pprelim1 - d r dt = 2 e 2 t i(sin t j-t 2 k r(0 =-i j-4 k 4...

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It is highly recommended to review the homework problems and the problems solved in class. The following exam was given in Fall 2002. PRELIM No calculators. Please make sure to give adequate reasons for all your answers. 1. a) Find the equation for the plane containing the points (1 , 0 , 1) and (0 , - 1 , 0) and perpendicular to the plane x - y + z = 0. b) Find a vector perpendicular to the lines L 1 : x = 3 , y = 2 t, z = 7 - t L 2 : x = - 6 + 2 s, y = 3 , z = 3 s c) Find an equation for the plane containing the lines L 1 : x = 1 - t, y = - 2 + 3 t, z = - 2 t L 2 : x = 2 + 3 s, y = - 9 s, z = 2 + 6 s 2. The vector r ( t ) = 7 i - e t j + sin( 3 t ) k is the position vector of a particle at time t . a) Find the particle’s velocity, speed, and acceleration at t = 0. b) Find the angle between the velocity and acceleration vectors at t = 0. c) For each t find a plane through the origin which contains the particle’s velocity and acceleration vectors.
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Unformatted text preview: d r dt = 2 e 2 t i-(sin t ) j-t 2 k r (0) =-i + j-4 k 4. a) Find the domain and range of the function f ( x, y ) = 1 p x 2 + y 2-9 . b) Which of the following properties does the domain satisfy: open, closed, neither open nor closed, bounded, unbounded? Why? c) Sketch the level curves f ( x, y ) = 1 4 and f ( x, y ) = 1 √ 7 . 5. Find the limit if it exists: a) lim ( x,y ) → (2 , 2) x + y 6 =4 x + y + 4 √ x + y-2 . b) lim ( x,y,z ) → (1 ,-1 ,-1) 2 xy + yz x 2 + z 2 . c) lim ( x,y ) → (0 , 0) xy-x 2 x 2 + y 2 . 6. Calculate ∂f ∂x , ∂f ∂y and ∂ 2 f ∂x∂y for f ( x, y ) = p x 2 + y 2 . 7. Consider the planes x-y + z = 0 and x + y-z = 0. Find a parametrization for a line parallel to both planes and one unit distance from each....
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