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MIDTERM1B - MATH 2401 FAll 2009 Midterm I September 23...

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MATH 2401, FAll 2009 Midterm I, September 23, Solutions Problem 1 (30 points). Calculations. (a)(6 pt) d dt [( e t i + t j ) ( e - t i - 3 t j )]. Solution: - 3 (b)(7 pt) d dt [( t 2 i + j ) × ( t 2 i - 3 t j )]. Solution: ( - 9 t 2 - 2 t ) k . (c)(9 pt) Let h ( r, θ, t ) = r 2 e 2 t sin ( θ - t ), calculate h r , h t and h rt . Solution: h r = 2 re 2 t sin ( θ - t ) . h t = 2 r 2 e 2 t sin ( θ - t ) - r 2 e 2 t cos ( θ - t ) . h rt = 4 re 2 t sin ( θ - t ) - 2 re 2 t cos ( θ - t ). (d)(8 pt) Set f ( x, y ) = x - y 3 x 3 - y 3 . Determine whether or not f has a limit at (1 , 1). Solution: Along x = 1, the limit is 1. Along y = 1, the limit is 1 3 . So it has no limit at (1 , 1). Problem 2 (25 pt) An object moves so that r ( t ) = 4 i + (1 + 3 t ) j + (9 - t 2 ) k , t 0 . (a)(6 pt) Compute the velocity, the acceleration and the speed of the ball at an arbitrary time t . Solution: v ( t ) = r 0 ( t ) = 3 j - 2 t k . v ( t ) = 9 + 4 t 2 , a ( t ) = - 2 k .
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(b) (4 pt) Find the time t 1 > 0 and the coordinates of the point P where the object hits the xy plane. Solution: t 1 = 3. P = P (4 , 10 , 0) . (c)(5 pt) Set up a definite integral equal to the length of the arc of the trajectory from the origin to the point P. Do not evaluate the integral.
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