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practice_exam1_solutions-1 - Math 223 Vector Calculus...

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Math 223 – Vector Calculus Practice Exam 1 Author: Arlo Caine Name: Solutions Directions: Read all questions carefully. Use a pencil and erase all unnecessary marks. Show all of your work in the space provided and display your answer on the line given if requested. You will lose points if you make an approximation and fail to indicate the approximation. Be careful to use proper notation to indicate vector versus scalar quantities as well. 1. Determine whether the following are true or false (T or F). You do not need to give your reasons. (a) T Two contours of f ( x, y ) with di ff erent heights never intersect. (b) T There is only one point in the yz -plane that is a distance 3 from the point (3 , 0 , 0). (c) T The graphs of f ( x, y ) = sin( xy ) and f ( x, y ) = sin( xy ) + 2 do not intersect. (d) T There is no linear function whose graph is a plane parallel to the xz -plane. (e) T The vectors 3 i - 2 j - 4 k and - 39 i + 26 j + 52 k are parallel. (f) T The quantity (( b × a ) · c ) a is a vector. (g) F If v and w are any two vectors then v + w = v + w . 2. Match the graph with the contour diagram.
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3. You are in a room 30 feet long with a heater at one end. In the morning the room is 65 F. You turn on the heater, which quickly warms the room up to 85 F. Let H ( x, t ) be the temperature x feet from the heater, t minutes after the heater is turned on. The figure shows the contour diagram for H .
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