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Practice Exam 1_answers

Practice Exam 1_answers - MATH 2401 Fall 2007 Practice Exam...

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MATH 2401, Fall 2007 Practice Exam 1, Solutions Problem 1 . Calculations. (a) d dt [(2 t i + t j ) ( t i - 3 j )] solution: 4 t - 3 2 t - 1 / 2 . (b) d dt [( cost i + sint j + t k ) × (3 i + 4 j + 5 k )] solution: (5 cost - 4) i + (5 sint + 3) j - (4 sint + 3 cost ) k (c) d dt [ e cos 2 t i + ln (1 + t 2 ) j + (1 - cost ) k ] solution: - 2 sin (2 t ) e cos 2 t i + 2 t 1+ t 2 j + sint k (d) Let w = f ( x, y, z ) = xz + e y 2 z + xy 2 z 3 , calculate w x , w y , w z , w xy and w yz solution: w x = z + 1 2 y 2 z 3 x - 1 / 2 , w y = 2 yze y 2 z + xz 3 , w z = x + y 2 e y 2 z + 3 2 xy 2 z , w xy = 1 2 z 3 / 2 x - 1 / 2 . w yz = 2 ye y 2 z + 2 y 3 ze y 2 z + 3 2 xz.
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(e) Set f ( x, y ) = x 2 - y 4 x 3 - y 4 . Determine whether or not f has a limit at (1 , 1). solution: Along x = 1, the limit is 1, while along y = 1, the limit is 2 / 3. So it has no limit at (1 , 1). Problem 2 A golf ball is hit at time t = 0. Its position vector as a function of time is given by r ( t ) = 2 t i + 3 t j + ( - t 2 + 4 t ) k . Notice that at
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Practice Exam 1_answers - MATH 2401 Fall 2007 Practice Exam...

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