MthEd/Math 300 Williams Fall 2011 Midterm Exam 3

Transcription

1 Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 3 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first column, circle the time period in which it occurred. If more than one century might apply, is in the case of Gauss, for example, I will accept either answer or both. One point each. 1. Hilbert delivered his famous Paris Problems speech. 1800s 1900s 2. L Hospital s calculus textbook appears. 1700s 3. Bishop George Berkeley questions the ghosts of departed quantities. 1700s 4. Gödel comes to the Institute for Advanced Study at Princeton 1900s 5. Cauchy and Weierstrass provide the formal definitions of limit that put calculus on a firmer logical foundation. 6. Two mathematicians who both died young publish work on the unsolvability of the quintic. 1800s 1800s 7. Paul Cohen shows the Continuum Hypothesis to be unproveable. 1900s 8. Despite urging from his father, János Bolyai continues to work on Euclid s 5 th Postulate. 1800s 9. Euler provides proof of many of Fermat s unproved assertions. 1700s 10. Cantor unsuccessfully attempts to prove the Continuum Hypothesis 1800s 11. Gauss gave a proof of the Fundamental Theorem of Algebra. 1700s 1800s 12. The techniques of calculus are developed to solve differential equations and to include multi-valued functions. 1700s 13. Eugenio Beltrami provided a model of hyperbolic geometry. 1800s 14. No one would pay attention to Paolo Ruffini s work on the quintic 1700s 1800s 15. Lambert worked on Euclid s 5 th Postulate. 1700s

2 Section 2: For each event, idea, or proof listed in the first column, fill in the name or names associated it. If more than one name might apply, fill in all of them. There are never more than three associated names. One point each. 16. First published correct and complete proof of the unsolvability of the quintic by radicals 17. Before Gauss, Bolyai, and Lobachevsky, studied quadrilaterals that had properties of rectangles but couldn t be proved to be rectangles. Abel Lambert, Saccheri 18. Posed the Brachistochrone problem. Johann Bernoulli 19. Gave three different proofs of the Fundamental Theorem of Algebra over his lifetime. 20. Provided a list of 23 problems that he felt needed to be solved in mathematics. 21. Proof that the power set of a set A is always larger than A. Gauss Hilbert Cantor 22. L Hospital s calculus book. L Hospital, Johann Bernoulli 23. Provided the mathematical machinery necessary to tell which polynomials were solvable by radicals. Galois 24. The most prolific mathematician ever. Euler 25. Proved the Incompleteness Theorem. Gödel There s too much space here, but not enough for another question. You may draw a picture of a duck if you wish.

3 Section 3: Short Answers. Five points each. 26. What is the catenary problem, who solved it, and what is its solution? A. What is the function that gives the shape of a cable hung between two fixed points? B. Johann Benoulli, Leibniz, Huygens y cosh( x) C What does it mean to say a polynomial equation is solvable by radicals? That there is a formula, containing only arithmetic operations along with powers and roots, and involving coefficients of the polynomial, that will give a solution to the equation.

4 28. What things did Cauchy do to rigorize calculus? (There were two main things.) First, he defined the operations of calculus, such as the derivative, in terms of a limit. So he based calculus ideas on limits. Second, he defined limits in terms of algebra and arithmetic, that is, absolute value, differences, and inequalities. 29. Why was the development of hyperbolic geometry of importance to the philosophy of mathematics? Prior to hyperbolic geometry s appearance, people believed that mathematics, and especially Euclid s geometry, was about describing reality, i.e, the physical world. When hyperbolic geometry appeared, it seemed that mathematics could arrive at conclusions that weren t necessarily about what was true in the physical world. It thus introduced the ideas that mathematics wasn t necessarily tied to reality or truth.

5 30. Which of the following long-standing problems in mathematics have been solved in the past 50 years? Circle the correct answers: Fermat s Last Theorem The Four Color Conjecture The Poincaré Conjecture Dang. More extra space, again. OK, if you didn t like the duck, in the space below you can draw Trogdor or a unicorn or something else of your choosing.

6 Section 4: Longer Questions. Ten points each. 31. This question deals with Karl Friedrich Gauss. a. Why did he never publish his discoveries in non-euclidean geometry? b. Briefly describe two of his mathematical accomplishments beyond his discoveries in non- Euclidean geometry. A. At the time, the followers of the influential philosopher Immanuel Kant believed that our knowledge of Euclidean geometry was inborn it was the only way we could understand geometry and the world. To suggest that there was another geometry would have entangled Gauss in debates with these followers of Kant. He didn t want to hear the howls of the Boatians. (Also, possibly, he didn t feel that his work was ready, that is, polished an completed enough, to publish.) B. The construction of the regular 17-gon The Fundamental Theorem of Calculus Work on congruences, including quadratic reciprocity The Prime Number theorem Linear interpolation The Gaussian distribution Others?

7 32. This question deals with Cantor s approach to infinite sets. a. In general, how did Cantor define what it meant to say that two sets were of the same size? Two sets are of the same size if there exists a one-to-one correspondence from one to the other, (that is, a one-to-one, onto function.) b. Name or describe three sets of numbers that have the same (infinite) size as the integers. even integers positive integers negative integers odd integers multiples of n for any integer n. rational numbers algebraic numbers surds c. Name two sets that have a larger (infinite) size than the integers. Real numbers positive real numbers negative real numbers ordered pairs of real numbers x a x b irrational numbers transcendental numbers complex numbers for any a, b. d. What is the Continuum Hypothesis? The statement that there is no infinite set of size between that of the integers and that of the real numbers.

8 33. Name some of Euler s mathematical accomplishments. proved many of Fermat s unproved conjectures (and proved a few of those conjectures were wrong). Königsburg Bridge problem modern notation for sines, cosines, others. proved n 2 6 ix i e cos x isin x e 1 Established that and Euler constant γ Developed the gamma function Was the most prolific mathematician ever Others?

9 34. Explain a little bit about the problems Berkeley and others saw in the way calculus was originally done. In other words, why did it need to be rigorized? Basically, everyone who took a derivative (found the slope of a tangent line, found a maximum or minimum point, etc.) did so by choosing a point and a second point a small distance away from it. Then, then found the difference of the y values of these points, and divided it by the small difference in x-values between the points. In order to do this division, they had to assume that this horizontal difference is non-zero. Then, as a last step, they assumed that this distance was either zero or negligible. This contradiction between seeing the difference as zero in order to do the division, and as nonzero or negligible in the last step, is what needed to be resolved, and of course, the answer was in the limit concept.

10 35. We ve given brief biographies and vignettes from the personal lives of several mathematicians in the modern era. Pick one mathematician and provide some details of his/her life. Answers vary.