2.1 day 1: step functions “miraculous staircase” loretto chapel, santa fe, nm two 360 o turns...
TRANSCRIPT
2.1 day 1: Step Functions
“Miraculous Staircase”Loretto Chapel, Santa Fe, NM
Two 360o turns without support!
The limit of a function refers to the value that the function approaches, not the actual value (if any).
2
lim 2x
f x
not 1
Properties of Limits:
Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.
(See page 58 for details.)
For a limit to exist, the function must approach the same value from both sides.
One-sided limits approach from either the left or right side only.
1 2 3 4
1
2
At x=1: 1
lim 0x
f x
1
lim 1x
f x
1 1f
left hand limit
right hand limit
value of the function
1
limxf x
does not exist because the left and right hand limits do not match!
At x=2: 2
lim 1x
f x
2
lim 1x
f x
2 2f
left hand limit
right hand limit
value of the function
2
lim 1xf x
because the left and right hand limits match.
1 2 3 4
1
2
At x=3: 3
lim 2x
f x
3
lim 2x
f x
3 2f
left hand limit
right hand limit
value of the function
3
lim 2xf x
because the left and right hand limits match.
1 2 3 4
1
2
“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-83 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
The greatest integer function is also called the floor function.
The notation for the floor function is:
y x
Some books use or . y x y x
The calculator “connects the dots” which covers up the discontinuities.
The TI-83 command for the floor function is int (x).
Graph the floor function for and .8 8x 4 4y
Y=
CATALOG I int(
int (x)
Go to Y=
Highlight the slant line to the left.
ENTER
GRAPH
The open and closed circles do not show, but we can see the discontinuities.
The TI-83 command for the floor function is int (x).
Graph the floor function for and .8 8x 4 4y
Press until the line appears dotted
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!