User:MiguelCaruncho
Hey, my name's Miguel and I'm from the Sauder School of Business hoping to major in Marketing in the future. I was born and raised in the Philippines and this is pretty much my first time living outside my home country so I'm pretty excited.
Pythagorean Theorem
The Pythagorean Theorem is a concept used in mathematics dating back to being utilised in Ancient Egyptian/Babylonian mathematics. The concept was used in a practical sense by these Ancient peoples and were not defined into a mathematical framework until the Greek mathematician Pythagoras "rediscovered" the concept and provided concrete mathematical proof to explain the phenomenon.
The theorem states that the area of the square with the hypotenuse as a side is equal to the sum of the areas of the squares having sides that are the other two on the same triangle. In mathematical terms this is given by the formula a^2 + b^2 = c^2. Where c is the length of the hypotenuse and a and b being the other two sides of the triangle.
This theorem has arguably the most amount of proofs (about 370 according to the book The Pythagorean Proposition), with the proof of similar triangles and the usage of squares, collinear points and shared angles being the most well known.
Calculus in Business and Economics
By its very definition, calculus is the math of change and what could be more in a state of constant flux than the world of business? Almost every single aspect of business is dynamic and ever-changing, thus, for us humans to be able to make sense of all this, we tend to use models and mathematical representations to not only understand what is happening, but also to predict what will happen. This is especially important considering the main aim for majority of businesses and organisations is profit, and when you can predict behaviour, you can pretty much predict the rate of change of your bank account.
First, let's take two very simple functions that we have actually used in class quite a bit, the cost and profit functions. Now while deceivingly simple, we have to note that costs and profits are not always as easy as the models we have encountered in the classroom and thus are not always just or anything like that. Once the way we arrive at our costs and profits becomes more complicated, the more the need to model it better arises, and this is where math, and eventually calculus come in. Businesses need to know at what cost they're operating at and if they are operating at the intended or even optimum amount of production. They also need to know if they are making money or not; how much and how fast they are earning, and at what point does it become not worth it to produce anymore. This is where calculus plays its role, by utilising manipulations such as optimising profits, minimising costs, understanding the plateaus of production (at which point does it not become worth it to continue production), finding out if we ever attain economies of scale and all that sort of jazz. It's all connected. While understandably a lot of these numbers can be foregone in a few cases, why would any sane profiteer choose to do so? Businesses operate under the assumption that more is better and finding out what limits you can stretch this definition to is what proves calculus to be a great aid in the business world.
Second, we can also use calculus in business to not only predict costs and profits, but also the extent of the possibility of profits in an area or even in a broader market. One of the models we studied in class was the model for exponential population growth. The function is great and all when put in the context of demographics or data gathering for banks of information but what if it was used in a different way? Let's think of using it as a predictor for the potential amount of people there will be in a certain geographical region. Taking into account the fact that it will take time for the individuals predicted to be born in that area to mature into an appropriate spending age, we can easily predict the amount of potential buyers or consumers there will be in an area for future financial planning. And it only gets better. If say the geographical region in question were to accept immigrants and the immigration rate was found to be a certain rate, we can easily just plug this into the model we have and continue on as usual. This not only allows us to calculate what we can do now, given the resources and constraints currently present, but also what we can do given the situation we predict in the future.
Lastly, in relation to predicting the potential for future profits is how calculus can predict what happens when we tinker with what we are currently selling. And by this, I do not mean tinkering with how many we produce or at what cost we produce at but rather things about the actual product, such as the price, augmenting certain aspects of it or anything like that. In fact, for tinkering with the price, we have a very defined economic term called the elasticity of a product. The elasticity of a product is basically the degree of responsiveness of the consumers to a products price, or pretty much anything you can model and predict as a substitute factor for price. This becomes invaluable for in business simply because of the repercussions of being able to model such a thing carries with it. If we know how much the customers will respond to a change in price for example, then we will know at what point the price will become to high for consumers to afford and at what price it will be too cheap for us to reap the maximum amount of profit we would be able to. Now understandably this all sounds very evil and manipulative but if a consumer is content with the price he/she buys a good at and we profit an amount that is acceptable or even optimal then ceteris paribus, there should be no problem. It all just boils down to finding the correct balance between how many consumers will decide to purchase the product at a certain price, and the price itself.
Now there many more examples of how calculus can be used in my field of study which is commerce, or business but I think I have outlined enough to show that it has a major impact in the volatile world of business. In all honesty, a long time ago I thought math beyond basic operations was gonna be absolutely useless in commerce but understanding principles and concepts in my other classes and tying it with mathematical models and functions has led me down a different trail of thought, most probably for the better.