What is the significance of the number e in calculus?

[JonF]

I just started calc 2. And my book defined e as the number such that: $$\lim_{h \rightarrow 0} \frac{e^h-1}{h}=1$$ I’m having trouble picturing what e is. Is there another definition of e? Is a actual physical ratio, (like pi or the golden ratio) or is it just some random number?

 

[AKG]

I just started calc 2. And my book defined e as the number such that: $$\lim_{h \rightarrow 0} \frac{e^h-1}{h}=1$$ I’m having trouble picturing what e is. Is there another definition of e? Is a actual physical ratio, (like pi or the golden ratio) or is it just some random number?

For one, the exponential function $f(x) = e^x$ is the same as all of its derivatives (and anti-derivatives if you ignore the constant). There are other definitions for e, perhaps look it up at wikipedia and mathworld.

 

[hello3719]

e = lim as h->infinity of (1+1/h)^h

 

[NateTG]

Another popular version is:
$$e=\sum_{i=0}^{\infty} \frac{1}{i!}$$

 

[JonF]

Thanks nate that was exactly what I was looking for, I can picture that one

 

[Gokul43201]

But that's (the series expansion) only saying how big e is (2.718281...). It doesn't tell you anything more about why e is useful or interesting.

 

[Goalie_Ca]

Other than the limits, and the mclaurin series, all i can think of at the moment is euler's (identity?)

e^(i*Pi) - 1 = 0

For a calc 2 perspective, you'll probably use all three.

Here's the easiest way to think of e.

d e^x / dx = e^x

The function is its own derivative (...and integral)

 

[arildno]

But that's (the series expansion) only saying how big e is (2.718281...). It doesn't tell you anything more about why e is useful or interesting.

If I'm not mistaken, I believe that the function Exp(x) defined by:
$$Exp(x)=\sum_{i=0}^{\infty}\frac{x^{i}}{i!}$$
is necessary in order to rigourously define the exponentiation process in general (for example, to introduce the concept of an irrational number raised to an irrational exponent (I think this is the rational way to do it..)).

Besides, the sequence of finite series approximations to Exp(1) converges quite fast.
So, Exp(1) is perhaps a form of e worth mentioning.

 

[Integral]

The magic of e lies in this expression

$$\int ^e _1 \frac {dx} x =1$$

so the area under the inverse curve between 1 and e is exactly 1 square unit.

 

[HallsofIvy]

The point of $\lim_{h \rightarrow 0} \frac{e^h-1}{h}=1$ is that it is easy to show that the derivative of a^x is
$(\lim_{h \rightarrow 0} \frac{a^h-1}{h})a^x$.

Since $\lim_{h \rightarrow 0} \frac{e^h-1}{h}=1$,
e^x has the nice property that its derivative is just e^x again.

Think of it this way: the derivative of a^x is C a^x where C is a constant (i.e. does not depend on x) the does depend on C. e is defined as the number for which that C is 1.