exam1_samp

# exam1_samp - r(0 = h 3 4 i ﬁnd the position at a general...

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Your Name Printed Clearly! Your Discussion Leader Name Discussion Period EXAM 1 SAMPLE BOYLAND CALCULUS 3 SPRING 10 1. (a) Find a vector perpendicular to the plane containing the points A = (1 , 0 , 0) , B = (2 , 0 , - 1) and C = (1 , 4 , 3). (b) Find the area of the triangle ABC . 2. (a) Find the vector and scalar projection of the vector 2 i +3 j + - 2 k onto the vector 3 i + - j +4 k . (b) Find the equation of the plane parallel to the plane x + 2 y + 5 z = 3 and passing through the point ( - 4 , 1 , 2). Page 1 of 3 Total points this page out of 20

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3. (a) Sketch the surface, labeling at least 4 points. 9 x 2 + y 2 + 4 z 2 = 16 (b) Find the equation of the tangent plane to the surface z = sin( xy ) at the point (1 , π/ 4 , 2 / 2) 4. A particle has constant acceleration a ( t ) = h 1 , 0 , 2 i and initial velocity v (0) = h 1 , - 2 , 3 i and initial position
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Unformatted text preview: r (0) = h , 3 , 4 i ﬁnd the position at a general time t and when t = 1. Page 2 of 3 Total points this page out of 20 5. Assume r ( t ) = h t 3 , 5 , 2 t 2 i . (a) Find the arc length for 0 ≤ t ≤ 1 (b) Find the unit tangent vector T ( t ) for t > 0. (c) Find the curvature κ ( t ) at a general point t > 0. (d) Find the parametric equation of the tangent line to the curve when t = 1 6. Find the linear approximation of the function f ( x,y ) = (4 x 2 + y 2 ) 1 / 3 at (1 , 2) and use it to approximate f (1 . 05 , 1 . 95). You must give a computed numerical answer, a fraction is OK. Page 3 of 3 Total points this page out of 20...
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