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Unformatted text preview: Math 1a, Section 1 Midterm Examination Solutions 1. (10 pts) Let b denote a fixed positive integer. Prove the following statement by induction: For every integer n ≥ 0, there exist nonnegative integers q and r such that, 0 ≤ r < b and n = qb + r. Proof For n < b , we let q = 0 and r = n . For n = b , we let q = 1 and r = 0. Suppose for some integer N ≥ b , there exists integer q N , r N such that ≤ r N < b (1) N = q N b + r N (2) Consider N +1. If r N = b 1, then let q N +1 = q N +1 and r N +1 = 0. If ≤ r N < b 1, we let q N +1 = q N and r N +1 = r N +1. Since r N < b 1, r N +1 < b . 2. (10 pts) The functions f ( x ) = 2 x 3 5 x 2 +2 x and g ( x ) = 2 x 2 3 x meet at 3 points. Find these points, and sketch the graphs of these functions on the same plot. The graphs of f ( x ) and g ( x ) bound two finite regions in the plane. Find the area of each of these regions. Proof Solve for f ( x ) g ( x ) = 2 x 3 7 x 2 + 5 x = x (2 x 5)( x 1)....
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This note was uploaded on 12/15/2009 for the course MA 1a taught by Professor Borodin,a during the Spring '08 term at Caltech.
 Spring '08
 Borodin,A
 Calculus, Linear Algebra, Algebra, Integers

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