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MIT18_014F10_pset3sols

MIT18_014F10_pset3sols - 18.014 Problem Set 3 Solutions...

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18.014 Problem Set 3 Solutions Total: 12 points Problem 1: Find all values of c for which c (a) x (1 x ) dx = 0. 0 c (b) 0 | x (1 x ) | dx = 0. Solution (4 points) (a) Computing, we get c c 1 1 x (1 x ) = ( x x 2 ) = 2 c 2 3 c 3 . 0 0 Setting the right hand side equal to zero, we get solutions 3 c = and c = 0 . 2 (b) Observe x (1 x ) 0 if 0 x 1 and x (1 x ) 0 if x 1 or x 0. There are three cases. If c < 0, then c c 1 1 0 | x (1 x ) | = 0 x (1 x ) = 2 c 2 + 3 c 3 which is never zero for c < 0. If 0 c 1, then c c 1 2 1 3 | x (1 x ) | = x (1 x ) = 2 c 3 c 0 0 which is zero only if c = 0 in the range 0 c 1. Finally, if c > 1, then c 1 c | x (1 x ) | = x (1 x ) + x (1 x ) . 0 0 1 The first integral is 6 1 , and the second integral is non-negative by the comparison theorem (Thm. 1.20) since x (1 x ) 0 on the interval (1 , c ). Hence, this integral is always positive and never zero. In summary, our integral is only zero when c = 0. 1

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2 Problem 2: Compute the area of the region S between the graphs of f ( x ) = x ( x 1) and g ( x ) = x over the interval [ 1 , 2]. Then make a sketch of the two graphs and indicate S by shading.
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