2.1 day 2: Step Functions “Miraculous Staircase” Loretto Chapel, Santa Fe, NM Two 360o turns without support! Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington
“Step functions” are sometimes used to describe real-life situations. Our book refers to one such function: This is the Greatest Integer Function. The TI-nspire contains the command , but it is important that you understand the function rather than just entering it in your calculator.
Greatest Integer Function:
Greatest Integer Function:
Greatest Integer Function:
Greatest Integer Function:
Greatest Integer Function: The greatest integer function is also called the floor function. The notation for the floor function is: This notation was introduced in 1962 by Kenneth E. Iverson. We will not use these notations. Some books use or . Recent by math standards!
Graph the floor function. Use Zoom – Standard. The TI-nspire command for the floor function is either int (x) or floor (x). Graph the floor function. Use Zoom – Standard. floor( F Notice that the calculator does not include the open and closed circles at the endpoints.
Least Integer Function:
Least Integer Function:
Least Integer Function:
Least Integer Function:
Least Integer Function: Don’t worry, there are not wall functions, front door functions, fireplace functions! The least integer function is also called the ceiling function. The notation for the ceiling function is: The TI-nspire command for the ceiling function is ceiling (x).
Using the Sandwich theorem to find
If we graph , it appears that
If we graph , it appears that We might try to prove this using the sandwich theorem as follows: Unfortunately, neither of these new limits are defined, since the left and right hand limits do not match. 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.
Note: The following proof assumes positive values of . You could do a similar proof for negative values. P(x,y) 1 (1,0) Unit Circle
T P(x,y) 1 O A (1,0) Unit Circle
T P(x,y) 1 O A (1,0) Unit Circle
T P(x,y) 1 O A (1,0) Unit Circle
T P(x,y) 1 O A (1,0) Unit Circle
T P(x,y) 1 O A (1,0) Unit Circle
T P(x,y) 1 O A (1,0) Unit Circle
T P(x,y) 1 O A (1,0) Unit Circle
T P(x,y) 1 O A (1,0) Unit Circle
multiply by two divide by Take the reciprocals, which reverses the inequalities. Switch ends.
By the sandwich theorem: p