"Basic Properties of Functions" PPT courseware on the concepts and properties of functions (the maximum (minimum) value of a function in Lesson 2)

"Basic Properties of Functions" PPT courseware on the concepts and properties of functions (the maximum (minimum) value of a function in Lesson 2)

Part One: Learning Objectives

1. Understand the concepts of maximum and minimum values ​​of functions and their geometric significance. (emphasis)

2. Can use the graph and monotonicity of functions to find the optimal value of some simple functions. (main difficulty)

3. Ability to use the optimal value of a function to solve relevant practical application problems. (emphasis)

4. Through the study of the content of this section, students can understand the role of combining ideas of numbers and shapes and classifying discussion ideas in solving the optimal value, and improve students' abilities of logical reasoning and mathematical operations. (main difficulty)

core competencies

1. Cultivate intuitive imagination and mathematical operation literacy with the help of the method of finding the optimal value of a function.

2. Use the maximum value of a function to solve practical problems and cultivate mathematical modeling literacy.

Basic properties of functions PPT, part 2: independent preview and exploration of new knowledge

A preliminary exploration of new knowledge

function maximum and minimum

maximum value minimum value

Conditional Suppose the domain of the function y=f(x) is I. If there is a real number M that satisfies: ∀x∈I, both

f(x) M f(x) M

∃x0∈I, such that

Conclusion M is the maximum value of the function y=f(x) M is the minimum value of the function y=f(x)

Geometric meaning The highest point on the graph of f(x) The lowest point on the graph of f(x)

Thinking: If function f(x)≤M, then M must be the maximum value of the function?

Tip: Not necessarily. Only when there is a point x0 in the domain of definition such that f(x0)=M, M is the maximum value of the function, otherwise it is not.

First try

1. The graph of function y=f(x) on [-2,2] is as shown in the figure, then the minimum and maximum values ​​of this function are ()

A． -1,0　B. 0,2

C. -1,2 D.12,2

2. Assume function f(x)=2x-1(x<0), then f(x)()

A． There is a maximum value

B． There is a minimum value

C. There is both a maximum value and a minimum value

D. There is neither maximum nor minimum value

3. Function f(x)=1x, x∈[1,2], then the maximum value of f(x) is ________ and the minimum value is ________.

Basic properties of functions PPT, the third part: cooperative exploration to improve literacy

Use the graph of the function to find the maximum value (range) of the function

[Example 1] It is known that the function f (x) = 3-x2, x∈[-1, 2], x-3, x∈2, 5].

(1) Draw the image of f(x) in the rectangular coordinate system;

(2) Write the monotonic interval and value range of the function based on the graph of the function.

[Solution] (1) The image is as shown in the figure:

(2) It can be seen from the figure that the monotonically increasing interval of f(x) is (-1,0), (2,5), the monotonically decreasing interval is (0,2), and the value range is [-1,3&# 093;．

regular method

How to find the optimal value of a function using images

1Draw the graph of function y＝fx;

2Observe the image and find the highest and lowest points of the image;

3Write the maximum value. The ordinate of the highest point is the maximum value of the function, and the ordinate of the lowest point is the minimum value of the function.

Use the monotonicity of the function to find the maximum value (range)

[Example 2] It is known that the function f(x)=2x+1x+1.

(1) Determine the monotonicity of the function on the interval (-1, +∞), and use the definition to prove your conclusion;

(2) Find the maximum and minimum values ​​of the function on the interval [2,4].

[ Solution

Then f(x1)-f(x2)＝2x1＋1x1＋1-2x2＋1x2＋1＝x1-x2x1＋1x2＋1,

Because -10, x2+1>0, x1-x2<0,

So f(x1)-f(x2)<0⇒f(x1)

So f(x) is an increasing function on (-1, +∞).

regular method

1. General steps for finding the maximum (minimum) value of a function using monotonicity

(1) Determine the monotonicity of the function.

(2) Use monotonicity to find the maximum (minimum) value.

2. The relationship between the maximum (small) value of the function and monotonicity

(1) If the function f(x) is an increasing (decreasing) function on the interval [a, b], then the minimum ( The largest) value is f(a), and the largest (smallest) value is f(b).

(2) If the function f(x) is an increasing (decreasing) function on the interval [a, b] and a decreasing (increasing) function on the interval [b, c], then The maximum (small) value of f(x) on the interval [a,c] is f(b), and the minimum (largest) value is the smaller (larger) of f(a) and f(c) one of.

Reminder: (1) Don’t forget to find the domain when searching for the maximum value.

(2) The maximum value on a closed interval. Directly substituting the two endpoint values ​​without judging monotonicity is the most likely error. Be careful when solving.

Class summary

1. The maximum (small) value of a function contains two meanings: one is existence, and the other is the maximum (small) value among all function values ​​in a given interval, which is reflected in the function graph. The graph of the function has the highest point or the lowest point. point.

2. Finding the maximum value of a function is similar to finding the range of a function. The commonly used method is:

(1) Graphic method, that is, draw the graph of the function and write the maximum value based on the highest point or lowest point of the graph;

(2) Monotonicity method, generally it is necessary to determine the monotonicity of the function first, and then find the optimal value according to the meaning of monotonicity;

(3) For quadratic functions, we can also use the combination method to study, and at the same time, we can flexibly use the idea of ​​combining numbers and shapes and the idea of ​​classification discussion to solve problems.

3. Through the study of the optimal value of functions, the idea of ​​combining numbers and shapes is penetrated, and the problem-solving awareness of recognizing numbers by shapes is established.

PPT on the basic properties of functions, part 4: Achieving standards in class and solidifying the bases

1. Thinking and analysis

(1) Any function has a maximum (minimum) value. ()

(2) The maximum value of function f(x) on [a, b] must be f(a) (or f(b)). ()

(3) The maximum value of the function must be greater than the minimum value. ()

2. Function y=x2-2x, the value range of x∈[0,3] is ()

A． [0,3]

B． [-1,0]

C. [-1, +∞)

D. [-1,3]

3. The maximum value of function y=ax+1 on the interval [1,3] is 4, then a=______.

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