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m233_E3sSP10 - Math 233 Exam 3 Page 1 Name ID This exam has...

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Page 1 Math 233 Exam 3 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 × 5 inch note card. 1. Find C F , where F ( x, y ) = y, x 2 and C goes along the parabola y = x 2 from (0 , 0) to (1 , 1). (a) - 1 (b) - 1 / 2 (c) - 1 / 3 (d) 0 (e) 1 / 4 (f) 1 / 2 (g) 2 / 3 (h) 3 / 4 (i) 5 / 6 (j) 1 (I) The vector field is not conservative ( ∂q ∂x = 2 x , whereas ∂p ∂y = 1 ), so the best approach in this case is to directly compute the curve integral. The curve can be parametrized as C ( t ) = ( t, t 2 ) as t goes from 0 to 1 , and the integral is 1 0 F ( C ( t )) · C ( t ) d t = 1 0 t 2 , t 2 · 1 , 2 t d t = 1 0 t 2 + 2 t 3 d t = t 3 3 + t 4 2 1 0 = 1 3 + 1 2 = 5 6 .
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Page 2 Math 233 Exam 3 2. Find C F , where F ( x, y ) = ye xy , xe xy and C goes along the parabola y = x 2 from (0 , 0) to (1 , 1). (a) 0 (b) ln 2 (c) 1 (d) e - 1 2 (e) 2 3 ( e - 1) (f) e - 1 (g) e 2 (h) e 2 2 (i) e 2 (j) e 3 (F) The vector field is conservative (both ∂q ∂x and ∂p ∂y equal e xy + xye xy ), so the FTC can be used. First integrate the vector field to get a potential function: f ( x, y ) = ye xy d x = e xy + g ( y ) and f ( x, y ) = xe xy d y = e xy + h ( x ) . The above results agree when g ( y ) = 0 and h ( x ) = 0 , making the potential function f ( x, y ) = e xy . [Note: at this point, you can double-check that f = F .] By the FTC, the integral is simply f (1 , 1) - f (0 , 0) = e - 1 .
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Page 3 Math 233 Exam 3 3. Find C G , where G ( x, y ) = - y x 2 + y 2 , x x 2 + y 2 and C is the path in the picture at the top of this page. (I) G is the very special vector field, which integrates to 2 π for any closed curve going around the origin once counterclockwise. 4. Find C F , where F ( x, y ) = x x 2 + y 2 , y x 2 + y 2 and C is the path in the picture at the top of this page. (i) 2 π (E) This vector field (which can also be written as 1 r 2 X ) is conservative. Since F is conservative and C is a closed curve, the integral is 0 .
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Page 4 Math 233 Exam 3
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