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S09mt1an - principal unit normal vector to the curve r t = 3 t i 5 j t 2 k at the point given by t = 2 5 ±ind the length of the curve r t = t 2

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Sample First Midterm Answer all seven questions. You must show your working to obtain full credit. Points may be deducted if you do not justify your Fnal answer. 1. ±ind an equation for the sphere centered at (4 , 2 , 3) which is tangent to the xz -plane. 2. Let a = i +4 j +2 k and b =2 i +5 j +3 k . ±ind vectors c parallel to a , and d perpendicular to a , such b = c + d . 3. ±ind an equation in x , y and z for the plane through the points P (1 , 4 , 3), Q (3 , 1 , 2) and R (5 , 1 , 0). Determine whether the points P , Q , R and S ( 1 , 1 , 5) all lie in the same plane. 4. ±ind the unit tangent vector, the equation of the tangent line, the curvature, and the
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Unformatted text preview: principal unit normal vector to the curve r ( t ) = 3 t i + 5 j + t 2 k at the point given by t = 2. 5. ±ind the length of the curve r ( t ) = t 2 i + (2 t 2 + 1) j + (3 − t 2 ) k from t = 1 to t = 3. 6. Let f ( x, y, z ) = ln( x 2 + y 2 − z 2 ). ±ind the domain and range of f . Describe in words, and draw a sketch of, the level set f ( x, y, z ) = 0. 7. ±ind the limit if it exists, or show that the limit does not exist: lim ( x,y ) → (0 , 0) xy 2 x 2 + y 2 ....
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This note was uploaded on 09/07/2011 for the course MATH 39578 taught by Professor Penner during the Spring '09 term at USC.

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