Final Exam

Professor Ilan Vardi

``The unexamined life is not worth living.'' Socrates, in Plato's Apology , 38A.

1. Does the earth go around the sun or vice--versa? Prove your answer.
2. Is matter made out of atoms or is it continuous? Prove your answer.
3. Give evidence to support the theory of evolution.
4. Is the earth round? Prove your answer. When was this first known?
5. Who first proposed the heliocentric theory? Who was the first to prove it?
6. (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)?
7. (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).
8. 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 (2n-1) pi/(2n) = [cos pi/(4n)]/[sin pi/(4n)]

9. 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.
10. What's your sign?

Once you've completed the final, check out the goals of the book.

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