Review guide for Final in Mathematics 436 - Spring 2006

1. A Renaissance artist is drawing a perspective picture of a tile floor. In the preliminary drawing one tile has been drawn along with a horizontal line representing the horizon. The plane of the painting is parallel to a side of the tile. Draw in perspective the 8 tiles surrounding this tile, explaining how you use the principles of perspective geometry (note: all you need is a straight-edge. No measurement is necessary).

2. TERMS TO KNOW. Know the definition of the term and its significance. Give an example of its use. What period of mathematics is the term associated with?

Rosetta stone

fluxion

differential

Euclid's 5 postulates

anthyphairesis

arithmos

Diophantine equation

commensurable

Euclid

Euclidean algorithm and its applications

congruence

Fibonacci sequence

Pell's equation

Differential equation

Golden Section

irrational number

Mesopotamia

Method of exhaustion

Old Babylonian

prime number

power series

Pythagoras

regular sexigesimal number

Thales

Theatesus

Inverse Square law

Laws of motion

Kepler's Three Laws

Binomial Series

Pascal's triangle

Squaring the Lune

Brachistochrone

3. CAST OF CHARACTERS - Be able to identify the approximate time period and the relation to the history of mathematics, and the major accomplishments of:

Ahmmose; Archimedes; Aristotle; Diophantus; Euclid; Eudoxos; Hypatia; Pappus; Plimpton; Ptolemy, Pythagoras; Qin; Thales; Theatesus; Newton; Leibniz; Descartes; Fermat; Euler; Lagrange; Brahmagupta; Liu Hui; Al-Khwarizmi; Fibonacci; Cardano; Kepler; Lagrange

4. POLYNOMIALS - Be able to discuss the contributions of the following to the solution of polynomial equations: Babylonians, Chinese and Arabic mathematicians, Tartaglia, Cardano, Ferrari, Bombelli, Viete

5. You should be familiar with well known works in the history of mathematics: Euclid's elements, Archimedes and the method of exhaustion, Diophantus's Arithmetika, Fibonacci's Liber Abaci Cardano's Ars Magna, Descartes's Geometry (geometric construction of sums, products, quotients, and square roots), Newton's Principia. What are these books famous for? When were they published?

6. How did Fermat compute the area under a parabola from the origin to a point x_0? (State the type of summation formula that he needed to know in order to make it work.) How did he modify his method to compute the area under a "higher hyperbola" from x_0 to infinity (which we would call an improper integral)?

4. SAMPLE PROBLEMS FOR THE FINAL EXAMINATION

(Refer also to the guide for the midterm exam)

1. Show that the cubic x^3+12x=12 has a unique real root and find it using Caradano's formulas to express it in radicals.

2. Spherical triangles are formed by three arcs of great circles on the unit sphere. Denote the angles of the spherical triangle by A,B,C and the lengths of the sides opposite the angles by a,b,c. The laws relating these are:

law of sines : sin A/ sin a=sin B/ sin b=sin C/ sin c

law of cosines for sides: cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C)

law of cosines for angles: cos(C) =-cos(A) cos(B) + sin(A) sin(B) cos(c)

Show that if two spherical triangles have the same angles, then they have the same side lengths, so any similar spherical triangles are congruent.

A spherical triangle on the unit sphere has angles Pi/4, Pi/3, Pi/2. What are the lengths of the sides of the triangle?

3. Leibniz's "transmutation theorem" is equivalent to what special case of what theorem found in calculus books today?

4. Find solutions to y^2-13x^2=1 in nonzero integers x,y.

5. Historian Judith Grabiner summarizes the history of the derivative as: "first used, then discovered, then developed and applied, and finally defined." Discuss this statement.

6. Hans Freudenthal once wrote a paper called "Should a teacher of mathematics know something about the history of mathematics?" How would you answer Freudenthal's question?

7. Match the terms in the first group with the related terms in the second group

Group I

fluxion

differential

derivative

Brachistochrone

cakravala

Berkeley

chord

Group II

Lagrange

Calculus Criticism

Ptolemy

Pell's equation

Newton

cycloid

Leibniz

8. Find the power series y(x)=a0+a1 x+ ... which satisfies the differential equation y'=2y+x, y(0)=0.

9. Derive Leibniz's formula for pi/4.

10. Trace the development of our current system of numbers (digits 0-9, method of representing numbers) from Egyptian and Babylonian times through India and Arabic mathematics. Who popularized the so called Arabic numerals?