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# fa07hw4 - 12 The exact value of I is I = integraldisplay 1...

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Unformatted text preview: 12. The exact value of I is I = integraldisplay 1 x 2 dx = x 3 3 vextendsingle vextendsingle vextendsingle vextendsingle 1 = 1 3 . The approximation is T 1 = ( 1 ) bracketleftBig 1 2 ( ) 2 + 1 2 ( 1 ) 2 bracketrightBig = 1 2 . The actual error is I- T 1 = - 1 6 . However, since f ( x ) = x 2 , then f primeprime ( x ) = 2 on [0,1], so the error estimate here gives | I- T 1 | ≤ 2 12 ( 1 ) 2 = 1 6 . Since this is the actual size of the error in this case, the constant “12” in the error estimate cannot be improved (i.e., cannot be made larger). 6. M 4 = 2 ( 3 . 8 + 6 . 7 + 8 + 5 . 2 ) = 47 . 4 5. T 4 = 2 2 [3 + 2 ( 5 + 8 + 7 ) + 3] = 46 T 8 = 1 2 [3 + 2 ( 3 . 8 + 5 + 6 . 7 + 8 + 8 + 7 + 5 . 2 ) + 3] = 46 . 7 36. Since sin x ≤ x for all x ≥ 0, thus sin x x ≤ 1. Then I = integraldisplay π sin x x dx = lim epsilon1 → + integraldisplay π epsilon1 sin x x dx ≤ integraldisplay π ( 1 ) dx = π....
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fa07hw4 - 12 The exact value of I is I = integraldisplay 1...

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