PS2_answers - | b One example f x,y = p | 1-x 2-y 2 | B6 f...

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Here are answers to most of the 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 2 Answers A3. a) L ( a,b,c ) ( x, y, z ) = f ( a, b, c ) + f x ( 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). A4. i) 1 . 08 ii) 359 60 iii) 1 . 125 A5. g x (5 , 1) = - 3, g t (5 , 1) = - 9. A7. α = - β 2 . B1. a) f x (1 , - 4) = 1, f x (0 , 0) = 0, f x (0 , 1) does not exist. b) The partial derivatives of f do not exist at all points where xy = 0, except (0 , 0). B2. a) x = y . B3. i) a) differentiable, b) continuous, c) no info ii) a) not differentiable, b) no info, c) at least one partial not continuous iii) a) differentiable, b) continuous, c) no info. iv) a) not differentiable, b) no info, c) at least one partial not continuous B4. i) does not exist ii) does not exist iii) does not exist iv) exists and equals 0 B5. a) One example
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Unformatted text preview: | b) One example f ( x,y ) = p | 1-x 2-y 2 | B6. f ( x,y ) = | x | . B7. u x ( 3 4 , 1) =-100 π e √ 2 , u t ( 3 4 , 1)-100 e √ 2 B8. u t represents the rate of change of the displacement with respect to time. i.e. if 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 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. 1 2 B9. The volume decreases by 12 . 5 π m 3 . B10. The approximate change in R is 55 . 123966 ... ohms. Actual, 55 . 1246 ... ohms. B11. 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 ). B12. b) c , c) u ( x,t ) = Ae-( x + ct ) 2 C1. C3....
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