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S09mt1an

# S09mt1an - principal unit normal vector to the curve r t =...

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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 final answer. 1. Find 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 . Find vectors c parallel to a , and d perpendicular to a , such b = c + d . 3. Find 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. Find 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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