# PS1_answers - we look at a ﬁxed point x on the string it...

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Here are some answers to some of the problem set problems. I have, of course, not included ones in which the question asks to prove or verify something. Also, for now, I have not had a chance to include pictures. Math 237 Problem Set 1 Answers A2. i) f x = - y 2 sin( xy ), f y = cos( xy ) - xy sin( xy ) ii) f x = ye y + z , f y = xe y + z + xye y + z , f z = xye y + z iii) g u = v uv + w 2 , g v = u uv + w 2 , g w = 2 w uv + w 2 A3. The tangent plane is z = 1 - 2( x - 1) - 4( y - 2). A4. The tangent plane is z = - 2 - 8( x - 1) + 3( y - 2). A5. a) L ( a,b,c ) ( x,y,z ) = f ( a,b,c ) + f ( a,b,c )( x - a ) + f y ( a,b,c )( y - b ) + f z ( a,b,c )( z - c ) b) i) L (3 , - 1) ( x,y ) = ( x - 3) + 2( y + 1) ii) L (0 ,π/ 4) ( x,y ) = 2 + 3 4 x + 2( y - π 4 ) iii) L (1 , 1 , - 1) ( x,y,z ) = 1 + ( x - 1) + 2( y - 1) + 3( z + 1) iv) L (2 , 1 , 3) ( x,y,z ) = 4( x - 2) - 3( y - 1) - ( z - 3). A6. i) Approx 1.08, ii) Approx 359 60 , iii) Approx 1 . 125 A7. g x (5 , 1) = 3, g t (5 , 1) = - 9 B1. u x ( 3 4 , 1) = - 100 π e 2 , u t ( 3 4 , 1) - 100 e 2 B2. u t ( x,t ) represents the rate of change of the displacement with respect to time. i.e. if
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Unformatted text preview: we look at a ﬁxed point x on the string, it measures the rate of change of the displacement of the string at that location at that time. u x ( x,t ) represents the rate of change of the displacement with respect to position. i.e. at a ﬁxed time, it measures the rate of change of the displacement of the string if you move along the string from some point. B4. The volume decreases by 25 2 π m 3 . B5. Approximate change in R is 55 . 123966 ... ohms. Actual, 55 . 1246 ... ohms. B6. The maximum elevation 12100 13 m. B7. The planes are z =-4-4( x-2)-2( y-1), and z = 4 5 + 4 5 ( x + 2 5 ) + 2 5 ( y + 1 5 ). 1...
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