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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.