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Unformatted text preview: Version 066 Exam03 Gilbert (56380) 1 This print-out should have 12 questions. Multiple-choice 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 two-sheeted 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 weve decided what the graphs of those plane slices should be. Now as slicing of shows, slices of this two-sheeted 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 cu.in./min. 2. rate = 60 cu.in./min. correct 3. rate = 44 cu.in./min. 4. rate = 48 cu.in./min. 5. rate = 56 cu.in./min. Explanation: The volume cylinder of a cylinder of height h and radius r is given by V ( r, h ) = r 2 h . Thus the rate of change of V with respect to t is V t = (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 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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- Spring '07
- Multivariable Calculus