M23Fall04a - = / 4. Plot a few points. 5. Consider the...

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Mathematics 23 Midterm Exam, September 30, 2004 Answer all questions and be sure to show all work . No notes, books, or calculators are allowed. 1. Let a = i + j + 2k and let b = - 3i - 2j + 4k . Are a and b orthogonal, parallel, or neither? Explain carefully. 2. a. Find the unit tangent vector T ( t ) to the curve r ( t ) = h 6 t 5 , 4 t 3 , 2 t i . b. Find the parametric equations of the tangent line to r ( t ) at the point (6 , 4 , 2). 3. a. Find an equation of the plane P containing the points (0 , 0 , 0), (0 , 1 , 0) and (0 , 1 , 1). b. Find parametric equations for the line through ( - 2 , 2 , 4) and perpendicular to the plane P . 4. a. Identify and sketch the surface in cylindrical coordinates given by r = 4 sin θ. Plot a few points. b. Identify and sketch the surface in spherical coordinates given by
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Unformatted text preview: = / 4. Plot a few points. 5. Consider the space curve r ( t ) = h t,t 2 + 1 , 1 i , t [-1 , 1]. a. Find the acceleration vector. b. Find the tangential component of the acceleration vector at the point (0 , 1 , 1). c. Find the normal component of acceleration. 6. A ball is thrown at an angle of forty ve degrees to the ground. If the ball lands 100 meters away, what was the initial speed of the ball? You may leave your answer in terms of g , acceleration due to gravity. 7. Let f ( x,y ) = x 2 2 x 2-y 2 . a. What is the domain of f ? b. Find lim ( x,y ) (0 , 0) f ( x,y ). Justify your answer with complete details. 1...
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This note was uploaded on 09/30/2009 for the course MATH 023 taught by Professor Napier during the Fall '08 term at Lehigh University .

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