Governors School for the Sciences

1
Governors 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?
But any answer can be justified! (Thats why the
SAT/ACT shouldnt ask about sequences)

• 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_

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,
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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, whats 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 PatternsWhats the pattern generating
function?

• 1, 2, 3, 4
• 2, 4, 6, 8
• 2, 5, 8, 11
• 1, 3, 7, 13

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Simple PatternsWhats 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!
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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.