Calculus: One and Several Variables

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Unformatted text preview: P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-05 JWDD027-Salas-v1 November 25, 2006 15:58 SECTION 5.2 229 CHAPTER 5 SECTION 5.2 1. L f ( P ) = 0( 1 4 ) + 1 2 ( 1 4 ) + 1( 1 2 ) = 5 8 , U f ( P ) = 1 2 ( 1 4 ) + 1( 1 4 ) + 2( 1 2 ) = 11 8 2. L f ( P ) = 2 3 ( 1 3 ) + 1 4 ( 5 12 ) + 0( 1 4 ) + ( 1)(1) = 97 144 , U f ( P ) = 1( 1 3 ) + 2 3 ( 5 12 ) + 1 4 ( 1 4 ) + 0(1) = 97 144 3. L f ( P ) = 1 4 ( 1 2 ) + 1 16 ( 1 4 ) + 0( 1 4 ) = 9 64 , U f ( P ) = 1( 1 2 ) + 1 4 ( 1 4 ) + 1 16 ( 1 4 ) = 37 64 4. L f ( P ) = 15 16 ( 1 4 ) + 3 4 ( 1 4 ) + 0( 1 2 ) = 27 64 , U f ( P ) = 1( 1 4 ) + 15 16 ( 1 4 ) + 3 4 ( 1 2 ) = 55 64 5. L f ( P ) = 1( 1 2 ) + 9 8 ( 1 2 ) = 17 16 , U f ( P ) = 9 8 ( 1 2 ) + 2( 1 2 ) = 25 16 6. L f ( P ) = 0( 1 25 ) + 1 5 ( 3 25 ) + 2 5 ( 5 25 ) + 3 5 ( 7 25 ) + 4 5 ( 9 25 ) = 14 25 , U f ( P ) = 1 5 ( 1 25 ) + 2 5 ( 3 25 ) + 3 5 ( 5 25 ) + 4 5 ( 7 25 ) + 1( 9 25 ) = 19 25 7. L f ( P ) = 1 16 ( 3 4 ) + 0( 1 2 ) + 1 16 ( 1 4 ) + 1 4 ( 1 2 ) = 3 16 , U f ( P ) = 1( 3 4 ) + 1 16 ( 1 2 ) + 1 4 ( 1 4 ) + 1( 1 2 ) = 43 32 8. L f ( P ) = 9 16 ( 1 4 ) + 1 16 ( 1 2 ) + 0( 1 2 ) + 1 16 ( 1 4 ) + 1 4 ( 1 2 ) = 5 16 , U f ( P ) = 1( 1 4 ) + 9 16 ( 1 2 ) + 1 16 ( 1 2 ) + 1 4 ( 1 4 ) + 1( 1 2 ) = 9 8 9. L f ( P ) = 0 ( 6 ) + 1 2 ( 3 ) + 0 ( 2 ) = 6 , U f ( P ) = 1 2 ( 6 ) + 1 ( 3 ) + 1 ( 2 ) = 11 12 10. L f ( P ) = 1 2 ( 3 ) + 0( 6 ) + ( 1)( 2 ) = 3 , U f ( P ) = 1( 3 ) + 1 2 ( 6 ) + 0( 2 ) = 5 12 11. (a) L f ( P ) U f ( P ) but 3 2 . (b) L f ( P ) 1 1 f ( x ) dx U f ( P ) but 3 2 6 . (c) L f ( P ) 1 1 f ( x ) dx U f ( P ) but 3 10 6 . 12. (a) L f ( P ) = ( x + 3)( x 1 x ) + ( x 1 + 3)( x 2 x 1 ) + + ( x n 1 + 3)( x n x n 1 ) , U f ( P ) = ( x 1 + 3)( x 1 x ) + ( x 2 + 3)( x 2 x 1 ) + + ( x n + 3)( x n x n 1 ) (b) For each index i x i 1 + 3 1 2 ( x i 1 + x i ) + 3 x i + 3 Multiplying by x i = x i x i 1 gives ( x i 1 + 3) x i 1 2 ( x 2 i x 2 i 1 ) + 3( x i x i 1 ) ( x i + 3) x i . Summing from i = 1 to i = n , we find that L f ( P ) 1 2 ( x 2 1 x 2 ) + 3( x 1 x ) + 1 2 ( x 2 n x n 1 2 ) + 3( x n x n 1 ) U f ( P ) P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-05 JWDD027-Salas-v1 November 25, 2006 15:58 230 SECTION 5.2 The middle sum collapses to 1 2 ( x n 2 x 2 ) + 3( x n x ) = 1 2 ( b 2 a 2 ) + 3( b a ) Thus b a ( x + 3) dx = 1 2 ( b 2 a 2 ) + 3( b a ) 13. (a) L f ( P ) = 3 x 1 ( x 1 x ) 3 x 2 ( x 2 x 1 ) 3 x n ( x n x n 1 ) , U f ( P ) = 3 x ( x 1 x ) 3 x 1 ( x 2 x 1 ) 3 x n 1 ( x n x n 1 ) (b) For each index i 3 x i 3 2 ( x i + x i 1 ) 3 x i 1 ....
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ch05[1] - P1 PBU/OVY P2 PBU/OVY QC PBU/OVY T1 PBU...

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