FA06Ma.9MTsolutions

FA06Ma.9MTsolutions - Math 1a Section 1 Midterm Examination...

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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 0 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 0 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). Points of intersection: x = 0 , 1 , 5 / 2. Area the two bounded regions are 5 / 2 0 | f ( x ) - g ( x ) | dx = 1 0 - 5 / 2 1 2 x 3 - 7 x 2 + 5 xdx = 2 / 3 + 63 / 32

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3. (10 pts) Find the following limits (you may assume that lim
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