The number e as a limit
We have defined the natural logarithm as an integral, an 'area' under the rectangular hyperbola: Then we can define number e as the positive number at which log(e)=1: We could approximate its value using rectangles. But there is another popular definition of number e as a limit (related with compound interest): These two definitions refer to the same number and we are going to see an intuitive approach to this. To study the limit you can consider, for each n, a series of rectangles whose bases are determined by the numbers It is not difficult to see that there are n rectangles and that the area of each rectangles is 1/n (You need to make some calculations). Then the total area is 1. The last term of this series is an approximation by default of number e. With more and more rectangles (bigger n) the approximation becomes better and better:
Then we conclude that Using the applet you can see that the 'convergence' to number e is slow (to get a good approximations you need a big n). There are better ways to calculate the value of e but this is a good approach to understand the idea of logarithm, integral and limit. REFERENCES
A. I. Markushevich, Areas and Logarithms, D.C. Heath and Company, 1963.
Füsum Akman, Proofs Without Words Under the Magic Curve. The Mathematical Association of America (Classroom capsules).
Serge Lang, A First Course in Calculus, Third Edition, AddisonWesley Publishing Company.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.
Michael Spivak, Calculus, Third Edition, PublishorPerish, Inc.
Otto Toeplitz, The Calculus, a genetic approach, The University of Chicago Press, 1963.
Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, SpringerVerlag New York Inc., 1980.
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We can study several properties of exponential functions, their derivatives and an introduction to the number e.
Using the integral of the equilateral hyperbola we can define a new function that is the natural logarithm function.
The natural logaritm can be defined using the integral of the rectangular hiperbola. In this page we are going to see an important property of this integral. Using this property you can justify that the logarithm of a product is the sum of the logarithms.
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Different hyperbolas allow us to define different logarithms functions and their inversas, exponentials functions.
Mercator published his famous series for the Logarithm Function in 1668. Euler discovered a practical series to calculate.
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Monotonic functions in a closed interval are integrable. In these cases we can bound the error we make when approximating the integral using rectangles.
If we consider the lower limit of integration a as fixed and if we can calculate the integral for different values of the upper limit of integration b then we can define a new function: an indefinite integral of f.
