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Unformatted text preview: Version 066 – Exam03 – Gilbert – (56380) 1 This printout should have 12 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points For which of the following quadratic rela tions is its graph a twosheeted hyperboloid? 1. z = y 2 − x 2 2. x 2 + y 2 − z 2 = 1 3. z 2 = x 2 + y 2 4. z 2 − x 2 − y 2 = 1 correct 5. 2 x 2 + y 2 + 3 z 2 = 1 6. z = x 2 + y 2 Explanation: The graphs of each of the given quadratic relations is a quadric surface in standard posi tion. We have to decide which quadric surface goes with which equation  a good way of do ing that is by taking plane slices parallel to the coordinate planes, i.e. , by setting respec tively x = a , y = a and z = a in the equations once we’ve decided what the graphs of those plane slices should be. Now as slicing of shows, slices of this twosheeted hyperboloid by z = a are circles for  a  sufficiently large, while the slices by x = 0 and y = 0 are hyperbolas opening up and down. Only the graph of z 2 − x 2 − y 2 = 1 has these properties. keywords: Surfaces, SurfacesExam, 002 10.0 points The radius of a right circular cylinder is decreasing at a rate of 2 inches per minute while the height is increasing at a rate of 4 inches per minute. Determine the rate of change of the volume when r = 3 and h = 8. 1. rate = − 52 π cu.in./min. 2. rate = − 60 π cu.in./min. correct 3. rate = − 44 π cu.in./min. 4. rate = − 48 π cu.in./min. 5. rate = − 56 π cu.in./min. Explanation: The volume cylinder of a cylinder of height h and radius r is given by V ( r, h ) = πr 2 h . Thus the rate of change of V with respect to t is ∂V ∂t = π (2 rh ∂r ∂t + r 2 ∂h ∂t ) But ∂r ∂t = − 2 , ∂h ∂t = 4 . Consequently, ∂V ∂t vextendsingle vextendsingle vextendsingle ( r =3 , h =8) = − 60 π cu.in./min. . 003 10.0 points Determine f y when Version 066 – Exam03 – Gilbert – (56380) 2 f ( x , y ) = 2 x − y x + 2 y ....
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This note was uploaded on 04/12/2011 for the course M 408d taught by Professor Sadler during the Spring '07 term at University of Texas.
 Spring '07
 Sadler
 Multivariable Calculus

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