The error bound formulas for trapezoidal rule and simpson's rule say that:Error Bound for the Trapezoid Rule: Suppose that Error Bound for the Trapezoid Rule: Suppose that | f ′ ′ ( x ) | ≤ k for some k ∈ R where a ≤ x ≤ b . Then | E T | ≤ k ( b − a ) 3 12 n 2 Error Bound for Simpson's Rule: Suppose that | f ( 4 ) ( x ) | ≤ k for some k ∈ R where a ≤ x ≤ b . Then | E S | ≤ k ( b − a ) 5 180 n 4 Using these formulas, is it possible to find functions where Trapezoid Rule is more accurate than Simpson's rule?

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The error bound formulas for trapezoidal rule and simpson&#039;s rule say that:Error Bound for the Trapezoid Rule: Suppose that                                Error Bound for the Trapezoid Rule: Suppose that                       |                          f                              ′                ′                                      (            x            )            |                    ≤          k           for some           k          ∈                      R                     where                                               a          ≤          x          ≤          b          .           Then                                                                     |                          E                              T                                      |                    ≤          k                                    (              b              −              a                              )                                  3                                                                    12                              n                                  2                                                                                                   Error Bound for Simpson&#039;s Rule: Suppose that                       |                          f                              (                4                )                                      (            x            )            |                    ≤          k           for some           k          ∈                      R                     where                                               a          ≤          x          ≤          b          .           Then                                                                     |                          E                              S                                      |                    ≤          k                                    (              b              −              a                              )                                  5                                                                    180                              n                                  4                                                                         Using these formulas, is it possible to find functions where Trapezoid Rule is more accurate than Simpson&#039;s rule?

pheniankang · Accepted Answer

Note that &quot;these formulas&quot; give only an error estimate. The actual error may be much smaller.In order to obtain an example where the formula for       E    S   produces a larger error than the formula for       E    T   consider the function  f  (  x  )  :=  sin  ⁡  (  ω  x  )    (  0  ≤  x  ≤  1  )     .You then have       |        f    ″    (  x  )      |    ≤      ω    2   and       |        f          (      4      )        (  x  )      |    ≤      ω    4  . This gives      |        E    T        |    ≤                    ω        2                    12                  n          2                      ,        |        E    S        |    ≤                    ω        4                    180                  n          4                         ,so that the estimate for       E    S   is worse than the estimate for       E    T   as soon as       ω    2    ≥  15      n    2  .

The error bound formulas for trapezoidal rule and simpson's rule say that: Error Bound for the Trap

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