One way to think about the function e^t is
to ask what properties it has. Probably the most important one, from some points of view
the defining property, is that it is its own derivative. Together with the added condition
that inputting zero returns 1, it’s the only function with this property. You can
illustrate what that means with a physical model: If e^t describes your position on the
number line as a function of time, then you start at 1.
What this equation says is that
your velocity, the derivative of position, is always equal your position. The farther
away from 0 you are, the faster you move. So even before knowing how to compute e^t
exactly, going from a specific time to a specific position, this ability to associate each position
with the velocity you must have at that position paints a very strong intuitive picture of
how the function must grow. You know you’ll be accelerating, at an accelerating rate,
with an all-around feeling of things getting out of hand quickly.
If we add a constant to this exponent, like
e^{2t}, the chain rule tells us the derivative is now 2 times itself. So at every point on
the number line, rather than attaching a vector corresponding to the number itself, first
double the magnitude, then attach it. Moving so that your position is always e^{2t} is
the same thing as moving in such a way that your velocity is always twice your position.
The implication of that 2 is that our runaway growth feels all the more out of control. If that constant was negative, say -0.5, then
your velocity vector is always -0.5 times your position vector, meaning you flip it
around 180-degrees, and scale its length by a half. Moving in such a way that your velocity
always matches this flipped and squished copy of the position vector, you’d go the other
direction, slowing down in exponential decay towards 0. What about if the constant was i? If your
position was always e^{i * t}, how would you move as that time t ticks forward? The derivative
of your position would now always be i times itself.
Multiplying by i has the effect of
rotating numbers 90-degrees, and as you might expect, things only make sense here if we
start thinking beyond the number line and in the complex plane. So even before you know how to compute e^{it},
you know that for any position this might give for some value of t, the velocity at
that time will be a 90-degree rotation of that position. Drawing this for all possible
positions you might come across, we get a vector field, whereas usual with vector field
we shrink things down to avoid clutter. At time t=0, e^{it} will be 1.
There’s only
one trajectory starting from that position where your velocity is always matching the
vector it’s passing through, a 90-degree rotation of position. It’s when you go around
the unit circle at a speed of 1 unit per second. So after pi seconds, you’ve traced a distance
of pi around; e^{i * pi} = -1. After tau seconds, you’ve gone full circle; e^{i * tau} = 1.
And more generally, e^{i * t} equals a number t radians around this circle. Nevertheless, something might still feel immoral
about putting an imaginary number up in that exponent. And you’d be right to question
that! What we write as e^t is a bit of a notational disaster, giving the number e and the idea
of repeated multiplication much more of an emphasis than they deserve. But my time is
up, so I’ll spare you my rant until the next video..
