CALCULUS Understanding Its Concepts and Methods

The number *e*

The constant
*e*
(the symbol honors Euler) is the base of the
natural logarithm. This means that
*e*
is the unique number with the property that
That
is, the area of the region bounded by the hyperbola
* y* = 1/*x* ,
the
*x*-axis,
and the vertical lines *x* = 1
and
*x* = *e*
is
1.

The associated exponential function
*e ^{x}*
has the important property that its value is equal to its slope at every
point:
The
numerical value of

The number
*
e *
appears in many mathematical contexts involving limits and derivatives. It is
also important in finance, in the context of continuously compounding
interest, because it occurs as the following
limit:

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.