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practicefinala - 18.02 PRACTICE FINAL EXAM A BJORN POONEN 1...

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18.02 PRACTICE FINAL EXAM A BJORN POONEN Please turn cell phones off completely and put them away. No books, notes, or electronic devices are permitted during this exam. Generally, you must show your work to receive credit. Name: Student ID number: Recitation instructor’s last name: Recitation time (e.g., 10am): 1 out of 25 2 out of 10 3 out of 10 4 out of 15 5 out of 20 6 out of 10 7 out of 10 8 out of 25 9 out of 20 10 out of 20 11 out of 30 12 out of 20 13 out of 15 14 out of 20 Total out of 250
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1) For each of (a)-(e) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please do not use the abbreviations T and F.) No explanations are required in this problem. (a) If a and b are vectors in R 3 such that a × b = 0 , then either a = 0 or b = 0 . (b) If the first row of a 3 × 3 matrix is the same as the second row, then the determinant of the matrix is 0. (c) If T is a unit tangent vector at P to a level curve of a differentiable function f ( x, y ), then the directional derivative of f at P in the direction T is 0. (d) Let P = P ( x, y ) and Q = Q ( x, y ) be continuously differentiable functions on the region R in R 2 where 1 < x 2 + y 2 < 4. If P dx + Q dy is an exact differential (i.e., the differential of some function on
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