I've spent the past twelve years as a professor. However, for five months last spring I spent part of a sabbatical working in the long-term forecasting group of an investment firm. I used a lot of math in those five months. (Admittedly, it was mostly a research-type position, and I gravitated toward the math-heavy problems.) Here are some problems I tackled on this job that required me to use math.
We have huge gaps in our set of stock prices because countries change currencies or come into existence or cease to exist or stocks move in and out of major indexes or name your reason. We need to know how these stocks rise and fall (or don't) with each other. How do you calculate a covariance matrix in the presence of missing data? The best solution often results in the matrix becoming singular. This is a big problem because your model requires you to invert it. Do you try one of the other solutions to avoid the singular matrix problem and accept the resulting drawbacks, or do you try to "fix" your matrix somehow? If the latter, what are the best ways to do that? Answering this question required a great deal of understanding of (well, to be honest, learning about) numerical issues in linear algebra.
We have a model that we're happy with that makes short-term predictions, and we have a model that we're happy with that makes long-term predictions. How about the medium term? How do we smooth our short-term predictions into our long-term predictions? To answer this question for us I had to, among other things, solve a couple of differential equations that resulted from trying variations on the logistic curve as models.
- Our model is giving weird, erratic results. Why? Does it have some
fundamental economic flaw? It takes me a couple of days to
determine that the answer to this one has to do with the eigenvalues
of the covariance matrix at the core of the model. Linear algebra
once again.
- I'm analyzing a set of economic indicators to determine their
predictive value. Are they random walks of some sort, or do recent
values have something to do with slightly less recent values? This
requires time series analysis.
- The woman sitting next to me is having trouble with her linear
regression. I can't remember now exactly what the difficulty was,
but I needed some statistics to solve the problem for her.
The job wasn't all math; I spent more time coding than anything else. But I did use linear algebra, numerical methods, optimization, statistics, differential equations, and even some calculus on this job.
So don't give up hope yet. Maybe they are in the research arm of a company, or maybe they require an advanced degree, but there are some jobs out there that require a good deal of math.
TL;DR: Learn by reading, doing, following online courses. Join groups/clubs. Work on projects. Find the area you really enjoy. Eventually you'll contribute to helping people out especially if you believe in whatever field you end up joining.
I love your awesome attitude about math and I wish there were more people like you. Well, there is a lot you can do to help people with math. A LOT! It really boils down to what do you want the world to be like and help it be more like that.
Do you want a greener/cleaner world? Go into environmental engineering.
Do you want a healthier world? Go into the biomedical sector.
Do you want a more technological advanced world? Go into computer science or some technology related engineering field.
Do you want people to be more educated? Go into teaching.
Do you want to do either of these or more but the current environment won't let you? Start a startup company.
I could go on and on and on with these recommendations. The thing is that mathematics is a very flexible subject. You can do with it whatever you want. Heck, there are even Simpsons and Futurama writers who are mathematicians: http://en.wikipedia.org/wiki/Jeff_Westbrook
Don't worry too much about helping people out. It kinda comes as a side effect of doing math. For example, studying graph theory back in the days of Euler may have sounded niche and a waste of time but if it were not for graph theory, no internet. (and certainly no facebook... although... i know people who think that wouldn't be such a bad thing)
Focus on finding something you really like and just roll with it. My recommendation... read, read, read. If you have difficulties taking those courses, read the books. A trick I used to do is that I looked at the syllabi of the courses I wish I could take and just got the textbooks and went through them chapter by chapter.
Learn as much as you can: set theory, graph theory, game theory, optimization, cryptography, differential equations, algebra... whatever you can. And try not to limit yourself to mathematics. We live in a complex interconnected world. Many ideas in math today come from economics and biology and physics and all sorts of places.
When you say you'd like something like taking classes... you know, there just isn't enough time and it is not always possible. But youtube exists. And all those MOOCs (Massive Online Open Courses) like Coursera, EDX, or Udacity. They have loads of courses, some more mathy than others. Even all those computer science Udacity courses are interesting because computer science in the end is all about math.
But learning is not enough, actually working on projects is way more helpful. If you have access to clubs or can get friends to work on some project, you'll learn so much more. Just a simple project like getting a little robot to follow a line on the floor or making a video game will challenge you mathematically and thus you'll get to see how you can apply math in the real world.
So, don't worry too much. Just by your attitude I can tell that you'll manage to do good in the world with math. Just learn, practice, and make awesome.
Best Answer
Here is a very general but broad class of applications: Suppose you have some quantity $q(t)$ that you want to model with respect to time, like maybe a population, or a chemical concentration, or an object's speed, or whatever. Quite often there will be a natural way to describe the quantity you're interested in by using a differential equation, i.e. an equation which relates the rate of change $\frac{dq}{dt}$ of the quantity to $q(t)$ itself.
Calculus can then be used to analyze the differential equation (which could be very complicated) and hopefully give a closed-form solution so that we can predict the quantity in the long term. If an explicit solution is found, calculus can again be used to analyze the solution to find maxima and minima, and all sorts of critical points of interest.
Differential equations aren't only useful for modelling quantities, but also positions. for example, in order to fully understand how a rocket ship blasts off into space, scientists need to take into account the fact that the burning of fuel means the mass is decreasing, and so the propulsion will cause a larger acceleration. This problem leads to solving a differential equation.