson plan of function 1 Find extreme value and derivative lesson plan of function, and solve extreme value and derivative lesson plan of equation function =0. When the extreme value and derivative teaching plan of the function =0:
(1) If the extreme value and derivative of the left function near x0 are greater than 0, and the extreme value and derivative of the right function are less than 0, then f(x0) is the largest.
(2) If the extreme value and derivative of the left function near x0 are less than 0, and the extreme value and derivative of the right function are greater than 0, then f(x0) is the minimum value.
Fourth > classroom exercises
1, find the extreme value of the function f(x)=3x-x3.
2. Thinking: It is known that the function f(x)=ax3+bx2-2x obtains the extreme value at x=-2 and x= 1.
Find the analytic formula and monotone interval of function f(x).
Class C students do 1 questions, and Class A and Class B students do 1 and 2 questions.
Thinking after class
1, if the function f(x)=x3-3bx+3b has the minimum value in (0, 1), find the range of the number b.
2. It is known that f(x)=x3+ax2+(a+b)x+ 1 has a maximum value and a minimum value, and the range of the number A is realistic.
Six > class summary
1, the definition of extreme value of function
2, function extremum solving steps
3. Necessary and sufficient conditions for a point to be an extreme point of a function.
Seven > homework P32 5 ① ④
Teaching reflection
The teaching content of this section is the extreme value of derivative. Based on the monotonicity of derivative in the last lesson, through the intuitive exploration of function diagram, the definition of extreme value of derivative is summarized, and the extreme value of function is obtained by using this definition. Teaching feedback mainly has problems in writing format. In order to unify the requirements, students were unwilling to accept this format at first, but with the display of several examples and exercises, students realized the simplicity of the list method. At the same time, in order to quickly judge whether the derivative is positive or negative, I ask students to factorize the derivative as much as possible. The difficulty of this lesson is the necessary and sufficient condition for a function to obtain an extreme value at a certain point. To illustrate this point, it is necessary to give more examples. In the process of solving, students also revealed that the precision of derivative of complex variable function is relatively low, and the process of finding the extreme value of function is still not standardized. It seems that these aspects should continue to strengthen the teaching plan of extreme value and derivative of function.
Discussion and evaluation
The overall design of teaching content is reasonable, focused and difficult to break through, which fully embodies the dual-subject classroom status of teacher-led and student-centered, fully mobilizes the enthusiasm of students, and teachers guide their thinking reasonably and clearly, so that students' mathematical thinking can be cultivated and improved. The teaching content is moderate in capacity and difficulty, in line with the learning situation, and pays attention to the individual differences of students, so that students of different degrees can get different results.
Geometric meaning of derivative
Teaching objectives
Knowledge and skills objectives:
The central task of this section is to study the geometric meaning and application of derivatives. The formation of the concept is divided into three levels:
(1) By reviewing the old knowledge of "two steps to obtain derivative" and "the relationship between average change rate and secant slope", the geometric meaning of average change rate is solved, and the way to solve the problem is found according to the formation of derivative concept.
(2) From the relationship between secant and tangent in a circle, it is extended to a general curve, and the tangent is defined intuitively by secant approximation.
(3) According to the relationship between secant and tangent, the geometric meaning of function derivative is explored by combining numbers, so that students can realize that the geometric meaning of derivative is the slope of tangent of function derivative. Namely:
Geometric meaning lesson plan of derivative = slope k of curve tangent at geometric meaning lesson plan of derivative.
On this basis, through examples and exercises, students can learn to use the geometric meaning of derivatives to explain real-life problems and deepen their understanding of derivatives. Feel the approximate thinking method in the learning process and understand the mathematical thinking method of "replacing music with straightness"
Process and method objectives:
(1) Students cultivate their ability to practice and perceive discovery through observation, perception and hands-on inquiry.
(2) By understanding the relationship between tangent and secant of a circle, students can explore general curves by analogy, improve their understanding of tangent, feel the idea of approximation, and realize that tangent is the essence of a local property, which is helpful to improve their mathematical thinking ability.
(3) Combining stratified inquiry with stratified practice, it is expected that students at all levels can try their best to stay ahead of teachers, solve problems independently, discover new knowledge and apply new knowledge by virtue of their own abilities.
