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221-angenent-07fafin - MATH 221 ——‘THE FINAL EXAM...

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Unformatted text preview: MATH 221 ——‘THE FINAL EXAM Your name: Zajj Daugherty Junwu Tu December 17, 2007 Your TA: (circle one) Problem -l on o A " w a V H A {U v OJ 4» H 4> A A M A 0' OJ 0' v V v I Rachel Davis Derek Garton (1) (8.) Sketch the graph of f(a:) = ln(1 + 1113), i.e. Cl find the domain of f E] find the intervals where f (x) > 0 and where f (x) < 0 -. [I find where f is increasing and where f is decreasing El find the inflection points on the graph of f. (b) Find the length of the piece of the graph of f($) = ln(1 + $3) on which 0 g a: g 1. You may leave an integral in your answer. 2 (2) A rectangle with vertical and horizontal sides has its lower left corner at the origin and its upper right corner on the curve y = (1 — a2)3. Which of such rectangles has the largest area? (3) (a) Find the volume of the solid obtained by rotating the region 91— ( ' 0351232, _ m’y)‘03ysx/1+Tc2 around the x-axis. (b) Find the volume of the solid you get when you rotate the same region IR around the y axis. (4) A 12 foot pole is attached to the floor at the point 0. As it falls over, it makes an angle 0(t) with the vertical line OB. A . B At the moment that 6 =—- 7r/3 the distance AB (length of the line segment AB) is increasing at 2 feet per second. How fast is the angle 6(t) increasing at that moment? Show th at {I} : hm / propefties to V L $22 {1:2 '— 4 (5) Use the “m“ do es not exist ' (6) (a) Let f(£13) : x + 3:2. Find a constant K such that {:13 — 1| < 5 => |f(:1:)— f(1)1 < K6. You may assume that 5 5 1. (b) Assuming you have done the previous problem show' that the function f is continuous at a: = 1. ...
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