The Cauchy-Schwarz Inequality u v u v isequivalent to the inequality we get by squaring bothsides: (u v)2 u2 v2.(a) In 2, with and ,this becomesProve this algebraically. [Hint: Subtract the left-handside from the right-hand side and show that thedifference must necessarily be nonnegative.](b) Prove the analogue of (a) in 3

Math 103 Week D 01/29-02/01 1/29 Apportionment: Problem of dividing congress seats to the states according to the population. (Division of identical, indivisible objects according to a plan) -5 Methods of Apportionment: Jefferson Method 1794-1842 Webster Method 1842-1852 Hamilton Method 1852-1900 Webster Method 1900-194 Hill-Huntington 1941-Current I. Hamilton Method: 1. Compute the Standard Divisor (# of people per seat) a. D =(total population) / (# of seats) 2. Standard Quota for each State (# of seats each state gets) a. Q = (state’s population) / (standard divisor) 3. Round down each Q and give that # to each State 4. Give the remaining seats one at a time to the States with the largest fractional parts. Example: Schools A-E are apportioning 109 computers based on pop.: A: 335 Q=335/20.16 = 16.14 or 16 B: 456 Q=456/20.16 = 21.96 or 21 (highest fraction=1) C: 298 Q=298/20.16 = 14.35 or 14 (highest fraction=1) D: 567 Q=567/20.16 = 27.31 or 27 E: 607 Q=607/20.16 = 29.24 or 29 Total: 2263 / 109 = 20.16 (d) (107, so 2 extra seats) In the example, school B & C get an extra seat each Why REJECT Hamilton -Census of 1790: Vermont (85,000) & Pennsylvania (430,000) were fighting over the last seat with a Q of 2.484