Here's an example of applied math in industry:

FedEx had candidate investor General
Dynamics (GD), and GD had sent two of their
guys, one aero engineer and one finance guy,
to help FedEx. FedEx needed the
GD investment.

At one point, the Board wanted some
revenue projections. I wasn't asked
to get involved and didn't want to,
but no one had any ideas
of how to do
the projections except just
draw a free hand line on a graph
based on hopes, dreams, intentions,
guesses, etc.

So, I thought: We know (1) the current
revenue. We know (2) the revenue
when the planned service to 90 US
cities with 33 airplanes is full.
So, the projections are roughly
how interpolate between (1) and (2).

For how the interpolation will go,
we have (3) current customers and
(4) the rest of the customers we
will have when we have (2) and are full.

Then say that the growth is
from current customers (3) talking
to the rest of the customers (4).
That is, the rate of growth is
directly proportional to the
number of (happy) customers (3)
talking to the rest of the
customers we will have (4).

So, let t denote time in days
with t = 0 corresponding to the
present. Let the revenue
at time t be y(t).
Let the revenue from (2), being full, be
b dollars a day. Then at time t,
the growth rate is y'(t),
that is, d/dt y(t). So,
we have that

y'(t) = k y(t) ( b - y(t) )

How 'bout that! Maybe it's
so simple it's almost just
a joke, but the rest of the
ideas floating around the office
are much worse!

So, sure, it's an initial value
problem for a first order,
linear ordinary differential
equation but, really, is so
simple can just use freshman
calculus directly and find the
closed form solution, right,
some exponentials I omit here.

So, on a Friday the Senior Vice
President (SVP) for Planning and I
got out some graph paper,
picked a value for k, and
drew the graph.

The next day, Saturday, I
was in my office, and at noon
I got a phone call asking if
I knew about the revenue projections.
Well, the SVP was traveling,
and, yes, I knew. So, I was
asked if I could come over to
the HQ offices and explain.
So, I got in my Camaro hot rod
and drove over.

When I got there, people
were unhappy. The two
GD guys were standing in
the hall with their bags
packed.

I was led to the graph,
given a time, and asked
to reproduce the value on the
graph at that time. So,
I had brought my HP scientific
calculator, punched the buttons,
and got the value on the graph.
I did that a few more times,
and people started to be happy
again.

It turned out that the Board meeting
had been that Saturday morning;
the graph had been presented;
and the two GD guys had asked how
it had been calculated.

Well, all the FedEx people there
tried to figure out how the
graph had been calculated and
went on for a few hours
without success, and
then GD guys lost patience,
got plane reservations back to
Texas, went to their rented
rooms and packed their bags,
and as a last chance returned
to the HQ offices to see if
FedEx had an explanation for the
graph. FedEx was about to die.

That's when I arrived and
explained the graph.

The FedEx guys stayed, and FedEx
was saved.

I never got thanked! Really
my impression was that the rest
of the top of FedEx had
concluded that I'd been
dangerous, maybe had 'hooked'
the company.

And I'd never been invited to the
Board meeting. So, I began to
conclude that my contribution
was so resented that the top
of FedEx would rather see
all of FedEx go under than
let me help.

The promised FedEx stock was
already 18 months late.
I was in Memphis while my
wife was in her Ph.D. program
at Johns Hopkins
near our home in Maryland.
So, to heck with FedEx --
I went home and
to Hopkins for my Ph.D.

So, that's an example of
applied math in US business!
E.g., I was the only one around
who still knew freshman calculus!
And, heck, I'd never even taken
freshman calculus, had studied it
on my own and started with
sophomore calculus!

A broad lesson is, for anything
very technical, have a tough
time just making a contribution
in the middle of a lot of
operational and management people
who forgot freshman calculus --
such people will feel out of control
and resentful.

So, instead, for technical contributions,
need an appropriate organizational
structure, maybe be in a research
division and out of the main line
operations or management,
maybe be a respected external vendor,
maybe be in a profession complete
with legal liability and licensing,
etc. And charge a LOT.

Of course, broadly, the flip side
of such technical incompetence,
at the level of just freshman
calculus, should be an opportunity!

There were several other cases where
I did some good technical work on
an important problem but resentment
by others and competitive office
politics kept me from getting credit
or actually blocked my contribution.
In the field of organizational behavior,
such politics is called goal subordination,
that is, someone finds it
in their interest in the company to
hurt the company, subordinate
the good of the company to their
own good.

Broadly, being a really good technical
contributor among a lot of less technical
people is a great way to create
enemies who will sabotage your career,
not friends who will help it.

So, net, usually have to make a technical
contribution from outside the
pecking order of the organization.
That is, be a vendor.

Still better, just sell a product
or service the customers will like
but where the customers need not
be aware of anything technical in the
work.