Final Exam
Professor Ilan Vardi
``The unexamined life is not worth living.''
Socrates, in Plato's Apology , 38A.
 Does the earth go around the sun or
viceversa? Prove your answer.
 Is matter made out of atoms or is it
continuous? Prove your answer.
 Give evidence to support the theory
of evolution.

Is the earth round? Prove your answer.
When was this first known?
 Who first proposed
the heliocentric theory? Who was the first to prove it?

(a) How long does it take for the earth to rotate once around its
axis?
(b) How long does it take for the earth to rotate once around the sun
(be as precise as possible)?
 (a) Show that the circumference of a
circle has length 2 pi r, where r is the radius of the
circle. (b) Give a proof
that \pi is greater than 3 but
less than 4 (this computes one digit of pi).

Prove the following theorems of Archimedes:
1^2 + 2^2 + 3^2 + ... + n^2 = n(n+1)(2n+1)/6,
sin pi/(2n) + sin 2 pi/(2n)
+... + sin (2n1) pi/(2n) = [cos pi/(4n)]/[sin pi/(4n)]

Devise observational
methods to (a) Compute the length of the year. (b) Compute
the dates for the astrological signs of the zodiac. (c) Order
according to size: earth, sun, moon.
 What's your sign?
Once you've completed the final, check out the
goals of the book.
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