Mock_Exam1 - -xy 2-x 2 yz what is the direction of the...

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Mock Exam 1 1
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1. a).Find the equation of the plane through (5,1,-2) perpendicular to n = < 2 , 4 , 3 > . Then find the angle between this plane and the one with equation 3x-4y+7z=5. b).Find the equation of the plane throught the three points P 1 (1 , - 2 , 3) , P 2 (4 , 1 , - 2) and P 3 ( - 2 , - 3 , 0). 2
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2. a).Find the curvature of the circular helix r ( t ) = a cos t i + a sin t j + ct k , a > 0 b).Find T , N , a T , a N for the curve above. 3
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3. Show that f ( x, y ) = xe y + x 2 y is differentiable everywhere and calculate its gradient. Then find the equation z = T ( x, y ) of the tangent plane at (2,0). 4
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4. If the temperature at any point in a homogeneous body is given by T
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Unformatted text preview: -xy 2-x 2 yz , what is the direction of the greatest drop in temperature at the point (1,-1,2)? 5 5. Find the maximum and minimum values of f ( x, y, z ) = x + 2 y + 3 z on the ellipse that is the intersection of the cylinder x 2 + y 2 = 2 and the plane y + z = 1. 6 6. Find the directional derivative of f ( x, y ) = e-x cos y at (0 , π/ 3) in the direction toward the origin. 7 7. Find the equation of the tangent plane and the normal line to the surface x 2 + y 2 + 2 z 2 = 23 at (1,2,3). 8...
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