"Monotonicity of Functions" PPT courseware on the concept and properties of functions (the definition and proof of monotonicity in lesson 1)

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"Monotonicity of Functions" PPT courseware on the concept and properties of functions (the definition and proof of monotonicity in lesson 1)

Part One: Learning Objectives

1. Understand the monotonicity of functions and their geometric significance, and be able to use function images to understand and study the monotonicity of functions. (emphasis)

2. Can use the definition of function monotonicity to judge (or prove) the monotonicity of some functions, and can find the monotonic intervals of some specific functions. (main difficulty)

3. Understand the concepts of maximum and minimum values ​​of functions, and be able to find the maximum value of some simple functions with the help of the graph and monotonicity of the function. (main difficulty)

core competencies

1. Cultivate mathematical abstraction, logical reasoning, and intuitive imagination with the help of monotonicity judgment and proof.

2. Use it to find monotonic intervals and maximum values ​​to cultivate mathematical operation literacy.

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

Monotonicity of functions PPT, part 2: independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. Definition of increasing and decreasing functions

Condition Generally, let the domain of function y=f(x) be A, and M⊆A: If for any x1, x2∈M, when x1>x2

Both f(x1)>f(x2) and f(x1)

Conclusion y=f(x) is an increasing function on M (also called monotonically increasing on M) y=f(x) is a decreasing function on M (also called monotonically decreasing on M)

Thinking 1: What are the characteristics of x1 and x2 in the definition of increasing (decreasing) function?

Tip: x1 and x2 in the definition have the following three characteristics

(1) Arbitrariness, that is, the word "arbitrary" in "arbitrarily take x1, x2" must not be removed, and the special cannot be used to replace the general when proving;

(2) There is a size, usually x1>x2;

(3) Belong to the same monotonic interval.

2. Monotonicity of functions and monotonic intervals

If the function y=f(x) is monotonically increasing or decreasing on M, then it is said that the function y=f(x) has on M (when M is an interval, M is called the monotonic interval of the function, which can also be called respectively is a monotonically increasing interval or a monotonically decreasing interval).

Thinking 2: Is the function y=1/x a decreasing function in the domain?

Tip: No. y=1/x decreases on (-∞, 0) and also decreases on (0, +∞), but it cannot be said that y=1/x decreases on (-∞, 0)∪(0, +∞) ．

3. The maximum value of the function

First try

1. The image of the function y=f(x) is as shown in the figure, and its increasing interval is ()

A． [-4,4]

B． [-4,-3]∪[1,4]

C. [－3,1]

D. [－3,4]

2. Among the following functions, which one is a decreasing function on the interval (0, +∞) ()

A． y＝-1x　B． y=x

C. y=x2 D. y=1-x

3. The image 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

4. The monotonic decreasing interval of function f(x)=x2-2x+3 is ________.

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

Prove (judge) the monotonicity of a function by definition method

[Example 1] Prove that the function f(x)=x+1x is a decreasing function on (0,1).

regular method

Steps to prove monotonicity of a function using definitions

1 value: Let x1 and x2 be any two values ​​in the interval, and x1>x2.

2. Difference deformation: Make the difference fx1-fx2, and transform it into a formula that is easy to judge positive and negative through factorization, general division, formula, rationalization and other means.

3 Fixed sign: Determine the sign of fx1-fx2.

4Conclusion: Judge monotonicity based on the symbols and definitions of fx1-fx2.

Reminder: Difference deformation is the key to proving monotonicity, and the result of the deformation is in the form of the product of several factors.

Find the monotonic interval of a function

[Example 2] Find the monotonic interval of the following function, and indicate whether the function is an increasing or decreasing function on its monotonic interval.

(1)f(x)＝-1x; (2)f(x)＝2x＋1, x≥1, 5-x, x<1;

(3)f(x)＝-x2+2|x|+3.

regular method

How to find the monotonic interval of a function

1 Use the monotonicity of the known function to find the monotonic interval of the function.

2 Use the function graph to find the monotonic interval of the function.

Reminder: 1. If the monotonic increasing interval or monotonic decreasing interval of the function to be obtained is not unique, the monotonic intervals of the function should be separated by ",".

2. Clarify the difference and connection between "monotone interval" and "monotone on the interval".

Application of function monotonicity

[Inquiry Questions]

1. If function f(x) is an increasing function on its domain, and f(a)>f(b), what relationship does a and b satisfy? What if function f(x) is a decreasing function?

Tip: If function f(x) is an increasing function on its domain, then when f(a)>f(b), a>b; if function f(x) is a decreasing function on its domain, then When f(a)>f(b), a

2. What are the factors that determine the monotonicity of the quadratic function f(x)=ax2+bx+c?

Tip: The direction of the opening and the position of the axis of symmetry, that is, the symbol of the letter a and the size of -b2a.

[Example 3] (1) If the function f(x)=-x2-2(a+1)x+3 is an increasing function on the interval (-∞, 3], then the value range of the real number a is ________.

(2) It is known that the function y=f(x) is an increasing function on (-∞, +∞), and f(2x-3)>f(5x-6), then the value range of the real number x is ________ ．

regular method

Application of function monotonicity

1. The "bidirectionality" of the definition of function monotonicity: The definition can be used to judge and prove the monotonicity of the function. In turn, if the monotonicity of the function is known, the value range of the parameters in the function can be determined.

2If a function is monotonic on the interval 'a, b', then the function is also monotonic on any subset of this monotonic interval.

Class summary

1. When defining monotonicity, we should emphasize the arbitrariness of x1 and x2 within their definition domain. Its essence is to transform the size comparison of infinite multiple function values ​​on the interval into the size comparison of two arbitrary values.

2. To prove the monotonicity of a function (using definitions), you must strictly follow the steps of setting elements, making differences, deformation, fixation, and conclusions. Especially in the deformation, you must pay attention to the use of factors, formulas, and other techniques until the symbol judgment is easy to come by. Only then.

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

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

(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;

4. 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.

Monotonicity of functions PPT, part 4: Achieve standards in class and solidify double bases

1. Thinking and analysis

(1) If the function y=f(x) has f(1)

(2) If the function y=f(x) is a decreasing function on the interval [1,3], then the monotonically decreasing interval of the function y=f(x) is [1,3] ;. ()

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

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

2. Among the following functions, which one is an increasing function on (0,2) ()

A． y＝1x　B． y=2x-1

C. y=1-2x D. y=(2x-1)2

3. The value range of function y=x2-2x, x∈[0,3] is ________.

4. Use the definition of function monotonicity to prove: f(x)=2xx-1 is a decreasing function on (1, +∞).

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For more information about the PPT courseware "Concept and Properties of Functions, Monotonicity of Functions, Definition and Proof of Monotonicity", please click on the Concept and Properties of Functions ppt Monotonicity of Functions ppt Definition and Proof of Monotonicity ppt tag.

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Update Time: 2024-11-20

This template belongs to Mathematics courseware People's Education High School Mathematics Edition B Compulsory Course 1 industry PPT template

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