Fencing Problem 1A 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 maximiz
Trang 1Functions And Mathematical
Models
Trang 2– Your height is a function of your age.
– The cost of mailing a package is a function of weight.
– The area of a circle is a function of its radius.
– The weight of an astronaut is a function of her elevation.
– The price of a commodity is a function of the demand for that
commodity.
– Your exam grade is a function of the time you spend studying!!!
A function is a “rule” that describes how one quantity depends on another For example
One of the most basic and important ideas in all mathematics is the concept of a function.
Trang 3Fencing 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
Trang 4Numerical Representation
Trang 5Width Length Area
Trang 6Graphical Representation
Trang 9Domain and Range
DomainRange
Trang 10Fencing 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
Trang 11Width Length Area
Trang 12Graphical Representation
Trang 13200 3 ( )
Algebraic Representation
Trang 14200 3 ( )
Trang 15Domain and Range
DomainRange
Trang 16Volume 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
l w
Trang 17Height, h Width, w Length, l Volume, V
Trang 18Graphical Representation
Trang 20What is the domain of the area function V ?
Trang 21Domain and Range
DomainRange
Trang 22Building Cheaper Can
A manufacturer wants to design a cylindrical can with a capacity of holding 354 cubic centimeters of liquid The manufacturer wishes
to minimize the amount of aluminum used to make each can
How should the manufacturer design the can to minimize the
amount of aluminum used?
r
h
Trang 23Radius, r Height, h Surface Area, S
Trang 24Graphical Representation
Trang 252 708 ( ) 2
Trang 26What is the domain of the surface area function S ?
Trang 27Domain and Range
DomainRange
Trang 28Mathematical Definition Of A Function
A function f is a rule that assigns to each
f
-3 -2 0 1 3
-12 5 23 35
h
f is a function h is not a function
Trang 29x y
Vertical Line Test
Trang 31Fencing 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
Trang 33Fencing 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
Trang 34200 3 ( )
Trang 35Volume of a Shipping Box
Estimate the value of h that gives the maximum
possible volume.
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
45
l w
Trang 37Building Cheaper Can
A manufacturer wants to design a cylindrical can with a capacity of holding 354 cubic centimeters of liquid The manufacturer wishes
to minimize the amount of aluminum used to make each can
How should the manufacturer design the can to minimize the
amount of aluminum used?
r
h
Trang 39Piecewise Defined Functions
Piecewise Defined Functions
Trang 40Piecewise Defined Functions
Piecewise Defined Functions
Trang 41Equations That Define Functions
Does the equation define y as a function of x?
Does the equation define y as a function of x?
-3 3
Trang 42Average Rate Of Change
Trang 43Average Rate Of Change
Trang 44Average Rate Of Change
Trang 45Average Rate Of Change
Concave upward
Trang 46Average Rate Of Change
Trang 47Average Rate Of Change
Concave upward
Trang 48Average Rate Of Change
Trang 49Average Rate Of Change
Trang 50a) Describe the total area covered as a function of its width w.
b) Estimate the value of w that gives the maximum possible area.
c) Find the domain and range of the area function
Trang 51Designing 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 rectangle are 60 inches and
20 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.
60
l w