Topicm15ec39d36e81bf78_1528449000663_0Topic

The monotonicity of functions - examples

Levelm15ec39d36e81bf78_1528449084556_0Level

Third

Core curriculumm15ec39d36e81bf78_1528449076687_0Core curriculum

V. Functions. The student:

4) reads from the graph of the function: the domain, the range, roots, monotonic intervals, intervals in which the function takes values not greater (not smaller) or smaller (not greater) than a given number, greatest and smallest values of the function (if they exist) in the closed interval and arguments for which the function takes greatest and smallest values.

Timingm15ec39d36e81bf78_1528449068082_0Timing

45 minutes

General objectivem15ec39d36e81bf78_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm15ec39d36e81bf78_1528449552113_0Specific objectives

1. Identifying the intervals of monotonicity of the function.

2. Creating monotonic functions.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm15ec39d36e81bf78_1528450430307_0Learning outcomes

The student:

- identifies the intervals of monotonicity of the function,

- creates monotonic functions.

Methodsm15ec39d36e81bf78_1528449534267_0Methods

1. Discussion.

2. Situational analysis.

Forms of workm15ec39d36e81bf78_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm15ec39d36e81bf78_1528450127855_0Introduction

The teacher introduces the subject of the lesson - identifying the intervals of monotonicitymonotonicitymonotonicity of the function and creating monotonic functions.

Students revise properties of the function monotonic in intervals.

Procedurem15ec39d36e81bf78_1528446435040_0Procedure

Task
Students work individually, using computers. Their task is to identify maximal intervals in which the function is increasing, decreasing or constant.

[Geogebra applet]

Task
In the drawing there is the function g.

Identify maximal intervals of monotonicitymonotonicitymonotonicity of this function.

[Illustration 1]

Discussion – what function is called non‑increasing or non‑decreasing.

Students give examples of plots of such functions.

Together they formulate definitions.

Definition of the non‑decreasing functionnon‑decreasing functionnon‑decreasing function
- Let f be a function defined in the intervalintervalinterval 〈a; b〉.
If for any xIndeks dolny 1₁, xIndeks dolny 2₂ ∈ 〈a; b〉 such that xIndeks dolny 1₁ < xIndeks dolny 2₂ the condition
f(xIndeks dolny 1₁) ≤ f(xIndeks dolny 2₂)
is met then we say the function is non‑decreasing in the intervalintervalinterval 〈a; b〉.

Definition of the non‑increasing functionnon‑increasing functionnon‑increasing function
- Let f be a function defined in the intervalintervalinterval 〈a; b〉.
If for any xIndeks dolny 1₁, xIndeks dolny 2₂ ∈ 〈a; b〉 such that xIndeks dolny 1₁ < xIndeks dolny 2₂ the condition
f(xIndeks dolny 1₁) ≥ f(xIndeks dolny 2₂)
is met then we say the function is non‑increasing in the intervalintervalinterval 〈a; b〉.

Students use obtained information in the exercises.

Task
Draw plots of any three non‑decreasing functions, defined for x ∈ 〈-5; 5〉.m15ec39d36e81bf78_1527752256679_0Draw plots of any three non‑decreasing functions, defined for x ∈ 〈-5; 5〉.

Task
Draw plots of any three non‑increasing functions, defined for x ∈ (2; 8).

Task
Using the plot of the function p give:

[Illustration 2]

1. The maximal intervalmaximal intervalmaximal interval in which the function is increasing.

2. The maximal intervalmaximal intervalmaximal interval in which the function is non‑increasing.

3. The maximal intervalmaximal intervalmaximal interval in which the function is decreasing.

4. The maximal intervalmaximal intervalmaximal interval in which the function is non‑decreasing.

An extra task:
Draw the plot of the function f(x) = (x - 2)Indeks górny 2² and determine the monotonicity of the function.m15ec39d36e81bf78_1527752263647_0An extra task:
Draw the plot of the function f(x) = (x - 2)Indeks górny 2² and determine the monotonicity of the function.

Lesson summarym15ec39d36e81bf78_1528450119332_0Lesson summary

Students do the revision exercises.

Then together they sum‑up the classes, by formulating the conclusions to memorise.

Definition of the non‑decreasing functionnon‑decreasing functionnon‑decreasing function
- Let f be a function defined in the intervalintervalinterval 〈a; b〉.
If for any xIndeks dolny 1₁, xIndeks dolny 2₂ ∈ 〈a; b〉 such that xIndeks dolny 1₁ < xIndeks dolny 2₂ the condition
f(xIndeks dolny 1₁) ≤ f(xIndeks dolny 2₂)
is met then we say the function is non‑decreasing in the intervalintervalinterval 〈a; b〉.

Definition of the non‑increasing functionnon‑increasing functionnon‑increasing function
- Let f be a function defined in the intervalintervalinterval 〈a; b〉.
If for any xIndeks dolny 1₁, xIndeks dolny 2₂ ∈ 〈a; b〉 such that xIndeks dolny 1₁ < xIndeks dolny 2₂ the condition
f(xIndeks dolny 1₁) ≥ f(xIndeks dolny 2₂)
is met then we say the function is non‑increasing in the intervalintervalinterval 〈a; b〉.

Selected words and expressions used in the lesson plan

intervalintervalinterval

maximal intervalmaximal intervalmaximal interval

monotonicitymonotonicitymonotonicity

non‑decreasing functionnon‑decreasing functionnon‑decreasing function

non‑increasing functionnon‑increasing functionnon‑increasing function