# What exactly is calculus?

I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is:

What exactly is calculus?

I know who invented it, the Leibniz controversy, etc., but I'm not exactly sure what it is. I think I heard it was used to calculate the area under a curve on a graph. If anyone can help me with this, I'd much appreciate it.

• Apr 20, 2017 at 17:47
• Apr 20, 2017 at 18:09
• There is a book from 1910 called "Calculus Made Easy" that is worth looking at: gutenberg.org/files/33283/33283-pdf.pdf Apr 21, 2017 at 14:50
• Better name is "Differential and integral calculus".
– IS4
Apr 21, 2017 at 18:48
• A related thread. Apr 22, 2017 at 3:36

In a nutshell, Calculus (as seen in most basic undergraduate courses) is the study of change and behaviour of functions and sequences. The three main points are:

• Limits: How sequences and functions behave when getting closer and closer to a desired point (geometrically, what happens when you "zoom in" near a point)
• Derivatives: How functions change over a parameter (geometrically, the "slope of a graph at a given point")
• Integrals: What's the cumulative effect of a function (geometrically, the "area under a graph")

And obviously (and maybe especially), how these relate to one another; the crowning jewel of Calculus is probably the Fundamental Theorem of Calculus, which truly lives up to its name and was developed by none other than Leibniz and Newton.

• It might help to connect your answer with the terms used in the question by giving gradients of curves as an example of derivatives, and areas under curves as an exampleo f the cumulative effect of a function. Apr 20, 2017 at 22:02
• do sequences and series (finite or infinite) come under the rubric of Calculus? i seem to remember doing those in my undergrad Calculus class. i know that convergence of a sequence or series is related to limits. are issues such as recursion and the inductive proof of closed-form expressions of a series, are those issue part of the discipline of Calculus? Apr 21, 2017 at 1:40
• The fundamental theorem of calculus wasn't discovered by Newton/Leibnitz. Apr 21, 2017 at 4:23
• Thanks for the tips. I'm impressed how my answer received so many views! Apr 21, 2017 at 19:47
• Does/should "the calculus" include things like so-called "discrete calculus" (or "calculus of finite differences")? I also presume "non-standard calculus" is really covered under that banner as non-standard integration and non-standard integration, so doesn't need listing separately, but if discrete calculus counts it probably deserves its own bullet point. Apr 21, 2017 at 23:07

In math, a "calculus" is a set of rules for computing/manipulating some set of objects. For instance the rules $\log AB = \log A+\log B$, etc, are the "logarithm calculus."

But commonly, "calculus" refers to "differential calculus" and "integral calculus." There is a set of rules (product rule, quotient rule, chain rule) for manipulating and computing derivatives. There is also a set of rules (integration by parts, trig substitution, etc.) for manipulating and computing integrals. So, at least etymologically, "calculus" refers to these two sets of rules.

But so far, this has been a bad answer. You really want to know what differential calculus and integral calculus have come to mean. So here goes:

There are a lot of formulas out there that involve multiplying two quantities. $D=RT$, area$= lw$, work = force x distance, etc. All of these formula are equivalent to finding the area of a rectangle. If you drive 30 miles per hour for 4 hours, your distance is $D = RT = 30\cdot 4 = 120$ miles. Easy-peasy.

But what if the speed varies during the drive. Now your $30 \times 4$ rectangle is warped. The left and right sides and bottom are still straight, but the top is all curvy. Still, the distance traveled is the area of the warped rectangle. Integral calculus cuts the area into infinitely many, infinitely tiny rectangles, computes the area of all of them and glues them back together to find the area.

How much work to lift a 10 pound rock to the top of a 200 foot cliff? $10 \times 200$ foot-pounds. But now what if the rock is ice and it melts on the way up, so that when it reaches the top it weighs only 1 pound?

Differential calculus is sort of the opposite problem. When a rock is falling, it starts at $0$ ft/sec., and accelerates. At each point in time, it is going a different speed. Differential calculus gives us a formula for that constantly changing speed. If you graph the position of the rock against time, then the speed is slope of that curve at each point in time.

In both integral and differential calculus, we do nothing more than take the formula for the area of a rectangle and the slope formula for a line and make them sit up and do tricks.

