Practice Exam.pdf - Math 10C B00 and C00 Practice Exam The...

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Math 10C B00 and C00 Practice Exam The Final Exam may or may not contain questions similar to those in this practice problem set. The actual Final Exam will be shorter than this Practice Exam. (1) Let u = h 1 , 2 , 3 i and v = h 8 , 6 , 7 i . Find (a) u · v and (b) u × v , where · is the dot product and × is the cross product. (2) Find the symmetric equations of the line that passes through (4 , 5 , 6) and is perpendicular to the plane 3 x - 3 y + z = 13. (3) Find the equation of the plane through (3 , 3 , 3) with normal vector n par- allel to the following line: x = 9 t + 11 , y = 8 t + 15 , z = 7 t + 23 , where t is any real number. (4) Find the equation of the plane that passes through the point (10 , 1 , 1) and contains the line: x = 5 t, y = 4 - 4 t, z = 3 t, where t is any real number. (5) Find the velocity and acceleration of the particle with position vector r ( t ) = h sin(2 t ) , t 4 , t ln t i , where t is any positive real number. (6) Let z = f ( x, y ) = 2 xy + 1, where x = e s + t and y = s 2 + t . Find ∂z ∂s . (7) Let w = f ( x, y, z ) = x sin( yz ). Find (a) the total differential dw ; (b) the directional derivative of f at (1 , π, 1) in the direction of v = 1 2 j + 1 2 k . (8) Find the equations of the tangent plane and normal line at the point (1 , 1 , 0) to the level surface : F ( x, y, z ) = x + z + y 2 e 2 xz

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