MthEd/Math 300 Williams Winter 2012 Review for Midterm Exam 2 PART 1

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1 MthEd/Math 300 Williams Winter 2012 Review for Midterm Exam 2 PART 1 1. In terms of the machine-scored sections of the test, you ll basically need to coordinate mathematical developments or events, people, and time periods/cultures. The lists below should be fairly comprehensive, but are not guaranteed to be so. People Simon Stevin Newton Leonardo of Pisa Regiomontanus Brahe Kepler Viète. Fermat Cardano Tartaglia Descartes Bombelli Leibniz Isaac Barrow John Napier Luca Pacioli Copernicus Omar Khayyam Al-Khwarizmi Qin Jiushao (p. 183) Brahmagupta (p ) Bhaskara (p ) Ferrari Bernoulli Brothers L Hospital Ancient and Medieval China Ancient and Medieval India Islamic Empire Events Use of Horner s Method for solving polynomial equations. Correction of Brahmagupta s work, including rules for arithmetic with 0. Publication of two influential books, one on arithmetic with Hindu/Arabic numerals, and one on algebra. Developed ideas of the calculus prior to the work of Newton and Leibniz. Introduced decimal fractions to Europe. Published On Triangles of Every Kind, the first introduction to trigonometry that treated it like a mathematical subject in its own right. Invented / developed analytic geometry. Invented / developed logarithms. Solution of cubic equation by radicals. Solution of quartic equation by radicals. Systematic study of cubic equations and their solutions by use of conics. Invented calculus. Publication of a book on algebra that went beyond a focus on solution to an analysis of equations. First solution of systems of linear equations by reduction. First solution of systems of two or more linear congruences. First solution of systems of two linear congruences. Development of calculus of variations. Solution of catenary and brachistochrone problems. First treatment of arithmetic with complex numbers. Claim that orbits of planets were elliptical. Argument for a heliocentric model of universe. Use of galley division and lattice multiplication Use of bamboo rods for computation. First calculus textbooks appear. First written appearance of zero as a number. Cultures / Time Periods European Medieval period European Renaissance / Early Modern period 17 th Century (and beginning of 18th)

2 2. Know the authors (if appropriate) and significance of the following manuscripts / books: Ars Magna (The Great Art) On Triangles of Every Kind The Analytic Art Geometrie Liber Abaci Sulbasutras Hisob al-jabr as l muqabalah A Description of the Marvelous Rule of Logarithms Liber Algorizmi The Nine Chapters 3. We know that several people were solving problems involving tangent lines, normal lines, areas under curves, arc length, and so forth, prior to the time of Newton and Leibniz. So why do they (Newton and Leibniz) get credit for inventing calculus? 4. From the time of ancient Babylon, people have been solving algebraic equations. What did Viete s book The Analytic Art contribute to algebra that was new and revolutionary? 5. Summarize two important mathematical contributions of Al Kwarizmi. 6. What were some of the important contributions of the Islamic Empire to mathematics? 7. We know that the Pythagorean Theorem and Pascal s triangle both showed up in the mathematical work of several cultures. Briefly discuss two other mathematical ideas or problems that were also significant in more than one culture. 8. What role did the Nine Chapters serve in Chinese mathematics? 9. Describe two of the mathematical accomplishments of Omar Khayyam. 10. How did the analytic geometry of Descartes and Fermat differ from modern analytic geometry? 11. Describe some of the mathematics found in the Sulba Sutras.

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MthEd/Math 300 Williams Fall 2011 Midterm Exam 2

Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 2 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first