• +1 since the other answers make no distinction between calculus and analysis. Apr 21, 2017 at 0:02

Another way of understanding calculus is that it is the science of refining approximations. The idea is that if we cannot calculate a value directly, we come up with a scheme that allows us to approximate the value as closely as we want. If that scheme is good enough that by sufficiently refining the approximation, we can eliminate all but one particular value as being the number we are after, then we have our answer.

There are a great many values that we cannot calculate directly, but which we can approximate. You mentioned one example: area. From our physical experience, we expect shapes to have a comparable quantity called area. If it takes the same amount of paint to cover two different shapes, then if I paint again, being careful of my thicknesses, it will still take the same amount of paint the second time. To quantify, we can define that a square with sidelength $1$ has area $1$. From this and the concepts that area should not change under rigid motions and that if a shape is divided into two shapes, then the sum of the areas of the parts should be the area of the whole, we can quickly calculate that the area of a rectangle has to be the width $w$ times the height $h$, provided that $w$ and $h$ are both rational values. With a little creativity, we can even show that it holds for some irrational values.

But by direct calculation, we can never show that the area of a rectangle is always its width times its height for all irrational values. And even worse, we cannot arrive at an area for any figure whose boundary is not made of line segments strung together. So we have to find a means to calculate them indirectly. That means is by refining approximations.

While Newton and Liebnitz truly do deserve their titles as the fathers of calculus for their joint invention of the Fundamental Theorem of Calculus, the basic ideas pre-date them - by nearly 2000 years. The key idea is attributed to Eudoxus, though it may predate him as well. That idea is this: If you are comparing two values $x$ and $y$, and can show that $x$ cannot be less than $y$ and $x$ cannot be greater than $y$, then it has to be that $x = y$. Pretty obvious. Let's apply it to finding areas:

Suppose you have some arbitrary shape $S$. We can't directly calculate the area of $S$, but we can cover it with a grid of squares of sidelength $\frac 1n$ for some natural number $n$. Now count the number $M_n$ of squares that overlap $S$ (both blue and tan squares), and the number $m_n$ of squares that are completely inside of $S$ (tan squares only). Since every square of the latter type is also of the former, it is always the case that $m_n \le M_n$. The total area of the covering squares is the sum of the areas of the individual squares, which we already know is $\frac 1{n^2}$, so it will be $\frac {M_n}{n^2}$, and similarly for the contained squares. If $S$ has an area , then it should be true that $$\frac {m_n}{n^2} \le \text{ area of }S \le \frac {M_n}{n^2}$$ for every $n$.

Now for most shapes $S$ we cannot exactly calculate its area this way - not unless $S$ happens to be the union of a bunch of squares. But for nice shapes, we can come up with approximations that are as good as we want. That is, if we say that we want to know the area to a tolerance of $\epsilon$ (epsilon is a traditional variable for this role) for any given $\epsilon > 0$, then by dint of effort we can produce a $n$ big enough that $$0 \le \frac {M_n}{n^2} - \frac {m_n}{n^2} < \epsilon$$ Since the actual area lies between the two, either value differs from the area by an amount less than $\epsilon$.

Now suppose there is a number $A$ that we think should be the area. Let $x < A < y$ for some values $x, y$, and let $\epsilon$ be the smaller of $A - x$ and $y - A$. If we can always find $n$ as above, then $$x = A - (A - x) \le A - \epsilon < \frac {m_n}{n^2} \le \text{area of }S$$ Hence the area cannot be $x$. And similarly, it cannot be $y$. So, per Eudoxus, the area has to be $A$. (If we cannot find such an $n$, then we were wrong about $A$ being the area.)

"Limits" are just a terminology used to describe this concept of refining approximations. Derivatives (slopes of tangent lines to curves) and integrals (areas of regions defined by curves) are two very common and very useful values that usually cannot be calculated directly, and which turn out to be closely related.

In a nutshell: calculus is about derivatives and integrals. A derivative generalizes the idea of slope to graphs that are not lines. For instance, you might look at the graph of $y = x^2$ and notice that it gets steeper as $x$ increases, but how can we make this observation precise? You might ask what the slope of the graph is at $x = 1$, for instance.

But what could the "slope at a point" mean? Rise-over-run gives the formula for the slope of a secant line, but what I really want is a formula for the slope of a tangent line, which would have a "rise" and "run" of zero.

