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2.1 day 2: Step Functions

2.1 day 2: Step Functions. “Miraculous Staircase” Loretto Chapel, Santa Fe, NM. Two 360 o turns without support!. Photo by Vickie Kelly, 2003. Greg Kelly, Hanford High School, Richland, Washington.

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2.1 day 2: Step Functions

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  1. 2.1 day 2: StepFunctions “Miraculous Staircase” Loretto Chapel, Santa Fe, NM Two 360o turns without support! Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington

  2. The TI-89 contains the command , but it is important that you understand the function rather than just entering it in your calculator. “Step functions” are sometimes used to describe real-life situations. Our book refers to one such function: This is the Greatest Integer Function.

  3. Greatest Integer Function:

  4. Greatest Integer Function:

  5. Greatest Integer Function:

  6. Greatest Integer Function:

  7. Some books use or . This notation was introduced in 1962 by Kenneth E. Iverson. We will not use these notations. Greatest Integer Function: The greatest integer function is also called the floor function. The notation for the floor function is: Recent by math standards!

  8. Graph the floor function for and . Y= The TI-89 command for the floor function is floor (x). CATALOG F floor( The older TI-89 calculator “connects the dots” which covers up the discontinuities. (The Titanium Edition does not do this.)

  9. ENTER Graph the floor function for and . Y= Go to GRAPH The TI-89 command for the floor function is floor (x). If you have the older TI-89 you could try this: Highlight the function. 2nd F6 Style 2:Dot The open and closed circles do not show, but we can see the discontinuities.

  10. Least Integer Function:

  11. Least Integer Function:

  12. Least Integer Function:

  13. Least Integer Function:

  14. Don’t worry, there are not wall functions, front door functions, fireplace functions! Least Integer Function: The least integer function is also called the ceiling function. The notation for the ceiling function is: The TI-89 command for the ceiling function is ceiling (x).

  15. Using the Sandwich theorem to find

  16. If we graph , it appears that

  17. 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. If we graph , it appears that We might try to prove this using the sandwich theorem as follows: 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.

  18. 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

  19. T P(x,y) 1 O A (1,0) Unit Circle

  20. T P(x,y) 1 O A (1,0) Unit Circle

  21. T P(x,y) 1 O A (1,0) Unit Circle

  22. T P(x,y) 1 O A (1,0) Unit Circle

  23. T P(x,y) 1 O A (1,0) Unit Circle

  24. T P(x,y) 1 O A (1,0) Unit Circle

  25. T P(x,y) 1 O A (1,0) Unit Circle

  26. T P(x,y) 1 O A (1,0) Unit Circle

  27. multiply by two divide by Take the reciprocals, which reverses the inequalities. Switch ends.

  28. By the sandwich theorem: p

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