Emotions, attitudes and values:
(1) Students can understand the dialectical relationship between approximation and accuracy by infiltrating the idea of approximation in the process of inquiry and replacing curve with straight; Experience the significance and value of transforming ideas in mathematics through limited understanding of infinity;
(2) Provide them with sufficient opportunities to engage in mathematical activities in teaching, such as inquiry activities, so that students can explore new knowledge independently and focus on teaching examples before speaking. Stimulate students' learning potential in activities, promote them to truly understand and master basic mathematical knowledge and skills and mathematical thinking methods, gain rich experience in mathematical activities, improve their comprehensive ability, learn to learn to learn, and further develop their emotions and attitudes in willpower, self-confidence and rational spirit.
Teaching emphases and difficulties
Emphasis: Understand and master the new definition of tangent, the geometric meaning of derivative and its application in solving practical problems, and experience the thinking method of combining numbers with shapes and replacing curves with straight ones.
Difficulties: Finding, understanding and applying the geometric meaning of derivatives.
teaching process
First, review questions
What is the definition of 1. derivative? What are the three steps to find the derivative? Find the derivative of the function y = x2 at x = 2.
Definition: The geometric meaning of function derivative in derivative teaching plan is the instantaneous change rate of function at this point.
Steps to find the derivative:
The first step is to find the geometric meaning teaching plan of the derivative of the average change rate;
Step 2: Find the geometric meaning of the derivative of instantaneous rate of change.
(that is, the geometric meaning of derivative teaching plan, the constant of average change rate approaching is point derivative)
2. Observe the geometric meaning of the derivative of the function. What is the graphical representation of the geometric meaning of the derivative of the average rate of change?
Student: The average change rate represents the slope of secant PQ. Geometric significance of derivative teaching plan
Teacher: This is the geometric meaning of the average change rate (the geometric meaning teaching plan of derivative).
3. What does the instantaneous rate of change (geometric meaning of derivative) mean?
As shown in figure 2- 1, let curve c be the image of function y=f(x), point P(x0, y0) be a point on curve c, and point Q (x0+δ x, y0+δ y) be any point adjacent to point p on curve c, and make secant PQ. When point Q approaches point P infinitely along curve C, it will be a secant.
Geometric meaning teaching plan of derivative
Follow-up: How to determine the tangent of curve C at point P? Because p is given, it is enough to find the slope of the tangent line according to the knowledge of the point oblique equation of the straight line in plane analytic geometry. Let the slope of secant PQ be the geometric meaning teaching plan of derivative and the slope of tangent PT be the geometric meaning teaching plan of derivative, so it is easy to know the geometric meaning teaching plan of secant PQ. Because the straight line PT at the limit position of secant PQ is tangent, the limit of secant PQ slope is the geometric meaning teaching plan of tangent PT slope derivative, that is, the geometric meaning teaching plan of derivative.
Understanding the geometric meaning of derivative from its definition.
Geometric meaning teaching plan of derivative
As can be seen from the above formula, the slope of the tangent of curve f(x) at point (x0, f(x0)) is the derivative f'(x0) of y = f (x) at point x0. Today we will discuss the geometric meaning of derivatives.
Class C students answer the question 1, and Class A and Class B students answer the question 2. On the basis of students' answers, the teacher focuses on the third question, and then gradually introduces the geometric meaning of derivatives.
Second, the new lesson
1, the geometric meaning of the derivative;
The geometric meaning of the derivative f'(x0) of the function y = f (x) at point x0 is the slope of the tangent of the curve y = f (x) at point (x0, f(x0)).
Namely: the geometric meaning of derivative teaching plan
Oral answer exercises:
(1) If the derivative of the function y = f (x) at the known point x0 is as follows: f'(x0)= 1, f'(x0)= 1, f'(x0)=- 1, f' (.
(c-level students do)
(2) It is known that the image of the function y = f (x) (as shown in Figure 2-2) is a straight line in the following three cases, and the derivative of the function at each point is determined by observation. (Students on the A and B floors do it)
Geometric meaning teaching plan of derivative
2. How to use derivatives to study the increase and decrease of functions?
Summary: nearby: instantaneous, increase or decrease: rate of change, that is, the instantaneous rate of change of the research function at this point, that is, the derivative. The plus or minus of the derivative is the increase or decrease of the corresponding function. If we make the tangent of this point, we can get the positive and negative of the tangent slope, that is, the positive and negative of the derivative, so as to judge the increase or decrease of the function and understand the derivative is an effective tool to study the increase or decrease and the change speed of the function.
At the same time, combined with the idea of replacing curve with straight line, the change of tangent near a certain point is the same as that of curve, and the increase or decrease of function can also be judged. Both of them reflect that derivative is an effective tool to study the increase and decrease of function and the speed of change.
Example 1 The geometric meaning of function derivative has a little geometric meaning of derivative. Find the geometric meaning of the derivative of this point and explain the increase and decrease of the function.
