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

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

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. (main difficulty)

2. Can use the definition of function monotonicity to judge (or prove) the monotonicity of some functions. (difficulty)

3. Can find the monotonic interval of some specific functions. (emphasis)

core competencies

1. Use the proof of monotonicity to develop logical reasoning skills.

2. Use finding monotonic intervals and applying monotonicity to solve problems to cultivate intuitive imagination and mathematical operation literacy.

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

A preliminary exploration of new knowledge

1. Definition of increasing and decreasing functions

Conditions Generally speaking, let the domain of function f(x) be I, and the interval D⊆I: if ∀x1, x2∈D, when x1

All have_____________ All have_____________

Conclusion Then it is said that the function f(x) is a ___ function on the interval D. Then it is said that the function f(x) is a ___ function on the interval D.

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

(3) Belong to the same monotonic interval.

2. Monotonicity of functions and monotonic intervals

If the function y=f(x) _____________ 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 the _________ of y=f(x).

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

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

First try

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

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

Find the monotonic interval of a function

[Example 1] 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.

[Solution](1) The monotonic interval of function f(x)=-1x is (-∞, 0), (0, +∞), which is between (-∞, 0), (0, +∞) are all increasing functions.

(2) When x≥1, f(x) is an increasing function. When x<1, f(x) is a decreasing function, so the monotonic interval of f(x) is (-∞, 1), [ ;1, +∞), and the function f(x) is a decreasing function on (-∞,1) and an increasing function on [1, +∞).

regular method

How to find the monotonic interval of a function

(1) Utilize the monotonicity of basic elementary functions, such as (1) and (2) in this example, where the monotonic interval of the piecewise function must be solved piecewise according to the value range of the independent variable of the function;

(2) Use the graph of the function, such as (3) in this example.

Reminder: 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 ",", such as (3) in this example.

Determination and proof of monotonicity of functions

[Example 2] Prove that the function f(x)=x+1x is a decreasing function on (0,1).

[Ideas Enlightenment] Let the element 0

��→Deformation code: fx1>fx2��→Conclusion subtraction function

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

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.

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.

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

1. Thinking and analysis

(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) 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, +∞). ()

2. As shown in the figure, the function y=f(x) defined on the interval [-5,5], then the following statement about the function f(x) is wrong ()

A． The function increases monotonically on the interval [-5,-3]

B． The function increases monotonically on the interval [1,4]

C. The function decreases monotonically on the interval [-3,1]∪[4,5]

D. The function is not monotonic on the interval [-5,5]

3. If the function f(x)=x2-2bx+2 is an increasing function on the interval [3,+∞), then the value range of b is ()

A． b=3

B． b≥3

C. b≤3

D. b≠3

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Update Time: 2024-07-01

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