Optimization

At its core, the concept of optimization in calculus involves the maximization and/or minimization values. Roughly speaking, this involves finding local maxima/minima of a function over real numbers. Doing so frequently requires the use of derivatives.

Consider, for example, the function \(f(x) = -x^2 + 6x + 9.\) We could perhaps pretend that this function plots the amount of revenue \(f(x)\) generated by a business against \(x\), the number of employees. Now, any business owner worth their salt will want to maximize their profits. So, how can we find the number of employees which returns the maximum profit?

The astute among you will have recognized by now that \(f(x)\) takes the form of a quadratic. From there, the value of \(x\) which results in the largest profits can be found by completing the square, or by using \(\frac{b}{2a}\) from the quadratic formula. However, this is calculus! We are no longer interested in simply completing the square or using the quadratic formula. We will instead bask in the glory of derivatives, and seek their blessings to complete the task at hand.

It will be accepted, without any proof, that for a function \(f(x)\) defined over the reals, local maxima and minima occur only at points where the where the derivative \(f'(x)\) equals zero. A quick Google search should be able to explain why. This does not mean that all points where the derivative is zero are either a local maxima or minima—consider, for example, the point \((0,0)\) of the function \(g(x) = x^3\).

With this knowledge in mind, we now have a plan for finding the local extrema of a single-variable function:

1. Calculate the function's derivative;
2. Identify \(x\)-values where \(f'(x) = 0\);
3. Verify that these points are truly local maxima/minima.

The last step may be done either by evaluating the function's second derivative, \(f''(x)\) at the \(x\)-value of the extrema, or by checking the sign of \(f'(x)\) at \(x\)-values near that of the extrema in question.

If it so happens that you have two variables to work with, you may want to consider finding a constraint that allows one variable to be written in terms of the other. Doing so would reduce the function to being expressed in terms of a single variable—much easier to manage!

If there isn't such a constraint, then optimizing by hand gets cumbersome (honestly, it's already troublesome enough as is for a single variable). When that occurs, you may want to utilize online technology, such as a 3-D graphing calculator, to find your extrema. The internet is your oyster, whether you like seafood or not.

"What does that have to do with manufacturing wooden crates?"

Well, let's assume that you're out to maximize your profits, like any business owner worth their salt. What that really implies is maximizing profit generated per unit time.

You may have figured this out already, but the price of wood manufacture a crate, the time required to manufacture a crate, and the selling price of a crate are all markedly defined. This means that we can partake in one of the greatest mathematical pleasures: modelling!

Let's say that \(C(a)\) represents the cost of the wood required to manufacture a crate, based on some variable \(a\). Let's also go ahead and let \(T(b)\) represents the time required to manufacture a crate in terms of some other varible \(b\), and let \(P(c)\) represent the selling price of a crate in terms of another nondescript variable \(c\).

In that case, the revenue generated per unit time is equal to \(\frac{P(c) - C(a)}{T(b)}\).

We'll leave it up to you to figure out what \(a\), \(b\), and \(c\) are, and to find expressions for those three functions. Hint: considering how many fields are up to you to fill in, some of these functions might be in terms of more than variable.

Best of luck, and happy optimizing!

Sustainability

What does it mean to be sustainable? Well, the word 'sustainability' has many definitions, and though they're all similar, they're also all different. In the context of this project, we will approach sustainability as a mindset. Simply put, it's a way of looking at and thinking about the world and its interactions—one which values growth, adaptability, and the ability to think long-term, above all else.

It's no secret that the Earth's environment is deteriorating. Recently, nature has not been kind to us. Floods, fires, and other sorts of natural disasters are all conspiring to exact their toll on the planet. The water levels are going to rise. In fact, scientists say they're already rising. Ben Shapiro, an American political pundit, once infamously proposed that homeowners could simply sell their properties in low-lying areas—it has since been pointed that such thinking is nothing more than folly. No, the solutions are not going to be anywhere as simple as what he would like to believe.

And so, it makes us wonder: what are we going to about this?

One of the avenues we've explored to encourage sustainability is the notion of eco-friendly sourcing. Eco-friendly products are becoming more and more prevalent. Surely, these wooden crates could join the fray! It's important to know where your products and materials are coming from. It's great to make money, but not at the expense of the environment.

Frankly, that's no good.

The idea of being sustainable is an intriguing prospect, but it isn't something we can do effortlessly. You might have to make some concessions. Sourcing wood from eco-friendly providers might cost more. Often, it definitely will. However, that may be fine. If we can still live fulfilling and hearty lives in the absence of something, did we really need whatever it was in the first place?

You may think that you, as an individual, have almost no ability to propogate this sort of change in thinking. And that's partially correct—but only partially. Change can't instantly occur on such a large scale.

To suggest such an idea would put us on par with Shapiro's underwater real estate fantasy.

Your actions will inspire others—your activism will move others. And this mindset will spread, and it will grow, until it becomes something far larger than any of us. Indeed, we're already starting to see it happen.

Perhaps, consider doing some research on your own to learn more about sustainable practices, and how a sustainable mindset could transform the future.