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Published in:
A Brief Historical Introduction Read chapter Read first chapter

2015 | OriginalPaper | Chapter

12. Differentiation

Authors : Miklós Laczkovich, Vera T. Sós

Published in: Real Analysis

Publisher: Springer New York

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Abstract

Consider a point that is moving on a line, and let s(t) denote the location of the point on the line at time t. Back when we talked about real-life problems that could lead to the definition of limits (see Chapter 9, p. 121), we saw that the definition ofinstantaneous velocity required taking the limit of the fraction \(\big(s(t) - s(t_{0})\big)/(t - t_{0})\) in t 0.

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previous chapter Various Important Classes of Functions (Elementary Functions)
next chapter Applications of Differentiation
Footnotes
1
So the calculus was correct; see page 5.
 
2
One such link was already outlined when we saw that differentiability implies continuity.
 
3
The other central constant is π. The relation between these two constants is given by \(e^{i\pi } = -1\), which is a special case of the identity (11.64).
 
4
Willebrord Snellius (1580–1626), Dutch mathematician.
 
5
Pierre de Fermat (1601–1665), French mathematician.
 
6
Michel Rolle (1652–1719), French mathematician.
 
7
That is, if f′ changes signs at the point a, meaning that it is nonpositive on a left-sided neighborhood of a and nonnegative on a right-sided neighborhood, or vice versa.
 
Literature
1.
Davidson, K.R., Dosig, A.P.: Real Analysis and Applications. Theory in Practice. Springer, New York (2010)
2.
Erdős, P., Surányi, J.: Topics in the Theory of Numbers. Springer, New York (2003)
3.
Euclid: The Thirteen Books of the Elements [Translated with introduction and commentary by Sir Thomas Heath]. Second Edition Unabridged. Dover, New York (1956)
4.
Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York (1975)
5.
Niven, I., Zuckerman, H.S., Montgomery, H.L.: An Introduction to the Theory of Numbers, 5th edn. Wiley, New York (1991)
6.
Rademacher, H., Toeplitz, O.: Von Zahlen und Figuren. Springer, Berlin (1933) [English translation: The Enjoyment of Mathematics]. Dover, New York (1990)
7.
Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)
8.
Zaidman, S.: Advanced Calculus. An Introduction to Mathematical Analysis. World Scientific, Singapore (1997)
Metadata
Title
Differentiation
Authors
Miklós Laczkovich
Vera T. Sós
Copyright Year
2015
Publisher
Springer New York
Book
Real Analysis

Print ISBN: 978-1-4939-2765-4

Electronic ISBN: 978-1-4939-2766-1

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-1-4939-2766-1_12

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