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140Bquiz9csols - Implicit di±erentiation gives x 2 y ′ 2...

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1 MATH 140B QUIZ#9 Solutions Due: Fri Nov 2, 2007 1. (This is #5 on Version B) i) True . x 2 y = V = 20 . ii) False . There is no top, so S = x 2 + 4 xy. iii) True . Rearranging the equation in part i) for y gives y = 20 /x 2 , and substituting this into S = x 2 + 4 xy, gives S ( x ) = x 2 + 4 x (20 /x 2 ) = x 2 + 80 /x. Therefore, C) is the answer. 2. (This is #1 on Version B) Implicit differentiation gives, - sin( xy ) · [ xy + 1 · y ] + 3 y 2 · y = - sin( x - π/ 2) . Evalutating this at P ( π, 1) NOW gives, 0 + 3 y = - 1 , so y = - 1 / 3 . Therefore, A) is the answer. 3. (This is #3 on Version B) i) True . ii) False . Missing dx dt at the end. Therefore, A) is the answer. 4. (This is #4 on Version B)
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Unformatted text preview: Implicit di±erentiation gives, x 2 y ′ +2 xy +2 y ′ = 3 x 2 . At P (1 , 0) this becomes, y ′ +0+2 y ′ = 3 , or 3 y ′ = 3 and hence y ′ = 1 . Therefore the tangent line in slope-point form is ( y-0) = 1( x-1) so y = x-1 . Therefore, E) is the answer. 5. (This is #2 on Version B) Di±erentiating with respect to t gives, dV dt = 4 πr 2 dr dt . We are given that dr dt = 1 . When r = 10 , this gives dV dt = 4 π (100)(1) = 400 π. Therefore, D) is the answer....
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