Traditionally, we resolve this paradox using limits (though it can also be done with "infinitesimals"). You'll see how limits work when you take calculus.

As it turns out, the answer we get to our question is that when $y = x^2$, $\frac{dy}{dx}|_{x = a} = 2a$. So: at $(0,0)$, the graph has a slope of $2(0) = 0$. At $(-1,1)$, the graph has a slope of $2(-1) = -2$. At $(2,4)$, the graph has a slope of $2(2) = 4$. This function $2x$ is called the derivative of the function $x^2$.

Intgeration, as you said, is about computing the area, usually between a graph and the $x$-axis, from $x = a$ to $x = b$. For instance, $$\int_1^4 2x\,dx$$ means "the area underneath the graph $y = 2x$, between the values $x = 1$ and $x = 4$". The fundamental theorem of calculus relates integrals to derivatives. In this particular case: because we know a function whose derivative is $2x$ (in this case, $x^2$), we can find this area by calculating $$\int_1^4 2x\,dx = \left. x^2\right|_1^4 = (4)^2 - (1)^2 = 15$$

• And the main idea in defining derivatives and integrals is that of "limit" - without diminishing their usefulness, derivatives and defined integrals are applications of limits. Apr 20, 2017 at 22:30
• @CatalinZara I disagree: it is possible to do differential calculus without limits, and calculus without limits is still calculus. Limits are the usual way to get there, but not by necessity. May 1, 2017 at 5:01
• I respect your right to not agree with me. May 2, 2017 at 0:31

As other answers have broken down some of the applications and the categories of calculus, I'll try to give a more intuitive and motivating explanation.

At its core, calculus is about trying to do meaningful computations with quantities that are infinitely large or infinitely small ("infinitesimals"). What math student, upon being introduced to the idea of infinity, hasn't wondered about $$\infty - \infty$$ or $$0 \cdot \infty$$? Or wondered what $$\frac{0}{0}$$ should be? As Siri will tell you:

Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn’t make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends.

Furthermore, there are similar ancient questions such as Zeno's Paradox---what happens if you take infinitely many steps that are infinitely small? Even the great Archimedes had grappled with such questions and worked with a "proto-calculus" of sorts, almost two millennia before Newton and Leibniz.

In it's original formulations by Newton and Leibniz, calculus was about trying to perform these sorts of computations and get meaningful answers. In Leibniz's version, $$\mathrm{d}x$$ and $$\mathrm{d}y$$ were literally meant to be infinitesimal quantities in the $$x$$- and $$y$$-directions that were smaller than any real number, but still non-zero (Newton had a similar sort of notation). Consider the slope of a curve at a single point, which would ordinarily be $$\frac{0}{0} = \frac{f(a) - f(a)}{a-a}$$. In Leibniz's calculus, we could compute it as $$\frac{\mathrm{d} y}{\mathrm{d}x}$$---the infinitesimal change in the $$y$$-direction divided by the infinitesimal change in the $$x$$-direction.

While this "infinitesimal" approach worked for nearly a hundred years, the logical inconsistencies grew more problematic. Thus Cauchy and other luminaries introduced the modern limit definitions of derivatives and integrals to give calculus a rigorous foundation. While modern calculus is phrased in terms limits, generally of a sequence of approximations that become arbitrarily precise, the entire field is fundamentally about trying to make sense of calculations using infinitely small and infinitely large quantities.

### Examples

• What happens if you add up $$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$$? Intuitively, it feels like the answer should be $$1$$, as you cover half the remaining distance with each additional term. However, you never actually get to $$1$$, no matter how many terms you add up. Furthermore, how can adding up infinitely many non-zero numbers give a finite result?
• How does a car speedometer work? What does it mean to be moving at $$60mph$$ when you haven't actually traveled $$60$$ miles or driven for an hour in the instant that you glance at the speedometer? It certainly makes sense to calculate an average speed over a finite distance, but that requires some finite amount of time as well. What does it mean to be traveling a certain speed at one instant? We can try to think of this as moving an "infinitesimal distance in an infinitesimal amount of time," but how can we actually calculate anything with that?
• Given an arbitrary geometric shape, we can approximate its length/area/volume with nice shapes, such as rectangles. If we allow ourselves to approximate a shape with lots and lots of rectangles, we seem to get a better approximation. What if we use infinitely many rectangles that are infinitely small? How can we use this to find the true area?

