Calculus: One and Several Variables

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-17 JWDD027-Salas-v1 December 7, 2006 16:38 866 SECTION 17.1 CHAPTER 17 SECTION 17.1 1. 3 ± i =1 3 ± j =1 2 i 1 3 j +1 = ² 3 ± i =1 2 i 1 ³ 3 ± j =1 3 j +1 =(1+2+4)(9+27+81)=819 2. 2+2 2 +3+3 2 +4+4 2 +5+5 2 =68 3. 4 ± i =1 3 ± j =1 ( i 2 +3 i )( j 2) = ´ 4 ± i =1 ( i 2 i ) µ 3 ± j =1 ( j 2) = (4 + 10 + 18 + 28)( 1+0+1)=0 4. 2 2 + 2 3 + 2 4 + 2 5 + 2 6 + 2 7 + 4 2 + 4 3 + 4 4 + 4 5 + 4 6 + 4 7 + 6 2 + 6 3 + 6 4 + 6 5 + 6 6 + 6 7 =19 4 35 . 5. m ± i =1 Δ x i x 1 x 2 + ··· x m =( x 1 x 0 )+( x 2 x 1 )+ +( x m x m 1 ) = x m x 0 = a 2 a 1 6. ( y 1 y 0 y 2 y 1 y n y n 1 )= y n y 0 = b 2 b 1 7. m ± i =1 n ± j =1 Δ x i Δ y j = ² m ± i =1 Δ x i ³ n ± j =1 Δ y j a 2 a 1 )( b 2 b 1 ) 8. n ± j =1 q ± k =1 Δ y j Δ z k = n ± j =1 Δ y j ² q ± k =1 Δ z k ³ b 2 b 1 c 2 c 1 ) 9. m ± i =1 ( x i + x i 1 x i = m ± i =1 ( x i + x i 1 )( x i x i 1 m ± i =1 ( x i 2 x 2 i 1 ) = x m 2 x 0 2 = a 2 2 a 1 2 10. n ± j =1 1 2 ( y j 2 + y j y j 1 + y j 1 2 y j = 1 2 n ± j =1 ( y j 3 y j 1 3 1 2 ( b 2 3 b 1 3 ) 11. m ± i =1 n ± j =1 ( x i + x i 1 x i Δ y j = ² m ± i =1 ( x i + x i 1 x i ³ n ± j =1 Δ y j · (Exercise 9) = ( a 2 2 a 1 2 ) ( b 2 b 1 ) 12. m ± i =1 n ± j =1 ( y i + y j 1 x i Δ y j = ² m ± i =1 Δ x i ³ n ± j =1 ( y j 2 y j 1 2 ) a 2 a 1 )( b 2 2 b 1 2 )
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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-17 JWDD027-Salas-v1 December 7, 2006 16:38 SECTION 17.2 867 13. m ± i =1 n ± j =1 (2Δ x i y j )=2 ² m ± i =1 Δ x i ³ ´ n ± j =1 1 µ 3 ² m ± i =1 1 ³ ´ n ± j =1 Δ y j µ =2 n ( a 2 a 1 ) 3 m ( b 2 b 1 ) 14. m ± i =1 n ± j =1 (3Δ x i y j )=3 m ± i =1 n ± j =1 Δ x i 2 m ± i =1 n ± j =1 Δ y j =3 n ( a 2 a 1 ) 2 m ( b 2 b 1 ). 15. m ± i =1 n ± j =1 q ± k =1 Δ x i Δ y j Δ z k = ² m ± i =1 Δ x i ³ n ± j =1 Δ y j ² q ± k =1 Δ z k ³ =( a 2 a 1 )( b 2 b 1 )( c 2 c 1 ) 16. m ± i =1 n ± j =1 q ± k =1 ( x i + x i 1 x i Δ y j Δ z k = m ± i =1 ( x i 2 x i 1 2 ) · n ± j =1 Δ y j ² q ± k =1 Δ z k ³ a 2 2 a 1 2 )( b 2 b 1 )( c 2 c 1 ) 17. n ± i =1 n ± j =1 n ± k =1 δ ijk a ijk = a 111 + a 222 + ··· + a nnn = n ± p =1 a ppp 18. Start with m ± i =1 n ± j =1 a ij . Take all the a ij (there are only a fnite number oF them) and order them in any order you chose. Call the frst one b 1 , the second b 2 , and so on. Then m ± i =1 n ± j =1 a ij = r ± p =1 b p where r = m × n. SECTION 17.2 1. L f ( P 1 4 ,U f ( P )=5 3 4 2. L f ( P f ( P 3. (a) L f ( P )= m ± i =1 n ± j =1 ( x i 1 +2 y j 1 x i Δ y j f ( P m ± i =1 n ± j =1 ( x i y j x i Δ y j (b) L f ( P ) m ± i =1 n ± j =1 ¸ x i 1 + x i 2 ´ y j 1 + y j 2 µ¹ Δ x i Δ y j U f ( P ) . The middle expression can be written m ± i =1 n ± j =1 1 2 ( x i 2 x 2 i 1 ) Δ y j + m ± i =1 n ± j =1 ( y j 2 y 2 j 1 ) Δ x i .
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