Fall 2007 - Cioaba's Class - Quiz 3

Fall 2007 - Cioaba's Class - Quiz 3 - x 2) + 8( y 1) = 0...

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Name: PID: Discussion Section No: Time: TA’s name: Quiz 3, Math 20C - Lecture D (Fall 2007) Duration: 20 minutes Please close all your notes and books, turn of your phones and put them away. To get Full credit you should support your answers. 1. (5 points) If f ( x, y ) = x 2 + 4 y 2 , Fnd the gradient vector f (2 , 1) and use it to Fnd the tangent line to the level curve f ( x, y ) = 8 at the point (2 , 1). Sketch the level curve, the tangent line, and the gradient vector. Solution. The gradient vector of f ( x, y ) is f ( x, y ) = a f x ( x, y ) , f y ( x, y ) A = a 2 x, 8 y A This implies that f (2 , 1) = a 4 , 8 A . The tangent line to the level curve f ( x, y ) = 8 is perpendicular to the gradient vector a 4 , 8 A and it passes through the point (2 , 1). Thus, its equation is 4(
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Unformatted text preview: x 2) + 8( y 1) = 0 which simpliFes to 8 y = 16 4 x or y = x 2 + 2. The level curve is an ellipse whose intersects are ( 8 , 0) and (0 , 2). The gradient vector is perpendicular to the tangent line at (2 , 1) and it points to the outside of the ellipse. 2. (5 points) Find all the second partial derivatives of the function u ( s, t ) = e-s sin t . Solution. The rst partial derivatives are the following u s = e-s sin t and u t = e-s cos t The second partial derivatives are u ss = e-s sin t, u tt = e-s sin t, u st = u ts = e-s cos t...
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This note was uploaded on 04/29/2008 for the course MATH 20C taught by Professor Helton during the Fall '08 term at UCSD.

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Fall 2007 - Cioaba's Class - Quiz 3 - x 2) + 8( y 1) = 0...

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