Geometric meaning teaching plan of derivative
The instantaneous change rate of the function at any point on the definition domain is 3, and the function increases monotonically on the definition domain. (At this time, the tangent of any point is the straight line itself, and the slope is the rate of change. )
3. Find the tangent equation of curve y = f (x) at point (x0, f(x0)) by derivative.
Example 2 Find the tangent equation of curve y = x2 at point m (2 2,4).
Solution: the geometric meaning of derivative teaching plan
∴y'|x=2=2×2=4.
The tangent equation at point m (2 2,4) is y-4 = 4 (x-2), that is, 4x-y-4 = 0.
From the above example, we can sum up two steps to find the tangent equation:
(1) Find the derivative f' (x0) of the function y = f (x) at the point x0.
(2) According to the oblique formula of the linear equation, the tangent equation is y-y0 = f' (x0) (x-x0).
Question: If the inclination of tangent PT at point (x0, f(x0)) is the geometric meaning of derivative, teach the geometric meaning of derivative and find the tangent equation. (Because the tangent is parallel to the Y axis and the derivative does not exist, the tangent equation cannot be solved by the above method. According to the definition of tangent, the geometric meaning of the derivative of tangent equation can be directly obtained.
(Answer by Class C first, then supplemented by A and B)
The geometric meaning of curve derivative is known. The geometric meaning of finding the derivative of a point in the teaching plan: (1) the slope of the tangent passing through point P;
(2) The tangent equation of point P. ..
Solution: the geometric meaning of (1) derivative,
Geometric meaning teaching plan of derivative
Y' | x = 2 = 22 = 4。 The slope of the tangent at point p is equal to 4.
(2) The geometric meaning teaching plan with the tangent equation of point P as the derivative is 12x-3y- 16 = 0.
Exercise: Find the tangent equation of parabola y = x 2+2 at point m (2,6).
Answer: y' = 2x, y' | x = 2 = 4 The tangent equation is 4x-y-2 = 0.
Students in class B do the questions and students in class A correct them.
Three. abstract
The geometric meaning of 1. derivative. (group c students answer)
2. Find the tangent equation of curve y=f(x) at point (x0, f(x0)) by derivative.
(group b students answer)
Fourth, homework
1. Finding the Geometric Meaning of Parabolic Derivatives The tangent equation of the teaching plan at point (1, 1).
2. Find the slope and tangent equation of parabola Y = 4x-x2 at point A (4,0) and point B (2,4).
3. Find the inclination of the tangent of the curve Y = 2x-x3 at each point (-1,-1).
* 4. Given parabola y = x2-4 and straight line y = x+2, find the coordinates of the intersection of straight line and parabola (1); (2) The tangent equation of parabola at the intersection;
(Group C students completed 1, 2 questions; Students in group B complete 1, 2 and 3; Group A students complete questions 2, 3 and 4)
Teaching reflection:
The content of this section is to learn the geometric meaning of derivative on the basis of learning the knowledge of "the problem of change rate and the concept of derivative" Because there is no design restriction in the new textbook, I try my best to let students draw by hand and feel the whole approximation process, so that students can deeply understand the geometric meaning of derivatives and the idea of "replacing music with directness".
This lesson focuses on two teaching points: "intuitively understanding the geometric meaning of derivatives by using function images" and "explaining practical problems by using the geometric meaning of derivatives". First, recall the practical and numerical significance of the derivative, and then naturally lead to the study of the geometric significance of the derivative from the perspective of graphics. Then, by analogy with the research idea of "average rate of change-instantaneous rate of change", the tangent of a point on the curve is defined by the idea of approximation, and then the students are guided to think from the perspective of the combination of numbers and shapes, and the geometric meaning of the derivative is obtained-"the derivative is the slope of the tangent of a point on the curve".
After completing the first stage of this lesson, the teacher pointed out that using the geometric meaning of derivative, when studying practical problems, the tangent approximation of this point can replace the curve near a certain point, that is, "directly replace the curve", so as to achieve the purpose of "describing complex objects with simple objects". Through the study of two examples, students can fully experience the relationship between derivative and tangent slope from different angles and feel the universality of derivative application. This kind of classroom focuses on students, and every knowledge and discovery is always managed by students themselves. In class, students are given enough time and space to think, so that students can organize discussions after activities such as hands-on operation and writing calculus. Our teacher only guides them at key points. Judging from the students' homework, the effect is better.
