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2.1 day 2: Step Functions
“Miraculous Staircase”Loretto Chapel, Santa Fe, NM
Two 360o turns without support!
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2003
“Step functions” are sometimes used to describe real-life situations.
Our book refers to one such function: int( )y x
This is the Greatest Integer Function.
The TI-89 contains the command , but it is important that you understand the function rather than just entering it in your calculator.
int( )x
Greatest Integer Function:
greatest integer that is xy
x y
0 00.5 0
0.75 01 1
Greatest Integer Function:
greatest integer that is xy
x y
0 00.5 0
0.75 01 1
1.5 12 2
Greatest Integer Function:
greatest integer that is xy
x y
0 00.5 0
0.75 01 1
1.5 12 2
Greatest Integer Function:
greatest integer that is xy
x y
0 00.5 0
0.75 01 1
1.5 12 2
This notation was introduced in 1962 by Kenneth E. Iverson.
Recent by math standards!
Greatest Integer Function:
greatest integer that is xy
The greatest integer function is also called the floor function.
The notation for the floor function is:
y x
We will not use these notations.
Some books use or . y x y x
The older TI-89 calculator “connects the dots” which covers up the discontinuities. (The Titanium Edition does not do this.)
The TI-89 command for the floor function is floor (x).
Graph the floor function for and .8 8x 4 4y
Y= floor x
CATALOG F floor(
Go to Y=
Highlight the function.
2nd F6 Style 2:Dot
ENTER
GRAPH
The open and closed circles do not show, but we can see the discontinuities.
The TI-89 command for the floor function is floor (x).
Graph the floor function for and .8 8x 4 4y
If you have the older TI-89 you could try this:
Least Integer Function:
least integer that is xy
x y
0 00.5 1
0.75 11 1
x y
0 00.5 1
0.75 11 1
1.5 22 2
Least Integer Function:
least integer that is xy
x y
0 00.5 1
0.75 11 1
1.5 22 2
Least Integer Function:
least integer that is xy
x y
0 00.5 1
0.75 11 1
1.5 22 2
Least Integer Function:
least integer that is xy
The least integer function is also called the ceiling function.
The notation for the ceiling function is:
y x
Least Integer Function:
least integer that is xy
The TI-89 command for the ceiling function is ceiling (x).
Don’t worry, there are not wall functions, front door functions, fireplace functions!
Using the Sandwich theorem to find 0
sinlimx
x
x
If we graph , it appears thatsin x
yx
0
sinlim 1x
x
x
If we graph , it appears thatsin x
yx
0
sinlim 1x
x
x
We might try to prove this using the sandwich theorem as follows:
sin 1 and sin 1x x
0 0 0
1 sin 1 lim lim lim
x x x
x
x x x
Unfortunately, neither of these new limits are defined, since the left and right hand limits do not match.
We will have to be more creative. Just see if you can follow this proof. Don’t worry that you wouldn’t have thought of it.
Unfortunately, neither of these new limits are defined, since the left and right hand limits do not match.
(1,0)
1
Unit Circle
cos
sin
P(x,y)
Note: The following proof assumes positive values of . You could do a similar proof for negative values.
(1,0)
1
Unit Circle
cos
sin
P(x,y)
T
AO
tan1
AT
tanAT
1, tan
(1,0)
1
Unit Circle
cos
sin
P(x,y)
T
AO
1, tan
(1,0)
1
Unit Circle
cos
sin
P(x,y)
T
AO
1, tan
Area AOP
(1,0)
1
Unit Circle
cos
sin
P(x,y)
T
AO
1, tan
Area AOP Area sector AOP
(1,0)
1
Unit Circle
cos
sin
P(x,y)
T
AO
1, tan
Area AOP Area sector AOP Area OAT
(1,0)
1
Unit Circle
cos
sin
P(x,y)
T
AO
1, tan
11 sin
2
Area AOP Area sector AOP Area OAT
(1,0)
1
Unit Circle
cos
sin
P(x,y)
T
AO
1, tan
11 sin
2
Area sector AOP
2
2r
2
2
Area AOP Area sector AOP Area OAT
(1,0)
1
Unit Circle
cos
sin
P(x,y)
T
AO
1, tan
11 sin
2
2
11 tan
2
Area AOP Area sector AOP Area OAT
11 sin
2
2
11 tan
2
sin tan multiply by two
sinsin
cos
11
sin cos
divide by sin
sin1 cos
Take the reciprocals, which reverses the inequalities.
sincos 1
Switch ends.
11 sin
2
2
11 tan
2
sin tan
sinsin
cos
11
sin cos
sin1 cos
sincos 1
0 0 0
sinlim cos lim lim1
0
sin1 lim 1
By the sandwich theorem:
0
sinlim 1
p
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