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Difference quotient

For broader coverage of this topic, see Finite difference.

In single-variable calculus, the difference quotient is usually the name for the expression

which when taken to the limit as h approaches 0 gives the derivative of the function f.[1][2][3][4] The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is (x + h) - x = h in this case).[5][6] The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h).[7][8]: 237 [9] The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.[9]

By a slight change in notation (and viewpoint), for an interval [a, b], the difference quotient

is called[5] the mean (or average) value of the derivative of f over the interval [a, b]. This name is justified by the mean value theorem, which states that for a differentiable function f, its derivative f′ reaches its mean value at some point in the interval.[5] Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates (a, f(a)) and (b, f(b)).[10]

Difference quotients are used as approximations in numerical differentiation,[8] but they have also been subject of criticism in this application.[11]

Difference quotients may also find relevance in applications involving Time discretization, where the width of the time step is used for the value of h.

The difference quotient is sometimes also called the Newton quotient[10][12][13][14] (after Isaac Newton) or Fermat's difference quotient (after Pierre de Fermat).[15]

  1. ^ Peter D. Lax; Maria Shea Terrell (2013). Calculus With Applications. Springer. p. 119. ISBN 978-1-4614-7946-8.
  2. ^ Shirley O. Hockett; David Bock (2005). Barron's how to Prepare for the AP Calculus. Barron's Educational Series. p. 44. ISBN 978-0-7641-2382-5.
  3. ^ Mark Ryan (2010). Calculus Essentials For Dummies. John Wiley & Sons. pp. 41–47. ISBN 978-0-470-64269-6.
  4. ^ Karla Neal; R. Gustafson; Jeff Hughes (2012). Precalculus. Cengage Learning. p. 133. ISBN 978-0-495-82662-0.
  5. ^ a b c Michael Comenetz (2002). Calculus: The Elements. World Scientific. pp. 71–76 and 151–161. ISBN 978-981-02-4904-5.
  6. ^ Moritz Pasch (2010). Essays on the Foundations of Mathematics by Moritz Pasch. Springer. p. 157. ISBN 978-90-481-9416-2.
  7. ^ Frank C. Wilson; Scott Adamson (2008). Applied Calculus. Cengage Learning. p. 177. ISBN 978-0-618-61104-1.
  8. ^ a b Tamara Lefcourt Ruby; James Sellers; Lisa Korf; Jeremy Van Horn; Mike Munn (2014). Kaplan AP Calculus AB & BC 2015. Kaplan Publishing. p. 299. ISBN 978-1-61865-686-5.
  9. ^ a b Thomas Hungerford; Douglas Shaw (2008). Contemporary Precalculus: A Graphing Approach. Cengage Learning. pp. 211–212. ISBN 978-0-495-10833-7.
  10. ^ a b Steven G. Krantz (2014). Foundations of Analysis. CRC Press. p. 127. ISBN 978-1-4822-2075-9.
  11. ^ Andreas Griewank; Andrea Walther (2008). Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, Second Edition. SIAM. pp. 2–. ISBN 978-0-89871-659-7.
  12. ^ Serge Lang (1968). Analysis 1. Addison-Wesley Publishing Company. p. 56.
  13. ^ Brian D. Hahn (1994). Fortran 90 for Scientists and Engineers. Elsevier. p. 276. ISBN 978-0-340-60034-4.
  14. ^ Christopher Clapham; James Nicholson (2009). The Concise Oxford Dictionary of Mathematics. Oxford University Press. p. 313. ISBN 978-0-19-157976-9.
  15. ^ Donald C. Benson, A Smoother Pebble: Mathematical Explorations, Oxford University Press, 2003, p. 176.

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