Coordinate representation of plane vector
First, the analysis of learning situation
This lesson is based on what students have learned, and it is also the consolidation and development of what they have learned before. But in terms of students' knowledge preparation, students have a good grasp of the relevant basic knowledge, so they should ask questions about the students' relevant knowledge in time when reviewing, and then consolidate and review this lesson. The difficulties students will encounter in this course are: the representation of number axis and coordinates; Coordinate representation of plane vector; Coordinate operation of plane vector.
Second, the requirements of the syllabus
1. will use coordinates to represent the addition, subtraction, multiplication and division of plane vectors.
2. Understand the condition that plane vector lines are represented by coordinates.
3. Master the coordinate expression of the product of quantity, and carry out the product operation of plane vector.
4. The included angle between two vectors can be expressed by coordinates, and the condition that the plane vector expressed by coordinates is vertical can be understood.
Third, the teaching process
(A) knowledge combing:
Solution of 1. vector coordinates
(1) If the starting point of the vector is the coordinate origin, then the ending coordinate is the vector coordinate.
(2) let A(x 1, y 1) and B(x2, y2), then
=_________________
| |=_______________
(2) Plane vector coordinate operation
1. Vector addition, subtraction, digital multiplication vector
Let = (x 1, y 1), = (x2, y2), then
+ = - = λ = .
2. Vector parallel coordinate representation
Let = (x 1, y 1), = (x2, y2), then ∨? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Test center 1. Coordinate operation of plane vector
Example 1. A (-2,4), B (3,-1) and C (-3,4) are known. Let (1) find 3+-3;
(2) Find the real number m, n that satisfies =m +n;
Lian: (20 15 Jiangsu, 6) known vector =(2, 1), =( 1, -2), if m +n =(9, -8).
(m, n∈R), then the value of m-n is.
Coordinate representation of plane vector * * * line of test center 2.
Example 2: Three vectors on a given plane = (3,2), = (- 1, 2), = (4, 1).
If (+k)∩(2-), find the value of k;
Lian: (20 15, Sichuan, 4) known vector = (1 2), = (1 0), = (3,4). If λ is a real number, (+λ) ∨, then λ = ().
Thinking: What are the manifestations of vector * * * lines? What are the necessary and sufficient conditions for two vector lines?
Method summary:
Two representations of 1. vector * * * line
Let a = (x 1, y 1), b = (x2, y2), ①a∨b? a =λb(b≠0)； ②a∨b？ X 1y2-x2y 1=0。 As for which form to use, it depends on the specific situation of the topic, which generally involves the application of coordinates.
2. The function of the necessary and sufficient conditions of two vector lines
Judge whether two vectors are * * * lines (parallel problem; In addition, using the necessary and sufficient conditions of two vector lines, equations (groups) can be listed and unknown values can be obtained.
Coordinate operation of plane vector product in test center 3
Example 3 "It is known that the side length of square ABCD is 1, and point E is the moving point on the side of AB.
The value of is; The value of is.
It is suggested that when solving the problem of vector product operation involving geometric figures, the coordinate representation of vector product is used to establish rectangular coordinate system for operation, which can make the product operation simple.
Lian: (20 14, Anhui, 13) Let = (1, 2), = (1, 1), =+K. If, the value of real number k is equal to ().
On the Necessary and Sufficient Conditions of Two Non-zero Vectors ⊥: = 0? .
Experience in solving problems:
(1) When the coordinates of the vector are known, it can be solved by the coordinate method, that is, if a = (x 1, y 1) and b = (x2, y2), a b = x 1x2+y 1y2.
(2) When solving the problem of vector quantity product operation involving geometric figures, using the coordinate representation of vector quantity product, rectangular coordinate system can be established for operation, which can make the operation of quantity product simple.
(3) What are the necessary and sufficient conditions for two nonzero vectors a⊥b: A ⊥ B = 0? x 1x2+y 1y2=0。
Test site 4: coordinate representation of plane vector modulus
Example 4: (20 15 Hunan, Li 8) It is known that points A, B and C move on the circle x2+y2= 1, but if the coordinate of point P is (2,0), the value of is ().
a6 b . 7 c . 8d . 9
Exercise: (20 16, Shanghai, 12)
In the plane rectangular coordinate system, it is known that A (1, 0), B(0,-1) and P are a moving point on the curve, so what is the range of values?
Experience in solving problems:
The method of finding vector modulus;
(1) formula method, using |a|= and (a b) 2 =| a | 2 2a b+| b | 2, the modular operation of vectors is transformed into scalar product operation;
(2) Geometric method, using parallelogram rule or triangle rule of vector addition and subtraction as vector, and then using cosine theorem and other methods to solve. ..
Verb (abbreviation of verb) homework (after-school exercises 1 and 2)