Mock Exam 1, Math. 406
Explain your answers carefully. Write complete sentences, not just formulas.
1. (10 points) Give the older Greek definition of equality (=“proportionality”) of ratios: a : b = c : d, where a, b, c, d are integral multiple of a unit length.
2
√
2. (10 points) Show that 2 is not a rational number p/q. Can you measure lengths
in geometry by using (ratios of) integral multiples of a unit length?
3. (10 points) Divide the diameter of circle into two parts of length a and b respectively,
and call S the separating points of the two segments. Draw a line perpendicular to
S and let P be the point of intersection of this line with the circle. Set x to be the
length of the segment SP . show geometrically that
x2 = ab
3
4. (10 points) What is Eudoxus’ principle? Give a Calculus explanation of it.
4
5. (10 points) What is the Archimedean Axiom (or the Archimedean-Eudoxusian Axiom) for two lengths `1 , `2 ?
6. (10 points) Let Pn be the polygons inscribed in the parabolic segment AP B in
Archimedes’ construction. Let A be the area of the triangle AP B. Knowing that:
1
1
a(Pn ) = A 1 + + · · · n
4
4
use an argument of double reductio ad absurdum to show that the area of the
parabolic sector AP B equals:
4
A
3
.
5
7. (10 points) Show that the area A of a frustrum of a cone whose base has radius r1
and height has radius r2 , equals:
A = π (r1 + r2 ) s
where s is the slant height (the lenght of the side).
6
8. (10 points) Let Pn be a regular polygon, with n sides, inscribed in a circle of radius
r. Let a(Pn ) be its area and p(Pn ) its perimeter. Show that:
a(Pn )
r
π
= cos( )
p(Pn )
2
2
7
9. (10 pts) Give the definition of convexity for surfaces and curves. Why is convexity
useful in calculating the surface area of a ”convex” object?
8
10. (10 points) Given a sphere of radius r, consider two similar regular polygons of
n sides, respectively inscribed in and circumscribed about a great circle: Pn and
Qn . Let Σ0n and Σ00n be the surfaces of the objects obtained by revolving Pn and
Qn around the diameter of the sphere. Show that:
a(Σ00n )
π
= sec2 ( )
a(Σ0n )
n