The function of one variable презентация

Содержание

1. The function of one variable
2. Literature: [1], v. І, p.
3. 1.The main definitions 2.The different ways of
4. Given two sets X and Y
5. Examples. D(y) - ? 1) f(x)=x3-4x+2 (polynomial
6. The most important ways of representing of
7. The analytic method:
8. 5) The demand function: q
9. The tabular method:
10. The graphical method:
11. The main ways of the graph transformations 1) Right-left translation: Example: y=(x-1)2
12. 2) Up-down translation: Example: Sketch the graph y=x2+4x+1 - ?
13. 3) Changing scale: stretching and shrinking Example: Sketch the graph
14. Example: Sketch the graph:
15. The main characteristics of behavior of the
16. The basic elementary functions: The power function;
17. 1) The power function: Some particular
18. : , . - odd; - increasing on ;
19. : - even;
20. : ,
21. 2) The exponential function: (a>1) (0
22. 3) The logarithmic function: y=logax
23. 4) The trigonometric functions:
24. b) y=cosx , . The function is even and periodic, period T=2π
25. c) y=tgx ,
26. d) y=сtgx , .
27. 5) The inverse trigonometric functions: а) y=arcsinx
28. b) y=arccosx , . The function is decreasing
29. c) y=arctgx, . The function is odd and increasing
30. d) y=arcсtgx, . The function is decreasing
31. Examples 3)
32. y=f(u), u=g(x) y=f(g(x)) –
33. An elementary function is a function of
34. The home work: Sketch the graph: a) y=x2-4x+5; b) y=3+2ln(x-1)

Literature: [1], v. І, p. 174-180, [2], part І, p. 40-57, [3], p. 174-181. Theme: The function of one variable

Слайд 1CALCULAS studies the relationships
that exists between one collection of
objects and

another

Слайд 2Literature: [1], v. І, p. 174-180, [2], part І, p. 40-57, [3],

p. 174-181.

Theme: The function of one variable

Слайд 31.The main definitions
2.The different ways of representing of the functions
3. The

main characteristics of behavior of the function
4. The basic elementary functions
5.The composite function
6.The elementary functions

Слайд 4Given two sets X and Y
Definition. A function is a

rule which assigns
to each element x of X one and only one
element y of Y.

Notation: y=f(x)

x - the independent variable
y - the dependent variable

The set X - the domain of the function (D(y))

The set of all corresponding values of y - the
range of the function (E(y))

Слайд 5Examples. D(y) - ?
1) f(x)=x3-4x+2 (polynomial of the third power)
2)

Слайд 6The most important ways of representing of the functions:
- the analytic

method;

- the tabular method;

- the graphical method

Слайд 7

The analytic method:
The function y=f(x) is represented analytically if
the

variables x and y are connected with each other
by equations

- an explicit function

- an implicit function

Examples

Слайд 85) The demand function:
q - price, p – demand
6)

Cost function V(x), income function D(x),
profit function P(x),
where x – the volume of production

Слайд 9The tabular method:

Слайд 10The graphical method:

Слайд 11The main ways of the graph transformations
1) Right-left translation:
Example: y=(x-1)2

Слайд 122) Up-down translation:
Example: Sketch the graph
y=x2+4x+1 - ?

Слайд 133) Changing scale: stretching and shrinking
Example: Sketch the graph

Слайд 14Example: Sketch the graph:

Слайд 15The main characteristics of behavior of the function
monotonic function (increasing

or decreasing):
- increasing:
- decreasing:
even or odd:
- even:
- odd:

periodicity: The periodic function is a function
that repeats its values in regular intervals

Слайд 16The basic elementary functions:
The power function;
The exponential function;
The logarithmic function;
The trigonometric

functions ( 4 );
The inverse trigonometric functions ( 4 ).

Слайд 171) The power function:
Some particular cases:

,
.
- even;
- decreasing

on [-∞;0];
- increasing on [0;+∞]

;

Слайд 18:

,
.
- odd;
- increasing on
;

Слайд 19:

- even;
- increasing on (-∞; 0),
- decreasing on

(0; +∞);

Слайд 20:

,
.
- odd;
- decreasing on
and
.

Слайд 212) The exponential function:
(a>1)
(0

decreasing,
if a>1 then the function is increasing.

Слайд 223) The logarithmic function:
y=logax

,
.
If

0if a>1 then the function is increasing.

