What is the importance of "e" and why is it important in Calculus?

Answer 1

The importance of "e" is to avoid "e's" as much as possible. We don't want any "e's" or "Errors" like on the way to and back from Mars! =<)

Answer 2

Of course exponents can be used to conveniently represent large numbers in mathematical notation. However, the natural constant "e" has been found to represent rates of change in nature. As a (poor) example a tree might grow linearly a foot a year for thirty years but a natural tree might grow at a more natural rate (where a small tree might grow relatively fast at the beginning on a percent basis but a large tree might grow more feet per year at a lower percent basis). The constant "e" in certain growth formulas more nearly matches natural growth (or shrinkage). A shirt that shrinks 10% each wash shrinks less and less each wash because it is smaller each cycle. Calculus can be used to manipulate data based on formulas employing "e" to obtain slopes (differential calculus) or total growth (integral calculus), etc.

Answer 3

e can be defined a few ways but it boils down to

1. It's the base where the exponential function is its own derivative.
2. It's the base where the exponential function will have a slope of 1 at x=0.

You can always change from 1 base to another. For example, here's how you would change y = e^x from a base of e to a base of 10.

y = exp(x)
ln(y) = x
log(y)/log(e) = x
log(y) = log(e)*x
y = 10^[log(e)*x]

NOTE: ln(y) = log(y)/log(e) works for converting to any base.

Now since you can change to any base, why do you want to use e? The answer. Because it doesn't look as messy. Would you rather write y = exp(x) or y = 10^[log(e)*x]? When you solve the equations for a lot of processes, e is the base that makes it look simpler.

Answer 4

It's the base of the natural log (ln). Instead of using normals logs (base 10) the natural log uses e which is 2.something. It simplifies exponential calculations and helps with solving complex derivatives and integrals.

Answer 5

In Calculus, other than the constant 0, the corresponding function f(x) = e^x is the only one which is its own derivative.

f(x) = e^x
f'(x) = e^x

Answer 6

In Calculus, other than the constant 0, the corresponding function f(x) = e^x is the only one which is its own derivative.

f(x) = e^x
f'(x) = e^x

Answer 7

http://www.math.toronto.edu/mathnet/answers/answers_13.html