What Is Calculus About?

What Is Calculus?

If you're willing to spend some time on this, go through section 1.1 of Strang's online textbook, available here. If not, read on; this lesson was inspired by that section.

Calculus is the study of change.

If I'm moving along a number line, then the change in my position can be measured by my velocity. Let's call my position $x$—but since my position depends on the current time $t$, my position is a function of $t$, so we should write my position as a function $x(t)$ to make that clear.

Using this equation, we can use position to find velocity:

We can also use velocity to find position:

So far, we've been using a simple equation to relate a quantity (position) and the rate of change of that quantity (velocity). You could use the same equation to relate the amount of water in your bathtub to how fast it's filling up, or a company's total amount of money to its quarterly profit. The only problem? It only works when the rate of change is constant.

If the rate of change is not constant, we need calculus. Let's call the rate of change $v$, except that the rate of change now depends on time, so we should write the rate of change as a function $v(t)$. Calculus helps us study the relationship between $x(t)$ and $v(t)$.

Specifically, calculus allows us to answer two hugely important questions:

(1) If we know $x(t)$, how do we find $v(t)$? (This is called differentiation.)

(2) If we know $v(t)$, how do we find $x(t)$? (This is called integration.)

These two concepts are big, but if you understand them well, you'll understand change, which is what calculus is all about.

Calculus literally means something like the "study of change". More precisely, and importantly, it describes functions through the lenses of the "very large" (essentially "infinite") and the "very small" (essentially "infinitesimal").

Roughly speaking, it draws upon three key perspectives:

1. Numerical === It's important to remember that a function is nothing more than a "machine" or "simple program": whenever you give it an input, it either "returns an error" (i.e. when the function is undefined), or spits out an output uniquely determined by the input. The last part is worth stressing: the output (if defined) is unique.

(Functions are also called "maps". Note that if you "input" a point on a physical world map, you get an "output" corresponding to a unique location in the world.)

For our purposes, we will be working mostly with "nice" functions, in the sense that plugging in "a lot" of inputs (and seeing the resulting outputs) will usually give us "a lot" of information about the function as a whole.

1. Algebraic === Algebraic manipulation is crucial, especially in formalizing the intuitive notions of "very large" and "very small". In particular, we'll work just as much with inequalities as with equations, if not more.

In addition, some important functions, like polynomials, are algebraically "simpler" than others. We'll often want to approximate a "complicated" function by such simpler functions, and indeed, this is one of the big problems addressed by calculus.

1. Graphical === The graph of a function allows us to visualize many "geometric" properties of functions that would be easy to miss otherwise. Most importantly, since the axes "continuously" represent the real numbers—each axis shows the real numbers "all bunched up" next to each other—graphs of functions will allow us to develop two key ideas behind calculus:

• the "instantaneous change" or "sensitivity" of a function (the "derivative");

• the "continuous sum" or "total accumulation" of a function (the definite integral).

Welcome to the heart of the subject of my heart: calculus! In the Expii calculus map, we plan to get you familiar to the fundamental concepts of calculus, such as limits, derivatives, integrals, and more. We will teach you not only computational definitions and give you practice with these, but we will also provide graphical intuition for each of these.

Join us in this journey and dive into the most fascinating subject of all of math $-$ the basis for a lot of advanced research math and the subject of math most closely related to science and engineering: calculus!

Welcome to calculus, probably the first of the advanced topics you will ever learn!

Calculus is fascinating and you can imagine why. With calculus, you can prove results that you could also prove with algebra, number, etc. but in some new, elegant fashions. You can also do what would be impossible with simple tools such as algebra. Calculus takes you beyond the limits of your regular high-school math. It's incredible. Learn about the basics and beyond here at Expii!

Calculus is one of the most powerful tools in mathematics, developed by Sir Isaac Newton and Gottfried Leibniz in the 17th century. This subject matter is key for understanding rates, i.e. how fast or slow something changes. This concept is central in physical life with the matter of velocity. Furthermore, integrals are used to reverse this process, and find the position of an object given its velocity.

These two concepts, the integral and derivative, are joined by a third that is often studied in precalculus: the limit. Limits observe the behavior of functions overtime, and are thus applicable to scientific applications such as population modeling.

Understanding why these underlining concepts work should make them easier to use. Calculus is practical enough that everybody should take at least one calculus course in high school and/or college.

Calculus

Calculus deals with the infinite. The three main concepts in calculus are

1. limits: what happens to a function $f(x)$as we get closer and closer to a certain value of $x$? $$\lim_{x\to x_0} f(x)$$ Is a function "continuous," or does it jump around? What is the behavior as $x$ goes to $\pm \infty$?

2. derivatives deal with infinitesimally small differences: what is the instantaneous rate of change? For example, velocity is the instantaneous rate of change over time. We can calculate an average velocity of a falling ball over an interval $$v=\frac{\Delta d}{\Delta t}=\frac{\text{distance}}{\text{time}},$$ but because the ball speeds up as it falls, this will only be an approximation to the instantaneous velocity. How can we get a well-defined quantity if we let the change in time go to 0? (It seems we get a division-by-0 error; the right thing to do is to take a limit as $\Delta t\to 0$: $$f'(x)=\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ )

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1. integrals: how can we sum infinitely many infinitely small quantities? $$\int_a^bf(x)\,dx = \lim_{n\to \infty}\sum_{k=0}^{n-1}\frac{1}{n}f\left(a+\frac{b-a}{n}k\right).$$

This comes up in the following:

(a) Calculating area. We know how to calculate areas of shapes built up from rectangles, but how do we calculate the area of arbitrary curvy shapes? Solution: break it up into more and more smaller rectangles that approximate the shape.

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(b) Calculating a quantity knowing its rate of change; for example, calculating distance traveled from knowing the velocity at every moment. Note that this makes the integral the opposite of the derivative, in a sense.

In a sense, calculus is just the continuous version of things like "sums" and "differences." When we deal with continuous mathematics, we have to develop some new intuition, though. It took a lot of time for people to develop the right definitions to deal with the infinite. See for example Zeno's paradoxes. It's a good exercise to think about how you would resolve these!

Calculus is essential in physics. For example, Newton's law $F=ma$ tells us that force is proportional to acceleration, the rate of change in velocity. So to find velocity we would have to integrate; to find distance we would have to integrate again.

Welcome to Calculus! This literally means "a pebble or stone used for counting," so you can imagine what math has come from and gotten to, right? Just kidding. Calculus is the study of change, as you might know. It involves derivatives, extensive formulas, and a lot of skill. Enjoy the ride!