People's Education High School Mathematics Edition A Compulsory Course 1

Description

"Basic Inequalities" Quadratic Functions, Equations and Inequalities PPT Courseware (First Lesson Basic Inequalities)

Part One: Learning Objectives

1. Understand the proof process of basic inequalities. (emphasis)

2. Be able to use basic inequalities to prove simple inequalities and compare the sizes of algebraic expressions.

core competencies

1. Cultivate logical reasoning skills through the proof of inequalities.

2. Use basic inequality forms to solve simple optimal value problems and improve your mathematical operation literacy.

Basic Inequalities PPT, Part 2: Independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. important inequalities

∀a, b∈R, there is a2+b2≥_______, if and only if _______, the equal sign is true.

2. basic inequalities

(1) Related concepts: When a and b are both positive numbers, a + b2 is called the arithmetic mean of the positive numbers a and b, and ab is called the geometric mean of the positive numbers a and b.

(2) Inequality: When a and b are any positive real numbers, the geometric mean of a and b is not greater than their arithmetic mean, that is, ab≤a+b2. The equal sign holds true if and only if _______.

First try

1. The condition for the establishment of the equal sign of the inequality a2+1≥2a is ()

A． a＝±1

B. a=1

C. a=-1

D. a=0

2. It is known that a, b∈(0,1), and a≠b, the largest of the following formulas is ()

A． a2＋b2

B. 2ab

C. 2ab

D. a+b

3． It is known that ab=1, a>0, b>0, then the minimum value of a+b is ()

A． 1B. 2

C. 4D. 8

Basic Inequalities PPT, Part 3: Collaborative exploration to improve literacy

Understanding basic inequalities

[Example 1] The following four derivation processes are given:

①∵a and b are positive real numbers, ∴ba+ab≥2ba·ab=2;

②∵a∈R, a≠0, ∴4a+a≥24a·a=4;

③∵x, y∈R, xy＜0, ∴xy＋yx＝－－xy＋－yx≤－2－xy－yx＝－2.

The correct derivation is ()

A． ①②B. ①③

C. ②③ D. ①②③

B　①∵a and b are positive real numbers, ∴ba and ab are positive real numbers, which meet the conditions of basic inequalities, so the derivation of ① is correct.

②∵a∈R, a≠0, does not meet the conditions of basic inequality,

∴4a＋a≥24a·a=4 is wrong.

③ From xy < 0, we get that xy and yx are both negative numbers. However, after the negative sign of the overall xy + yx is raised during the derivation process, -xy and -yx both become positive numbers, which meets the conditions of mean inequality, so ③ is correct. ]

regular method

1. The basic inequality ab≤a+b2 (a>0, b>0) reflects the relationship between the sum and product of two positive numbers.

2. To accurately grasp basic inequalities, we must grasp the following two aspects: (1) The condition for the establishment of the theorem is that a and b are both positive numbers. (2) The meaning of "if and only if": when a=b, the equality of ab≤a+b2 holds, that is, a=b⇒a+b2=ab; only when a=b, the equality of a+b2≥ab holds, That is, a+b2=ab⇒a=b.

Basic Inequalities PPT, Part 4: Complying with Standards and Solidifying Basics in Class

1. Thinking and analysis

(1) For any a, b∈R, a2+b2≥2ab, a+b≥2ab are all true. ()

(2) If a≠0, then a+1a≥2a·1a=2.()

(3) If a>0, b>0, then ab≤a+b22.()

[Tips] (1) For any a, b∈R, a2+b2≥2ab holds. When a and b are both positive numbers, the inequality a+b≥2ab holds.

(2) Only when a>0, according to the basic inequality, the inequality a+1a≥2a·1a=2 holds.

(3) Because ab≤a+b2, so ab≤a+b22.

2. Assuming a>b>0, then the following inequalities must be true ()

A． a-b<0

B. 0

C.ab

D. ab>a+b

3． Inequality 9x-2+(x-2)≥6 (where x>2) The condition for the establishment of the equal sign is ()

A． x=3

B. x=-3

C. x=5

D. x=-5

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Update Time: 2024-06-15

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