prelim2_fall2005solutions

# prelim2_fall2005solutions - Math 192 Prelim 2 7:30-9:00 You...

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Unformatted text preview: Math 192, Prelim 2 October 27, 2005. 7:30-9:00 You are allowed one 8 . 5 × 11 sheet of paper with information on both sides. You are NOT allowed calculators, the text, or any other book or notes. SHOW ALL WORK! 1) (10pts) TRUE or FALSE Instructions: On the first page of your booklet, label five lines a), b), c), d), e). On each line, answer the corresponding question by writing either TRUE or FALSE, or leave the line blank. No abbreviations, please . A correct answer is worth 2pts, an incorrect answer-1pt, no answer 0pts. a) Let f ( x, y ) = 2 xy- x 2 + 4 y and ~v = 6 ~ i + 2 ~ j . TRUE or FALSE: At (1 , 2), ~v gives the direction of greatest increase. FALSE. ∇ f = h 2 y- 2 x, 2 x + 4 i and so ∇ f (1 , 2) = h 2 , 6 i gives the direction of greatest increase. This is not a positive scalar multiple of ~v = h 6 , 2 i , so ~v does not give the direction of greatest increase. b) Let R be the region defined by a ≤ x ≤ b , g 1 ( x ) ≤ y ≤ g 2 ( x ). TRUE or FALSE: Z Z R f ( x, y ) dA = Z b a Z g 2 ( x ) g 1 ( x ) f ( x, y ) dydx = Z g 2 ( x ) g 1 ( x ) Z b a f ( x, y ) dxdy. FALSE. This is just Fubini’s Theorem applied incorrectly. The first equality is true, however the second equality is false. The third integral in the expression gives a function of x as its result while the other two end with numerical values. c) TRUE or FALSE: The centroid of a region R must lie in the region R . FALSE. Take the ring created by { ( x, y ) : 1 ≤ x 2 + y 2 ≤ 2 } . Its centroid is at (0 , 0) which is not in the region. d) TRUE or FALSE: The point (0,0) is a local minimum for the function shown below....
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prelim2_fall2005solutions - Math 192 Prelim 2 7:30-9:00 You...

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