Слайд 234) The trigonometric functions:

,
.
The function is odd and periodic,

period T=2π

Слайд 24b) y=cosx

,
.
The function is even and periodic, period T=2π

Слайд 25c) y=tgx

,
.
The function is odd and periodic, period

T=π

Слайд 26d) y=сtgx

,
.

The function is odd and periodic, period T=π

Слайд 275) The inverse trigonometric functions:
а) y=arcsinx

,
.
The function is odd and

increasing

Слайд 28b) y=arccosx

,
.
The function is decreasing

Слайд 29c) y=arctgx,
.
The function is odd and increasing

Слайд 30d) y=arcсtgx,
.
The function is decreasing

Слайд 31Examples

3)

Слайд 32 y=f(u), u=g(x)
y=f(g(x)) – the composite function
u – the

intermediate variable,
f(u) - external function,
g(x) - internal function

Слайд 33An elementary function is a function of one
variable built from

a finite number of the basic
elementary function and constants through
composition and combinations using the four
elementary operations (+ – × ÷).

Examples:

Слайд 34The home work:

Sketch the graph:
a) y=x2-4x+5;
b) y=3+2ln(x-1)

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The function of one variable презентация

Содержание

Слайд 1CALCULAS studies the relationships
that exists between one collection of
objects and

Слайд 2Literature: [1], v. І, p. 174-180, [2], part І, p. 40-57, [3],

Слайд 31.The main definitions
2.The different ways of representing of the functions
3. The

Слайд 4Given two sets X and Y
Definition. A function is a

Слайд 5Examples. D(y) - ?
1) f(x)=x3-4x+2 (polynomial of the third power)
2)

Слайд 6The most important ways of representing of the functions:
- the analytic

Слайд 7

The analytic method:
The function y=f(x) is represented analytically if
the

Слайд 85) The demand function:
q - price, p – demand
6)

Слайд 9The tabular method:

Слайд 10The graphical method:

Слайд 11The main ways of the graph transformations
1) Right-left translation:
Example: y=(x-1)2

Слайд 122) Up-down translation:
Example: Sketch the graph
y=x2+4x+1 - ?

Слайд 133) Changing scale: stretching and shrinking
Example: Sketch the graph

Слайд 14Example: Sketch the graph:

Слайд 15The main characteristics of behavior of the function
monotonic function (increasing

Слайд 16The basic elementary functions:
The power function;
The exponential function;
The logarithmic function;
The trigonometric

Слайд 171) The power function:
Some particular cases:

,
.
- even;
- decreasing

Слайд 18:

,
.
- odd;
- increasing on
;

Слайд 19:

- even;
- increasing on (-∞; 0),
- decreasing on

Слайд 20:

,
.
- odd;
- decreasing on
and
.

Слайд 212) The exponential function:
(a>1)
(0

Слайд 223) The logarithmic function:
y=logax

,
.
If

Слайд 234) The trigonometric functions:

,
.
The function is odd and periodic,

Слайд 24b) y=cosx

,
.
The function is even and periodic, period T=2π

Слайд 25c) y=tgx

,
.
The function is odd and periodic, period

Слайд 26d) y=сtgx

,
.

The function is odd and periodic, period T=π

Слайд 275) The inverse trigonometric functions:
а) y=arcsinx

,
.
The function is odd and

Слайд 28b) y=arccosx

,
.
The function is decreasing

Слайд 29c) y=arctgx,
.
The function is odd and increasing

Слайд 30d) y=arcсtgx,
.
The function is decreasing

Слайд 31Examples

3)

Слайд 32 y=f(u), u=g(x)
y=f(g(x)) – the composite function
u – the

Слайд 33An elementary function is a function of one
variable built from

Слайд 34The home work:

Sketch the graph:
a) y=x2-4x+5;
b) y=3+2ln(x-1)

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The function of one variable презентация

Содержание

Слайд 1CALCULAS studies the relationshipsthat exists between one collection of objects and

Слайд 2Literature: [1], v. І, p. 174-180, [2], part І, p. 40-57, [3],

Слайд 31.The main definitions2.The different ways of representing of the functions3. The

Слайд 4Given two sets X and Y Definition. A function is a

Слайд 5Examples. D(y) - ?1) f(x)=x3-4x+2 (polynomial of the third power)2)

Слайд 6The most important ways of representing of the functions:- the analytic

Слайд 7The analytic method: The function y=f(x) is represented analytically if the

Слайд 85) The demand function: q - price, p – demand6)

Слайд 9The tabular method:

Слайд 10The graphical method:

Слайд 11The main ways of the graph transformations1) Right-left translation:Example: y=(x-1)2

Слайд 122) Up-down translation:Example: Sketch the graph y=x2+4x+1 - ?

