By Nick Vasilyev
Let us solve a differential equation. Suppose we want to find the function that is its own derivative. Then we will write y=sigma x^n/n!. That’s it, eureka! The infinite summation is its own derivative.
For those of you who don’t know, the exclamation mark is the factorial that you can recall from Algebra 2. There is a right way and a wrong way to read it. The wrong way is to shout “n” at the top of your lungs. Hip, young, professors read it as “n-bang:” they just say bang. Bang, bang, bang, bang, the terms of the infinite summation all fall into the correct place because n/n equals unity.
For the students taking Calculus, I admire that you are taking the time to learn the advanced concepts deeply. It is rare that you can take a high-brow mathematics course and say to yourself “I really understand what is going on in this course.”
Mathematics is painful and abstract, but it is the crowning jewel in modern Sciences. Students that are successful in Calculus do more than solve homework problems: you are earning the power to solve problems in scientific research and analyze Biological, Social, and Physical models of real-world situations. People that have a knowledge of Calculus can set up equations to real-world problems and can test the reasonableness of their answers.
The teaching and presentation of mathematics will be an exclusive club after the Calculus series. It is difficult to teach and understand pure mathematics, abstract geometry, and methods to solve hard problems without first understanding simple things deeply. It helps to listen to the experience of others and read histories to avoid pitfalls. Your calculus class will clear misconceptions you may have on how to solve it. The problems you are working on are deeply rooted in the solution of Physics problems.
The research professor Isaac Newton and the polymath Liebnitz came to the conclusion that Calculus is the best way to do Engineering and Astronomy. The real goal of your Calculus class is to take Differential Equations and use the knowledge of Calculus. Calculus has been around for 300 years, unlike Operations Research, which sparked with WWII. We can solve the classical differential equations that motivated Newton and Liebnitz and we can solve modern problems.
A way of thinking that is nurtured in calculus is a desire for exactness. In homework, it is more exact to think “I want pi” rather than “I have 3.1414.” It’s possible that a question may have 311/99 as an answer, yet many problems in Calculus are solved exactly by pi. We’re not clicking away on calculators in high-brow mathematics, we will solve the really concrete problems using the transcendental number pi. The practical use of pi is that twice pi is the periodicity of waves when modeled by cos(x). Recall that in finding the time elapsed for successive crests to reach a point, say by y=cos(3*x), that this will happen every time the input for sin(blob) is zero or pi.
We want to see pi radians in trig functions. An interesting thing about pi is that it doesn’t solve X^3-31=0, in fact, it doesn’t solve any polynomial of any degree. This is why pi is snug in Early Transcendentals, by definition of transcendental. A transcendental function cannot be made out of powers of x in a few steps. A transcendental number cannot be a solution to any polynomial.
Using transcendental numbers like the sine of 2 or 1/e allows us to solve hard problems in operations research, we just want rules of thumb to manage resources. Suppose that you are interviewing 10 people and want to hire one manager on the spot. How do you get the best, rule of thumb, hire out of ten interviews with no callbacks? The answer may surprise you, it is ten times 1/e. You will interview 10/e people (use scissors) and then hire the next candidate that is better than the first 10/e.
The breadth and scope of Calculus is not confined to being an effective manager. The problems and techniques that have been accumulated and put into layman’s terms over 300 years are taught to you in Calculus I and II over the course of two semesters. You are getting the very finest in Mathematics in courses below Advanced Calculus I: the material has been seasoned with age, tempered with experience.
Another cool thing about mathematics is the social events. There is a lot of engaging history: mathematicians have been pals and poisonous foes. You can’t teach the relationships between people that refined the Calculus during class. The important thing is to be aware that your math class is all about the interaction between your brain, your class, and you.
Derivatives and Integrals are the tools of the trade, some of them will have you move mountains from one place to another using a vacuum cleaner. Other tools will have you pierce to the heart of a low hanging fruit and collect the pulp. This is what you can learn through emailing hard problems, working together, and striking when the iron is hot. The most rewarding skill in solving Calculus problems is knowing where to strike.
Placing the right tool in the right place is the most elegant aspect of a collegiate mind. The question becomes, where can I go to get the tools I need? I can only indicate to the Queen of the Sciences: Calculus.
So what do Newton, your teacher, and Salman Khan have in common? It’s sunny that those who can show students mathematics in the clearest way have seen the light before. After solving problems, whole college courses are taught on how to get to those elusive answers.
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