Winter 2008 - Enright's Class - Quiz 1

Winter 2008 - Enright's Class - Quiz 1 - x ) = R a x f ( t...

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20E Quiz Solutions February 7, 2008 1) Determine the area of the parallelogram generated by the two vectors A = (1 , 2 , 3) and B = (1 , 0 , 2) . Answer The area is || A × B || . A × B = det ± ± ± ± ± ± i j k 1 2 3 1 0 2 ± ± ± ± ± ± = i (4 - 0) - j (2 - 3) + k ( - 2) = (4 , 1 , - 2) . So Area = || A × B || = p 4 2 + 1 2 + ( - 2) 2 = 21 . 2) Determine the volume of the parallepiped generated by the three vectors A = (1 , 2 , 3) , B = (1 , 0 , 2) and C = (1 , 3 , 0) . Answer V olume = ± ± ± ± ± ± det ± ± ± ± ± ± 1 2 3 1 0 2 1 3 0 ± ± ± ± ± ± ± ± ± ± ± ± = | 1(0 - 6) - 2(0 - 2) + 3(3 - 0) | = 7 3) Suppose φ = (cos t, sin t, 5 t ) is a curve for 1 t π/ 2 . Compute it’s length. Answer Compute: φ 0 ( t ) = ( - sin t, cos t, 5) . Then: L = Z π/ 2 1 || φ 0 ( t ) || dt = Z π/ 2 1 p ( - sin t ) 2 + cos 2 t + 25 = Z π/ 2 1 26 = 26 ² π 2 - 1 ³ . 4) Determine the equation for the tangent plane to the function f ( x, y, z ) = x 2 + y 2 + 5 z 2 at the point (1 , 1 , 1 , 7) . Answer 1
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Compute: f = (2 x, 2 y, 10 z ) . Then using the equation for the tangent hyperplane, w = w 0 + ∂f ∂x ( x 0 , y 0 , z 0 )( x - x 0 ) + ∂f ∂y ( x 0 , y 0 , z 0 )( y - y 0 ) + ∂f ∂z ( x 0 , y 0 , z 0 )( z - z 0 ) , we get the equation w = 7 + 2( x - 1) + 2( y - 1) + 10( z - 1) . 5) Answer as TRUE/FALSE: (a) Suppose that f ( t ) is a continuous function on the t axis. Set g (
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Unformatted text preview: x ) = R a x f ( t ) dt. Then d dx g ( x ) =-f ( x ) . (b) Suppose f ( x, y, z ) = ( y, x, z ) . Then the total derivative is 1 1 1 . Answer (a) is true. This follows from the fundamental theorem of calculus: d dx Z a x f ( t ) dt =-d dx Z x a f ( t ) dt =-f ( x ) . (b) is also true. This follows from the dention of the total derivative: If f ( x, y, z ) = ( f 1 ( x, y, z ) , f 2 ( x, y, z ) , f 3 ( x, y, z )) (as in the setup of this problem), then D ( f ) = f 1 x f 1 y f 1 z f 2 x f 2 y f 2 z f 3 x f 3 y f 3 z . In this case of the problem, this is 1 1 1 . 2...
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Winter 2008 - Enright's Class - Quiz 1 - x ) = R a x f ( t...

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