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1. 2. 3. Ab 4. 5. 6. 7. 8. 9 History of Mathematics MATH 629—100 Homework -Helenistic Period Texas A&M University 1. In what sense can one consider Archimedes to be the inventor of calculus? Be sure to clarify what you mean by the term “inventor”. Use the lecture notes and the text as your sources. 2. Prove the Archimedes result that the area of a section of a parabola equals 4 times the area of the inscribed 3 triangle of the same height. (You may use calculus here.) n 3. With respect to a circle of radius r,Further,letb1, b2,…,bn denote the regular inscribed 6 ⋅ 1…6 ⋅ 2 … polygons, similarly, B1, B2 Bn for the circumscribed polygons. Prove the following formulae give the n relations between the perimeters (pn and Pn and areas (an and An of these 6 ⋅ 2 polygons. 2pnPn a. Pn+1 = pn+1 = pnPn+1 pn + Pn 2an+1An b. an+1 = anAn An+1 = an+1 + An 4. Of the many innovations in Euclid, which one or two do you find are the most innovative? π α sinβ 5. Prove Aristarchus’ theorem that if 0 < α < β < , then sin > . (You may use any 2 α β mathematical techniques you know. Calculus helps. Can you prove it using only geometrical methods?) π α α 6. A result known to and used by Aristarchus is that if 0 < α < , then sin decreases and tan 2 α α increases. Show these propositions. 7. Using induction, prove that 1 = 1 3 + 5 = 23 7 + 9 + 11 = 33 13 + 15 + 17 + 19 = 43 and so on. 8. Using the parabola, show how to find 3 2 . (Hint. You will need two parabolas with orthogonal axes.) Can you generalize to find the cube root of any number? 9. Prove a generalization of the Pythagorean theorem ála Pappus using triangles instead of paralellograms. [You need to get the correct statement first. This will come by mimicing the Pappus proof for triangles.] 10. Prove that the Pappus generalization of the Pythagorean Theorem yields the Pythagorean Theorem in the case of a right triangle △ABC. That is the “parallelogram” constructed in the Pappus proof is indeed a square upon the hypotenuse. (Hint. Here’s an approach. Show KM ⊥ BC BD ⊥ BC and then |BD| = |BC|.To do this, discover an importance congruence. ) K D E A C B M 11. Give a proof of the Zenodorus theorem that if a circle and regular polygon have the same perimeter, the circle has the greater area. (Hint. One part of the above problem helps.) 12. Find two numbers in mean proportion between a and b. 13. Find two numbers in mean proportion between 5 and 9. 14. A sequence an is in arithmetic progression if an+1 − an = d, is constant for all n.Ifd > 0, the sequence is increasing, and if d < 0, the sequence is decreasing. Show than if a given arithmetic sequence is decreasing then 2 a1 + a2 + ⋯ + an − an+1 + an+2 + ⋯ + a2n = −n d (Hypsicles, 2nd century BCE) 15. Apply the Heron method to approximate the cube root of 45. How accurate is the approximation? 16. Apply the Heron method to approximate the cube root of 1450. How accurate is the approximation? 17. Show that the solution to any cubic equation can be determined by intersecting a hyperbola with a parabola. 18. Prove that any two triangles inscribed in a circle having the same base have the same vertex. (This is Proposition III -21 of Euclid. See the diagram below.) Vertices of the inscribed triangles D C B A Show that ACB = BDA 19. From the previous problem prove the conclusion that the maximum angle subtended from a point on a line to a segment of a line must occur at the point where a circle passing through the endpoints of the segment is tangent to the line. Line segment Subtended angle Φ Point Optimal point Line.
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