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Math.111HW5(2008-2009,Fall)

Math.111HW5(2008-2009,Fall) - HOMEWORK V Due on January...

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HOMEWORK V Due on January 5,2009 for sections 01,04,05 Due on January 6,2009 for sections 03,06 Due on January 7,2009 for sections 02,07 Bring the homework to the first lecture on the due date. 1) Suppose is twice differentiable on an interval (i.e., f ( f I x) ′′ exists on ).Suppose that the points 0 and 2 belong to and that f ( I I 0) f (1) 0 = = and f (2) 1 = .Prove that: a) 1 1 f (c ) 2 = for some point in . 1 c I b) 2 1 f (c ) 2 ′′ > for some point in . 2 c I c) 3 1 f (c ) 111 = for some point in . 3 c I 2) Do the fcns. a) 1 f (x) sin x = on and b) (0, ), 2 2 x g(x) x e = have absolute maximim or minumum values?Justify your answer. 3) If and for 1 x ,how small can f ( possibly be? f (1) 10 = f (x) 2 4 4) 4) Find the length of the longest beam that can be carried horizontally around the corners from hallway of width a m to hallway of width b m.(See the figure,and assume that the beam has no width.) a m b m 5) Sketch the graph of a function which has the following properties: x x f (0) 1,f ( 1) 0,f (2) 1,lim f (x) 2, lim f (x) 1,f (x) 0 →∞ →−∞ = ± = = = = − > on ( ) ,0 −∞
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