Sample Exam 3 Solution on Calculus III

# Z f x x z f z x z f z f x if we use subscript

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z 0 F x x z F z x z F / z F / x . If we use subscript notation for the partial derivatives, we set g(y,z) = F(h(y,z), y, z). Then 0 = g z (y,z) = F x (h(y,z), y, z)h z (y, z) + F z (h(y,z), y, z), which implies that x/ z = h z (y,z) = - F z (h(y,z), y, z)/F x (h(y,z), y, z).

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TEST3/MAC2313 Page 5 of 5 _________________________________________________________________ 9. (10 pts.) It turns out that h(y) = sin(y)/y does not have an elementary antiderivative. Despite that you can evaluate the following iterated integral. Do this by reversing the order of integration and then evaluating the new iterated integral you obtain. [Hint: It helps to sketch the region of integration, R.] π 0 π x sin( y ) y dydx π 0 y 0 sin( y ) y dxdy π 0 sin( y ) y y 0 1 dx dy π 0 y sin( y ) y dy π 0 sin( y ) dy cos( π ) ( cos(0)) 2. Note: Although the integral appears to be improper, it really isn’t. Why?? _________________________________________________________________ 10. (10 pts.) (a) Let R be the region bounded by the curves defined by y = x 2 and y = x + 2. Write an iterated double integral that gives the area of the region, but do not attempt to evaluate the iterated integral. area ( R ) R 1 dA 2 1 x 2 x 2 1 dydx (b) Set up, but do not attempt to evaluate the iterated double integral that will give the numerical value for the volume of the solid bounded by the cylinders in 3-space defined by x 2 + y 2 = 1 and y 2 + z 2 = 1. [Hint: You should only need to sketch the xy-cylindrical stuff to obtain the limits of integration.] // Here the top surface is z = (1 - y 2 ) 1/2 and the bottom surface is z = -(1 - y 2 ) 1/2 . V R (1 y 2 ) 1/2 ( (1 y 2 ) 1/2 ) dA 1 1 (1 x 2 ) 1/2 (1 x 2 ) 1/2 2(1 y 2 ) 1/2 dydx 1 1 (1 y 2 ) 1/2 (1 y 2 ) 1/2 2(1 y 2 ) 1/2 dxdy _________________________________________________________________ Silly 10 Point Bonus: Suppose f(x,y) is differentiable at an interior point (x 0 ,y 0 ) in its domain. Pretend there are at least three distinct unit vectors u satisfying the following equation: D u f(x 0 ,y 0 ) = 0. Does it follow as a consequence that this equation must be true for all unit vectors? Proof?? Where???
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