1322_midterm3_2004 - MAT 1322 Winter 2004 MIDTERM TEST 3...

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MAT 1322 Winter 2004 MIDTERM TEST 3 Version A 1. [2 points, 8.8 #7] Find the MacLaurin series c 0 + c 1 x + c 2 x 2 + c 3 x 3 + ··· of the function 1 9+6 x . The coefficient c 3 and the radius of convergence R is A. c 3 = 20 / 81 ,R = 9B . c 3 = 5 / 162 =9 C. c 3 = 5 / 72 =3 / 2D . c 3 = 5 / 162 / 2 E. c 3 = 27 / 128 / 4F . c 3 = 20 / 81 / 4 Solution . 1 9+6 x = 1 3 1+ 2 x 3 = 1 3 ( 1+ 2 x 3 ) ) 1 / 2 = 1 3 (1 + ( 1 2 ) 1! ( 2 x 3 )+ ( 1 2 )( 3 2 ) 2! ( 2 x 3 ) 2 + ( 1 2 )( 3 2 )( 5 2 ) 3! ( 2 x 3 ) 3 + c 3 = 1 3 ( 1 2 )( 3 2 )( 5 2 ) 3! ( 2 3 ) 3 = 5 162 Convergent for | 2 x 3 | < 1 i.e. | x | < 3 2 . R = 3 2 . Answer. D 2. [2 points, 11.4, #13] Find the linear approximation of the function f ( x, y )= p 11 + x 2 y 2 at the point (3 , 2). A. 4 + 1 2 ( x 3) 3 4 ( y 2) B. 5 + 1 2 ( x 3) 1 2 ( y 2) C. 4 + 3 4 ( x 3) 1 2 ( y 2) D. 5 + 3 5 ( x 3) 2 5 ( y 2) E. 4 + 3 5 ( x 3) 2 5 ( y 2) F. 5 + 2 3 ( x 3) 1 3 ( y 2) Solution . f ( x, y ) f (3 , 2) + f x (3 , 2)( x 3) + f y (3 , 2)( y 2) f ( x, y )=(11+ x 2 y 2 ) 1 / 2 , f x ( x, y x 2 y 2 ) 1 / 2 x, f y ( x, y x 2 y 2 ) 1 / 2 ( y ) , f (3 , 2) = 4 ,f x (3 , 2) = 3 4 y (3 , 2) = 1 2 , f ( x, y ) 4+ 3 4 ( x 3) 1 2 ( y 2) Answer. C 1
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3. [2 points, 8.7 #43] Use multiplication or long division of power series to Fnd the Frst three nonzero terms of the MacLaurin series of e 2 x 2 1+ x 2 . A. 1 + x 2 + x 4 B. 1 + x 2 +5 x 4 C. 1 + 3 x 2 x 4 D. 1 + 3 x 2 + x 4 E. 1 x 2 +2 x 4 ±. 1 x 2 4 x 4 Solution . e 2 x 2 = (1 + (2 x 2 ) 1! + (2 x 2 ) 2 2! + ··· )=(1+2 x 2 x 4 + ) 1 1+ x 2 = (1 x 2 + x 4 + ) e 2 x 2 1+ x 2 = (1 + 2 x 2 x 4 ) ( x 2 x 4 )+( x 4 )+ =1+ x 2 + x 4 + Answer. A 4. [2 points, 11.3 #17,21,23] Let f ( x, y, z )= x x + y + z i n d f x (2 , 1 , 3).
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This note was uploaded on 02/23/2010 for the course MAT MAT1332 taught by Professor Arianemasuda during the Fall '09 term at University of Ottawa.

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1322_midterm3_2004 - MAT 1322 Winter 2004 MIDTERM TEST 3...

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