Governor’s School for the Sciences

1. Governor’s School for the Sciences Mathematics Day 1

2. Schedule • Introductions • Hometown: Good, Bad or Ugly • MOTD • A Joke • Some Math! • Forms and Tests • Lab

3. Course Structure • Instructors • Lecture, Work and Labs • Point System • Project • AfterClass

4. Where are you from? Give us the Good, the Bad or the Ugly!

5. MOTD: Abu al-Khwarizmi • 780-850 AD • Hindu-Arabic numerals • First to use 0 • “Algorithm” derived from his name

6. Why did Homeland Security arrest the mathematician?

7. Why did Homeland Security arrest the mathematician? • They found out he was a member of the al Gebra network. • They found a calculus book, a compass and chalk in his briefcase: weapons of math instruction.

8. All About Patterns • Patterns in number sequences • Patterns in letters/words • Patterns in geometry Mathematics is about finding and studying patterns.

9. Pattern “Black Box” Nature gives us a pattern we need to create a matching box.

10. Simple PatternsWhat comes next? • 1, 2, 3, 4, ___ • 1, 2, 3, 5, ___ • 1, 2, 4, 8, ___ • 2, 4, 6, 8, ___ • 1, 3, 2, 4, ___ • 3, 5, 9, 17, ___

11. Simple PatternsWhat comes next? • 1, 2, 3, 4, _5_ • 1, 2, 3, 5, __ • 1, 2, 4, 8, __ • 2, 4, 6, 8, __ • 1, 3, 2, 4, __ • 3, 5, 9, 17, __

12. Simple PatternsWhat comes next? • 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_ • 1, 2, 4, 8, __ • 2, 4, 6, 8, __ • 1, 3, 2, 4, __ • 3, 5, 9, 17, __

13. Simple PatternsWhat comes next? • 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_ • 1, 2, 4, 8, _16_ • 2, 4, 6, 8, __ • 1, 3, 2, 4, __ • 3, 5, 9, 17, __

14. Simple PatternsWhat comes next? • 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_ • 1, 2, 4, 8, _16_ • 2, 4, 6, 8, _10_ • 1, 3, 2, 4, __ • 3, 5, 9, 17, __

15. Simple PatternsWhat comes next? • 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_ • 1, 2, 4, 8, _16_ • 2, 4, 6, 8, _10_ • 1, 3, 2, 4, _3_ • 3, 5, 9, 17, __

16. Simple PatternsWhat comes next? • 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_ • 1, 2, 4, 8, _16_ • 2, 4, 6, 8, _10_ • 1, 3, 2, 4, _3_ • 3, 5, 9, 17, _33_

17. Simple PatternsWhat comes next? • 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_ • 1, 2, 4, 8, _16_ • 2, 4, 6, 8, _10_ • 1, 3, 2, 4, _3_ • 3, 5, 9, 17, _33_ But any answer can be justified! (That’s why the SAT/ACT shouldn’t ask about sequences)

18. Easiest way to generate a pattern of numbers is to use a function f (call it the pattern generating function ) and compute f(1), f(2), f(3), f(4), etc. For example: if f(x) = 2x – 1, we get: 1, 3, 5, 7, 9, … For f(x) = x2 – 3x + 2, we get 0, 0, 2, 6, 12, …

19. Now, suppose we have a sequence of numbers: A(1), A(2), A(3), A(4), … Is there a pattern generating function, I.e. a function f s.t. A(n) = f(n), for n = 1, 2, 3, …? If so, how do we find it? Since there are multiple choices, what’s the simplest answer?

20. Polynomial Interpolation • General polynomial of degree n p(x) = a0 + a1x + a2x2 + … + anxn • Finding p means finding a0, a1, a2, …, an • After picking n there are 3 approaches: • Vandermode Matrix • Nested Form/Divided Differences • MATLAB (computer lab)

21. Time for Boardwork!

22. Simple PatternsWhat’s the pattern generating function? • 1, 2, 3, 4: • 2, 4, 6, 8: • 2, 5, 8, 11: • 1, 3, 7, 13:

23. Simple PatternsWhat’s the pattern generating function? • 1, 2, 3, 4: A(n) = n • 2, 4, 6, 8: A(n) = 2n • 2, 5, 8, 11: A(n) = 3n-1 • 1, 3, 7, 13: A(n) = n2 – n + 1

24. Test Time!

25. References • Introduction to Scientific Computing by Charles Van Loan, Ch. 2 (Poly & Trig Interp), Ch. 7 (Least Squares) [Handout] • mathworld.wolfram.com/topics/Interpolation.html (Pointers to other methods) • Any book on ‘Numerical Analysis’ or ‘Numerical Methods’ should contain a chapter or 2 on interpolation.