"Basic Properties of Functions" Concept and Properties of Functions PPT (Monotonicity of Functions in Lesson 1)

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"Basic Properties of Functions" Concept and Properties of Functions PPT (Monotonicity of Functions in Lesson 1)

Part One: Learning Objectives

Understand the concept of monotonicity of functions and be able to use definitions to judge or prove the monotonicity of functions.

Able to find the monotonic interval of a function with the help of graphics and definitions

Ability to find parameters or solve parametric inequalities based on the monotonicity of functions

Basic properties of functions PPT, part 2: independent learning

Problem guide

Preview the textbook P76-P79 and think about the following questions:

1. What are the concepts of increasing and decreasing functions?

2. What is the relationship between the monotonicity of a function and the monotonic interval?

A preliminary exploration of new knowledge

1. The concepts of increasing and decreasing functions

Generally speaking, let the domain of function f(x) be I and the interval D⊆I:

(1) If ∀x1, x2∈D, when x1

In particular, when the function f(x) _________ on its domain, we call it an increasing function.

(2) If ∀x1, x2∈D, when x1 <

In particular, when the function f(x) _________ on its domain, we call it a decreasing function.

■Instructions from famous teachers

(1) Three characteristics of x1 and x2 in the definition of increasing and decreasing functions

① Arbitrariness: The symbol "∀" in the definition cannot be removed, and special cannot be used instead of general when applied;

②With size: generally let x1

③Same interval: x1 and x2 belong to the same monotonic interval.

(2) The mutual inference relationship between increasing and decreasing functions, independent variables, and function values

①x1

②x1f(x2), opposite sign ⇔ subtracting function.

2. Monotonicity of functions and monotonic intervals

If the function y=f(x) is _________ or _________ on the interval D, then the function y=f(x) is said to have (strict) monotonicity in this interval, and the interval D is called y=f(x) of_________.

■Instructions from famous teachers

Two properties of monotonicity

(1) "Holistic" property: Monotone functions have the same properties in the same monotonic interval.

(2) "Local" property: refers to that a function can have different monotonicity in different intervals of the domain of definition.

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) All functions are monotonic in their domain. ()

(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) If the function f(x) is a decreasing function on R, then f(-3)>f(3). ()

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

(5) If the function f(x) decreases monotonically on (-∞, 0) and (0, +∞), then f(x) decreases monotonically on (-∞, 0) ∪ (0, +∞). ()

The graph of function y=f(x) on the interval [-2,2] is as shown in the figure, then the increasing interval of this function is ()

A． [-2,0]B． [0,1]

C. [-2,1] D． [-1,1]

Basic properties of functions PPT, the third part: interactive teaching and practice

Determination and proof of monotonicity of functions

Prove that the function f(x)=x+4x is an increasing function on (2, +∞).

[Proof] ∀x1, x2∈(2, +∞), and x1

Then f(x1)-f(x2)=x1+4x1-x2-4x2

=(x1-x2)+4(x2-x1)x1x2

(Variable question method) If the function in this example does not change, try to judge the monotonicity of f(x) on (0, 2).

regular method

Steps to prove monotonicity of a function using definitions

[Note]Difference deformation is the key to prove the monotonicity of the function, and the result of the deformation is mostly in the form of the product of several factors.

Basic properties of functions PPT, Part 4: Feedback on achievement of standards

1. The subtraction interval of function y=x2-6x is ()

A． (-∞, 2] B.[2, +∞)

C. [3, +∞) D. (-∞, 3]

2. Suppose (a, b), (c, d) are all monotonic increasing intervals of f(x), and x1∈(a, b), x2∈(c, d), x1

A． f(x1)f(x2)

C. f(x1)＝f(x2) D. Can not be sure

3. If f(x) is monotonically decreasing on R, and f(x-2)

4. As shown in the figure, the graphs of functions y=f(x) and y=g(x) are respectively. Try to write the monotonically increasing intervals of functions y=f(x) and y=g(x).

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Update Time: 2024-10-02

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

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