Governor’s School for the Sciences

MathematicsDay 1

Schedule

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

Course Structure

• Instructors• Lecture, Work and Labs• Point System• Project• AfterClass

Where are you from?

Give us the Good, the Bad or the Ugly!

Tnmap.gif

MOTD: Abu al-Khwarizmi

• 780-850 AD• Hindu-Arabic

numerals• First to use 0• “Algorithm”

derived from his name

Why did Homeland Security arrest the mathematician?

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.

All About Patterns

Patterns in number sequencesPatterns in letters/wordsPatterns in geometry

Mathematics is about finding and studying patterns.

Pattern “Black Box”

Nature gives us a pattern we need to create a matching box.

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, ___

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, __

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, __

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, __

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, __

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, __

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_

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)

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, …

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?

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:1. Vandermode Matrix2. Nested Form/Divided Differences3. MATLAB (computer lab)

Time for Boardwork!

Simple PatternsWhat’s the pattern generating function?

• 1, 2, 3, 4:

• 2, 4, 6, 8:

• 2, 5, 8, 11:

• 1, 3, 7, 13:

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

Test Time!

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.