Chức năng và mô hình toán học
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (397.27 KB, 51 trang )
Functions And Mathematical
Functions And Mathematical
Models
Models
–
Your height is a function of your age.
Your height is a function of your age.
–
The cost of mailing a package is a function of weight.
The cost of mailing a package is a function of weight.
–
The area of a circle is a function of its radius.
The area of a circle is a function of its radius.
–
The weight of an astronaut is a function of her elevation.
The weight of an astronaut is a function of her elevation.
–
The price of a commodity is a function of the demand for that
The price of a commodity is a function of the demand for that
commodity.
commodity.
–
Your exam grade is a function of the time you spend studying!!!
Your exam grade is a function of the time you spend studying!!!
A function is a “rule” that describes how one quantity
A function is a “rule” that describes how one quantity
depends on another. For example
depends on another. For example
One of the most basic and important ideas in all
One of the most basic and important ideas in all
mathematics is the concept of a function.
mathematics is the concept of a function.
Fencing Problem 1
Fencing Problem 1
A rancher has 200 feet of chain-link fencing to enclose a
rectangular field as demonstrated below.
How should the rancher cut the chain-link fencing in order to
maximize the enclosed area?
200 feet
l
w
Numerical Representation
Numerical Representation
Widt h Length Area
0 100 0
5 95 475
10 90 900
15 85 1275
20 80 1600
25 75 1875
30 70 2100
35 65 2275
40 60 2400
45 55 2475
50 50 2500
55 45 2475
60 40 2400
65 35 2275
70 30 2100
75 25 1875
80 20 1600
85 15 1275
90 10 900
95 5 475
100 0 0
105 -5 -525
Graphical Representation
Graphical Representation
( )
( ) 100A w w w= −
“The value of the function A
at w”
OR
“A of w”
Dependent
variable
Independent
variable
Algebraic Representation
Algebraic Representation
( )
( ) 100A w w w= −
What is the domain of the area function A ?
OR
What are the possible values for the width w?
[ ]
{ }
0,100 | 0 100Domain w w
= = ≤ ≤
[
]
0
100
Domain and Range
Domain and Range
Domain
Range
Fencing Problem 2
Fencing Problem 2
A rancher has 200 feet of chain-link fencing to enclose two
adjacent rectangular corrals as demonstrated below.
How should the rancher cut the chain-link fencing in order to
maximize the enclosed area?
200 feet
l
w
Width
Width
Length
Length
Area
Area
0
0
100
100
0
0
5
5
92.5
92.5
462.5
462.5
10
10
85
85
850
850
15
15
77.5
77.5
1162.5
1162.5
20
20
70
70
1400
1400
25
25
62.5
62.5
1562.5
1562.5
30
30
55
55
1650
1650
35
35
47.5
47.5
1662.5
1662.5
40
40
40
40
1600
1600
45
45
32.5
32.5
1462.5
1462.5
50
50
25
25
1250
1250
55
55
17.5
17.5
962.5
962.5
60
60
10
10
600
600
65
65
2.5
2.5
162.5
162.5
70
70
-5
-5
-350
-350
Graphical Representation
Graphical Representation
200 3
( )
2
w
A w w
−
=
÷
“The value of the function A
at w”
OR
“A of w”
Dependent
variable
Independent
variable
Algebraic Representation
Algebraic Representation
200 3
( )
2
w
A w w
−
=
÷
What is the domain of the area function A ?
OR
What are the possible values for the width w?
200 200
0, | 0
3 3
Domain w w
= = ≤ ≤
[
]
0
200
3
Domain and Range
Domain and Range
Domain
Range
Volume of a Shipping Box
Volume of a Shipping Box
An overnight shipping company is designing a closed box by
cutting along the solid lines and folding along the broken lines
on the rectangular piece of cardboard as shown in the figure
below. The length and width of the rectangular cardboard are 45
inches and 15 inches, respectively.
a) Find the volume when h = 3, h = 5, and h = 7.
b) Find the volume of the shipping box as a function of its height h.
c) Estimate the value of h that gives the maximum possible volume.
45
15
h
l
w
Height,
Height,
h
h
Width,
Width,
w
w
Length,
Length,
l
l
Volume,
Volume,
V
V
0
0
15
15
22.5
22.5
0
0
1
1
13
13
21
21
273
273
2
2
11
11
19.5
19.5
429
429
3
3
9
9
18
18
486
486
4
4
7
7
16.5
16.5
462
462
5
5
5
5
15
15
375
375
6
6
3
3
13.5
13.5
243
243
7
7
1
1
12
12
84
84
8
8
-1
-1
10.5
10.5
-84
-84
Graphical Representation
Graphical Representation
( )
45 3
( ) 15 2
2
h
V h h h
−
= −
÷
“The value of the function V
at h”
OR
“V of h”
Dependent
variable
Independent
variable
What is the domain of the area function V ?
OR
What are the possible values for x?
[ ]
{ }
0,7.5 | 0 7.5Domain h h
= = ≤ ≤
[ ]
0
7.5
( )
45 3
( ) 15 2
2
h
V h h h
−
= −
÷
Domain and Range
Domain and Range
Domain
Range
Building Cheaper Can
Building Cheaper Can
A manufacturer wants to design a cylindrical can with a capacity of
A manufacturer wants to design a cylindrical can with a capacity of
holding 354 cubic centimeters of liquid. The manufacturer wishes
holding 354 cubic centimeters of liquid. The manufacturer wishes
to minimize the amount of aluminum used to make each can.
to minimize the amount of aluminum used to make each can.
How should the manufacturer design the can to minimize the
How should the manufacturer design the can to minimize the
amount of aluminum used?
amount of aluminum used?
r
h
Radius,
Radius,
r
r
Height,
Height,
h
h
Surface Area,
Surface Area,
S
S
0
0
Error
Error
Error
Error
0.5
0.5
450.7267988
450.7267988
1417.570796
1417.570796
1
1
112.6816997
112.6816997
714.2831853
714.2831853
2
2
28.17042493
28.17042493
379.1327412
379.1327412
3
3
12.52018886
12.52018886
292.5486678
292.5486678
4
4
7.042606232
7.042606232
277.5309649
277.5309649
5
5
4.507267988
4.507267988
298.6796327
298.6796327
6
6
3.130047214
3.130047214
344.1946711
344.1946711
7
7
2.299626525
2.299626525
409.0189372
409.0189372
8
8
1.760651558
1.760651558
490.6238597
490.6238597
9
9
1.391132095
1.391132095
587.6046765
587.6046765
10
10
1.126816997
1.126816997
699.1185307
699.1185307
11
11
0.931253717
0.931253717
824.6290585
824.6290585
12
12
0.782511804
0.782511804
963.7786842
963.7786842
13
13
0.66675562
0.66675562
1116.319855
1116.319855
14
14
0.574906631
0.574906631
1282.075749
1282.075749
15
15
0.500807554
0.500807554
1460.916694
1460.916694
Graphical Representation
Graphical Representation
2
708
( ) 2S r r
r
π
= +
“The value of the function S
at r”
OR
“S of r”
Dependent
variable
Independent
variable
