Calculus: One and Several Variables

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-16 JWDD027-Salas-v1 December 7, 2006 16:36 788 SECTION 16.1 CHAPTER 16 SECTION 16.1 1. f =(6 x y ) i +(1 x ) j2 . f =(2 Ax + By ) i +( Bx +2 Cy ) j 3. f = e xy [( xy +1) i + x 2 j ] 4. f = 1 ( x 2 + y 2 ) 2 [( y 2 x 2 xy ) i y 2 x 2 2 xy ) j ] 5. f = ± 2 y 2 sin( x 2 +1)+4 x 2 y 2 cos( x 2 ² i +4 xy sin( x 2 j 6. f = 2 x x 2 + y 2 i + 2 y x 2 + y 2 j 7. f =( e x y + e y x ) i e x y e y x ) j e x y + e y x )( i j ) 8. f = AD BC ( Cx + Dy ) 2 [ y i x j ] 9. f z 2 xy ) i x 2 yz ) j y 2 zx ) k 10. f = x ³ x 2 + y 2 + z 2 i + y ³ x 2 + y 2 + z 2 j + z ³ x 2 + y 2 + z 2 k 11. f = e z (2 xy i + x 2 j x 2 y k ) 12. f = ´ xyz x + y + z + ln( x + y + z ) µ i + ´ xyz x + y + z + xz ln( x + y + z ) µ j + ´ xyz x + y + z + xy ln( x + y + z ) µ k 13. f = e x +2 y cos ( z 2 +1 ) i e x +2 y cos ( z 2 ) j 2 ze x +2 y sin ( z 2 ) k 14. f = e yz 2 /x 3 3 2 x 4 i + z 2 x 3 j + 2 x 3 k · 15. f = ´ 2 y cos(2 xy )+ 2 x µ i x cos(2 xy ) j + 1 z k 16. f = 2 xy z 3 z 4 · i + x 2 z j x 2 y z 2 +12 xz 3 · k 17. f =(4 x 3 y ) i +(8 y 3 x ) j ;a t ( 2 , 3) , f = i +18 j 18. f = 1 ( x y ) 2 ( 2 y i x j ) , f (3 , 1) = 1 2 i + 3 2 j 19. f = 2 x x 2 + y 2 i + 2 y x 2 + y 2 j t ( 2 , 1) , f = 4 5 i + 2 5 j

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-16 JWDD027-Salas-v1 December 7, 2006 16:36 SECTION 16.1 789 20. f = ± tan 1 ( y/x ) xy x 2 + y 2 ² i + ± x 2 x 2 + y 2 ² j , f (1 , 1) = ± π 4 1 2 ² i + 1 2 j 21. f = (sin xy + xy cos xy ) i + x 2 cos xy j ;a t ( 1 ,π/ 2) , f = i 22. f = e ( x 2 + y 2 ) [( y 2 x 2 y ) i +( x 2 xy 2 ) j ] , f (1 , 1) = e 2 ( i j ) 23. f = e x sin( z +2 y ) i e x cos( z y ) j + e x z y ) k ; at (0 4 4) , f = 1 2 2( i j + k ) 24. f = cos πz i cos j π ( x y )sin k , f ± 1 , 0 , 1 2 ² = π k 25. f = i y ³ y 2 + z 2 j z ³ y 2 + z 2 k t ( 2 , 3 , 4) , f = i + 3 5 j 4 5 k 26. f = sin( xyz 2 )( yz 2 i + xz 2 j xyz k ) , f ± π, 1 4 , 1 ² = 2 2 ± 1 4 i + π j π 2 k ² 27. (a) f (0 , 2) = 4 i (b) f ( 1 4 1 6 π ) = ´ 1 1+ 3 2 2 µ i + ´ 1 2 3 2 µ j (c) f (1 ,e )=(1 2 e ) i 2 j 28. (a) f (1 , 2 , 3) = 1 8 2 i + 1 2 2 j 27 8 2 k (b) f (1 , 2 , 3) = 5 18 i + 1 9 j + 1 18 k (c) f (1 2 6) = 3 2 i + π 12 e 2 j + k 29. For the function f ( x,y )=3 x 2 xy + y, we have f ( x + h ) f ( x )= f ( x + h 1 ,y + h 2 ) f ( ) =3( x + h 1 ) 2 ( x + h 1 )( y + h 2 )+( y + h 2 ) 3 x 2 xy + y · = [(6 x y ) i +(1 x ) j ] · ( h 1 i + h 2 j )+3 h 2 1 h 1 h 2 = [(6 x y ) i x ) j ] · h +3 h 2 1 h 1 h 2 The remainder g ( h h 2 1 h 1 h 2 =(3 h 1 i h 1 j ) · ( h 1 i + h 2 j ) , and | g ( h ) | ± h ± = ± 3 h 1 i h 1 j ±·± h ±· cos θ ± h ± ≤± 3 h 1 i h 1 j ± Since ± 3 h 1 i h 1 j ±→ 0a s h 0 it follows that f =(6 x y ) i x ) j 30. f ( x + h ) f ( x )=[( x y ) i +(2 x y ) j ] · [ h 1 i + h 2 j ]+ 1 2 h 2 1 h 1 h 2 + h 2 2 ; g ( h 1 2 h 2 1 h 1 h 2 + h 2 2 is o ( h ).
P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-16 JWDD027-Salas-v1 December 7, 2006 16:36 790 SECTION 16.1 31. For the function f ( x,y,z )= x 2 y + y 2 z + z 2 x, we have f ( x + h ) f ( x f ( x + h 1 ,y + h 2 ,z + h 3 ) f ( ) =( x + h 1 ) 2 ( y + h 2 )+( y + h 2 ) 2 ( z + h 3 z + h 3 ) 2 ( x + h 1 ) ( x 2 y + y 2 z + z 2 x ) = ( 2 xy + z 2 ) h 1 + ( 2 yz + x 2 ) h 2 + ( 2

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This note was uploaded on 04/30/2011 for the course MATH 1431 taught by Professor Any during the Spring '08 term at University of Houston.

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ch16[1] - P1: PBU/OVY JWDD027-16 P2: PBU/OVY...

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