Слайд 133) Changing scale: stretching and shrinkingExample: Sketch the graph

Слайд 14Example: Sketch the graph:

Слайд 15The main characteristics of behavior of the function monotonic function (increasing

Слайд 16The basic elementary functions:The power function;The exponential function;The logarithmic function;The trigonometric

Слайд 171) The power function: Some particular cases:, .- even;- decreasing

Слайд 18: , .- odd;- increasing on ;

Слайд 19: - even;- increasing on (-∞; 0), - decreasing on

Слайд 20: , .- odd;- decreasing on and .

Слайд 212) The exponential function:(a>1)(0

Слайд 223) The logarithmic function: y=logax , . If

Слайд 234) The trigonometric functions: , .The function is odd and periodic,

Слайд 24b) y=cosx, .The function is even and periodic, period T=2π

Слайд 25c) y=tgx , .The function is odd and periodic, period

Слайд 26d) y=сtgx, .The function is odd and periodic, period T=π

Слайд 275) The inverse trigonometric functions:а) y=arcsinx, .The function is odd and

Слайд 28b) y=arccosx, .The function is decreasing

Слайд 29c) y=arctgx, .The function is odd and increasing

Слайд 30d) y=arcсtgx, .The function is decreasing

Слайд 31Examples3)

Слайд 32 y=f(u), u=g(x) y=f(g(x)) – the composite functionu – the

Слайд 33An elementary function is a function of one variable built from

Слайд 34The home work:Sketch the graph: a) y=x2-4x+5; b) y=3+2ln(x-1)

Похожие презентации

Обратная связь

Что такое ThePresentation.ru?

Слайд 1CALCULAS studies the relationships
that exists between one collection of
objects and

Слайд 31.The main definitions
2.The different ways of representing of the functions
3. The

Слайд 4Given two sets X and Y
Definition. A function is a

Слайд 5Examples. D(y) - ?
1) f(x)=x3-4x+2 (polynomial of the third power)
2)

Слайд 6The most important ways of representing of the functions:
- the analytic

Слайд 7

The analytic method:
The function y=f(x) is represented analytically if
the

Слайд 85) The demand function:
q - price, p – demand
6)

Слайд 11The main ways of the graph transformations
1) Right-left translation:
Example: y=(x-1)2

Слайд 122) Up-down translation:
Example: Sketch the graph
y=x2+4x+1 - ?

Слайд 133) Changing scale: stretching and shrinking
Example: Sketch the graph

Слайд 15The main characteristics of behavior of the function
monotonic function (increasing

Слайд 16The basic elementary functions:
The power function;
The exponential function;
The logarithmic function;
The trigonometric

Слайд 171) The power function:
Some particular cases:

,
.
- even;
- decreasing

Слайд 18:

,
.
- odd;
- increasing on
;

Слайд 19:

- even;
- increasing on (-∞; 0),
- decreasing on

Слайд 20:

,
.
- odd;
- decreasing on
and
.

Слайд 212) The exponential function:
(a>1)
(0

Слайд 223) The logarithmic function:
y=logax

,
.
If

Слайд 234) The trigonometric functions:

,
.
The function is odd and periodic,

Слайд 24b) y=cosx

,
.
The function is even and periodic, period T=2π

Слайд 25c) y=tgx

,
.
The function is odd and periodic, period

Слайд 26d) y=сtgx

,
.

The function is odd and periodic, period T=π

Слайд 275) The inverse trigonometric functions:
а) y=arcsinx

,
.
The function is odd and

Слайд 28b) y=arccosx

,
.
The function is decreasing

Слайд 29c) y=arctgx,
.
The function is odd and increasing

Слайд 30d) y=arcсtgx,
.
The function is decreasing

Слайд 31Examples

3)

Слайд 32 y=f(u), u=g(x)
y=f(g(x)) – the composite function
u – the

Слайд 33An elementary function is a function of one
variable built from

Слайд 34The home work:

Sketch the graph:
a) y=x2-4x+5;
b) y=3+2ln(x-1)