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Answers and Explanations

## The Number e in Calculus

In calculus there is the notion of a *derivative* of a function,
which is a measure of its rate of change with respect to changes in
its input. When one differentiates an exponential function of the form
f(x) = a^x one gets
f(x+h) - f(x)
f'(x) = lim -------------
h->0 h
x + h x
a - a
= lim -------------
h->0 h
x h x
a a - a
= lim ------------
h->0 h
h
x a - 1
= a lim ------
h->0 h

which is a^x times a number (the limit of
(a^h-1)/h as h goes to 0) which is a constant
independent of x, depending only on the base a
of the exponential.

Therefore, the derivative of an exponential function is just a constant times
the function value. What the constant is depends on what the base of the
exponential function is.

*The number e is that value of the base
which yields the constant 1*, so that the derivative of the function
f(x) = e^x is actually equal to e^x itself.

Moreover, the derivative of the more general function f(x) = B
e^(Rx) is R B e^(Rx). If we were
using some other number a as the base instead of e, there would be
an additional constant out front.

Not only does this make it more natural and convenient to use e
as the base in exponentials and logarithms, but it relates back to the
compound interest interpretation given earlier. If a bank account is
growing under compound interest, with an interest rate of R per
unit time, that means that at any instant in time the growth rate is
R times the current balance. We've already seen that the
balance at the end of the period is B e^R where B is the
beginning balance. More generally,
to figure out the balance somewhere in the middle of the time period,
let t denote the fraction of the time period that has passed;
it turns out that the balance at that time is
B e^(Rt).
The derivative of this function is B R e^(Rt) which is
R times the function value; this is just another way of saying
that the growth rate is R times the current balance.

This page last updated: September 1, 1997

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Philip Spencer

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