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Error Terms

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More Calculus of a Single Variable

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Abstract

Any inequality A ≤ B can be restated, at least in principle, as an equality \(A = B - E\) where E ≥ 0 is an error term. Of course, if A ≤ B is complicated, then the error term probably cannot be known exactly. But quite often in calculus, something can be said about the error term. When this is the case, interesting and useful things usually follow.

From error to error one discovers the entire truth.

—Sigmund Freud

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References

  1. Burk, F.: Geometric, logarithmic, and arithmetic mean inequality. Am. Math. Mon. 94, 527–528 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cerone, P., Dragomir, S.S.: Mathematical Inequalities: A Perspective. CRC, New York (2011)

    Google Scholar 

  3. Courant, R., John, F.: Introduction to Calculus and Analysis, vol. I. Springer, New York (1989)

    Book  Google Scholar 

  4. Cruz-Uribe, D., Neugebauer, C.J.: An elementary proof of error estimates for the trapezoidal rule. Math. Mag. 76, 303–306 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Drazen, D.: Note on Simpson’s rule. Am. Math. Mon. 76, 929–930 (1969)

    Article  Google Scholar 

  6. Fazekas, E., Jr., Mercer, P.R.: Elementary proofs of error estimates for the midpoint and Simpson’s rules. Math. Mag. 82, 365–370 (2009)

    Article  MATH  Google Scholar 

  7. Fink, A.M.: Estimating the defect in Jensen’s inequality. Publ. Math. Debr. 69, 451–455 (2006)

    MathSciNet  MATH  Google Scholar 

  8. Glaister, P.: Error analysis of quadrature rules. Int. J. Math. Educ. Sci. Techno. 35, 424–432 (2004)

    Article  Google Scholar 

  9. Greenwell, R.N.: Why Simpson’s rule gives exact answers for cubics. Math. Gaz. 83, 508 (1999)

    Article  Google Scholar 

  10. Hai, D.D., Smith, R.C.: An elementary proof of the error estimates in Simpson’s rule. Math. Mag. 81, 295–300 (2008)

    MATH  Google Scholar 

  11. Hardy, G.H.: A Course of Pure Mathematics, 9th edn. Cambridge University Press, Cambridge (1948)

    Google Scholar 

  12. Jennings, W.: On the remainders of certain quadrature formulas. Am. Math. Mon. 72, 530–531 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  13. Johnsonbaugh, R.: The Trapezoid rule, Stirling’s formula, and Euler’s constant. Am. Math. Mon. 88, 696–698 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Leach, E.B.: The remainder term in numerical integration formulas. Am. Math. Mon. 68, 273–275 (1961)

    Article  MathSciNet  Google Scholar 

  15. Levrie, P.: Stirred, not shaken, by Stirling’s formula. Math. Mag. 84, 208–211 (2011)

    Article  MATH  Google Scholar 

  16. Lin, T.P.: The power mean and the logarithmic mean. Am. Math. Mon. 81, 879–883 (1974)

    Article  MATH  Google Scholar 

  17. Mercer, A.McD.: An “error term” for the Ky Fan Inequality. J. Math. Anal. Appl. 220, 774–777 (1998)

    Google Scholar 

  18. Mercer, A.McD.: Short proofs of Jensen’s and Levinson’s inequalities. Math. Gaz. 94, 492–494 (2010)

    Google Scholar 

  19. Mercer, P.R.: Error estimates for numerical integration rules. Coll. Math. J. 36, 27–34 (2005)

    Article  Google Scholar 

  20. Mercer, P.R.: Error terms for Steffensen’s, Young’s, and Chebyshev’s inequalities. J. Math. Ineq. 2, 479–486 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mercer, P.R., Sesay, A.A.: Error terms for Jensen’s and Levinson’s inequalities. Math. Gaz. 97, 19–22 (2013)

    Google Scholar 

  22. Niizki, S., Araki, M.: Simple and clear proofs of Stirling’s formula. Int. J. Math. Educ. Sci. Technol. 41, 555–558 (2010)

    Article  Google Scholar 

  23. Ostrowski, A.M.: On an integral inequality. Aequ. Math. 4, 358–373 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  24. Perisastry, M., Murty, V.N.: Bounds for the ratio of the arithmetic mean to the geometric mean. Coll. Math. J. 13, 160–161 (1982)

    Google Scholar 

  25. Pecaric, J., Peric, I., Srivastava, H,M.: A family of the Cauchy type mean-value theorems. J. Math. Anal. Appl. 306, 730–739 (2005)

    Google Scholar 

  26. Pinker, A., Shafer, R.E.: Problem 209. Coll. Math. J. 14, 353–356 (1983)

    Google Scholar 

  27. Pinkham, R.: Simpson’s rule with constant weights. Coll. Math. J. 32, 91–93 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ralston, A., Rabinowitz, P.: A First Course in Numerical Analysis. McGraw-Hill, New York (1978)

    MATH  Google Scholar 

  29. Richert, A.: A non-Simpsonian use of parabolas in numerical integration. Am. Math. Mon. 92, 425–426 (1985)

    Article  MathSciNet  Google Scholar 

  30. Sandomierski, F.: Unified proofs of the error estimates for the midpoint, trapezoidal, and Simpson’s rules. Math. Mag. 86, 261–264 (2013)

    Article  MATH  Google Scholar 

  31. Steele, J.M.: The Cauchy-Schwarz Master Class. Mathematical Association of America/Cambridge University Press, Washington DC - Cambridge/New York (2004)

    Book  MATH  Google Scholar 

  32. Supowit, K.J.: Understanding the extra power of the Newton-Cotes formula for even degree. Math. Mag. 70, 292–293 (1997)

    Article  MathSciNet  Google Scholar 

  33. Young, R.M.: Euler’s constant. Math. Gaz. 75, 187–190 (1991)

    Article  MATH  Google Scholar 

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Mercer, P.R. (2014). Error Terms. In: More Calculus of a Single Variable. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1926-0_14

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