"Monotonicity of Functions" Concept and Properties of Functions PPT (Monotonicity of Functions and Average Rate of Change of Functions in Lesson 1)

"Monotonicity of Functions" Concept and Properties of Functions PPT (Monotonicity of Functions and Average Rate of Change 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 graphs and definitions

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

Monotonicity of functions PPT, part 2: independent learning

Problem guide

Preview the contents of textbook P95-P100 and think about the following questions:

1. What is the concept of increasing function?

2. What is the concept of decreasing function?

3. What is the monotonic interval of a function?

A preliminary exploration of new knowledge

1. The concepts of increasing and decreasing functions

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

(1) If for any x1, x2∈I, when x1 < 1) shown;

(2) If for any x1, x2∈I, when x1

In both cases, the function is said to be monotonic on I (when I is an interval, I is called the _____________ of the function, which can also be called _______________ or ____________________ respectively).

■Instructions from famous teachers

(1) 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

③Belong to the same monotonic interval.

(2) When two or more monotonic intervals appear in a function, they cannot be connected with "∪" but should be connected with "and". For example, the function y=1x decreases monotonically on (-∞, 0) and (0, +∞), but it cannot be expressed as: the function y=1x decreases monotonically on (-∞, 0) ∪ (0, +∞).

2. average rate of change of function

(1)Slope of straight line

Generally speaking, given any two points A(x1, y1) and B(x2, y2) in the plane rectangular coordinate system, when x1≠x2, __________ is called the slope of straight line AB; when x1=x2, The slope of straight line AB is called __________.

The slope of line AB reflects the degree of inclination of the line relative to __________.

If we record Δx=x2-x1, and the corresponding Δy=y2-y1, then when Δx≠0, the slope can be recorded as ________.

(2) Average rate of change

Generally, when x1≠x2, it is said that ΔfΔx=_____________

It is the average rate of change of function y=f(x) on the interval [x1, x2] (when x1x2).

3. y=f(x) is a necessary and sufficient condition for an increasing function (decreasing function) on I

Generally, if I is a subset of the domain of the function y=f(x), for any x1, x2∈I and x1≠x2, note y1=f(x1), y2=f(x2), ΔyΔx=y2 -y1x2-x1 (i.e. ΔfΔx=f(x2)-f(x1)x2-x1), then:

(1) The necessary and sufficient condition that y=f(x) is an increasing function on I is that ________ is always true on I;

(2) The necessary and sufficient condition that y=f(x) is a decreasing function on I is that ________ is always true on I.

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) monotonically decreases on (-∞, 0) and (0, +∞), then the monotonically decreasing interval of f(x) is (-∞, 0) ∪ (0, +∞) ． ()

The image 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]

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

If y=(2k-1)x+b is a decreasing function on R, then there is ()

A． k>12 B. k>－12

C. k<12 D． k<-12

Monotonicity of functions PPT, Part 3: Interactive lecture and practice

Determination and proof of monotonicity of functions

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

Interactive exploration

(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.

Track training

1. Which of the following four functions is an increasing function on (-∞, 0) is ()

① y＝|

A． ①②B. ②③

C. ③④ D. ①④

2. It is known that the function f(x)=2-x/x+1, prove that the function f(x) is a decreasing function on (-1, +∞).

Find the monotonic interval of a function

Draw the graph of the function y=-x2+2|x+3, and point out the monotonic interval of the function.

Interactive exploration

(Variable conditions) Change "y＝-x2＋2|x|＋3" in this example to "y＝-x2＋2x+3|". How to solve it?

Monotonicity 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)

B． 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 are the images of functions y=f(x) and y=g(x) respectively. Try to write the monotonically increasing intervals of functions y=f(x) and y=g(x).

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