There are several subdisciplines in calculus, which I'll summarise simply below. But what they have in common is the study of how one quantity changes when another changes by a small amount.

Differential calculus computes gradients of tangents to curves or surfaces, integral calculus computes the sizes of regions (including the areas of 2D regions), and it turns out differentiation reverses integration. (To see why, imagine expanding an area slightly with a narrow strip you can approximate as a rectangle, and changing a function slightly along the direction of its input increasing. That function might as well be the cumulative area left of a moving cutoff.) Newton co-invented calculus because he wanted to study motion this way. The distance traveled over time is the area under a graph of velocity.

Intrinsic calculus measures curve lengths. It's like applying Pythagoras to lots of small right-angled triangles, then adding their hypotenuses to make an arc. Relativity can be restated as a claim that a certain measure of paths through space and time don't change when you do the equivalent of rotating axes.

Maybe useful:

• George Exner, Inside Calculus, Springer (2000), Introduction - Propaganda: For Students, page vii:

Calculus is really two things: a tool to be used for solving problems for many other disciplines, and a field of study all its own.

Calculus as a tool cares deeply about ways to find the largest value of a function, or obtain relationships between rates of change of some related variables, or obtain graphs of motion of physical objects.

The study of calculus itself is really the internal, supporting structure, for all of the above tools and techniques.

These "structure" is made of three basic concepts: number, function, and limit.

In algebra, the central focus was lines, which change consistently. Slope was a big focal point, from its use in the fundamental equations of lines (slope-intercept, point-slope, etc), to working on parallel/perpendicular lines, to using lines for models of life and extrapolation/forecasting. While you also should have touched on more complex functions in geometry and conics, you didn't spend nearly the time on any of those as you did on lines. At the same time you've been building up some notation and tools, especially during the latter part of algebra/pre-calculus, expanding your scope (such as inverses, logarithms, rational functions) and setting a larger foundation that will be helpful in the next steps (function notation, matrices, transformations).

Well in life most functions indeed aren't linear. For that matter, they often aren't even any of the other major geometric shapes and algebraic functions. When you look at these nonlinear functions, their "slopes" vary by where you are on the graph (in time\space). Life is like a roller coaster, sometimes up, sometimes down. No precision solutions to understand and predict the functions. Not as simple as those nice steady lines: You can get an estimate of the rate of change at each spot, basically how it is sloping, by picking a pair of points in a given area and using the old $(y_2-y_1)/(x_2-x_1)$, but because the graph continually changes slope at every point, the locations use aren't perfectly representative of the location you're interested in. You need to take that idea further, and that's what the foundations of calculus are.

And you quickly come to see that this cousin to slope, rates of change (the derivative) actually shows up in a wide range of giant applications. They're central to physics and to the full span of sciences; after all, physical properties vary over space and time, so rates of change are everywhere. You can also use the same rate of change concept to calculate the distance along curved functions. And then the next concept, working backwards (antiderivatives/integrals), allows you calculate the area of shapes/solids. Rates of change are fairly central to most every math beyond here, from differential equations to numerical solution methods.

So calculus really is all about getting the precise rate of change. But just as obsessing over the form of line equations may have at first seem rather limited and pointless, until you started to apply them more... likewise calculating the precise rate of change is much more monumental and useful than it probably sounds at first.

I consider calculus to be the study of "infintessimals."

As in, what happens to a secant line if you take smaller and smaller distances from the starting point. (It approaches the tangent line, which is its "derivative.")

Or what happens to your calculation of the area under a curve as you approximate it using "thinner" and thinner rectangles until they become infinite thin? (The approximation becomes progressively more accurate until you have a useful calculation.)

This is not a "standard" definition, and I am going to gloss over many details, but I think it provides some insight.

I consider the fundamental insight of calculus as a mathematical discovery (i.e., the basis for the branches of mathematics that you will eventually study) to be the idea that in some cases, an "infinitely close approximation" of a value that can't be directly computed (using pre-calculus methods) actual is equal to the value itself. So the study of calculus is essentially the study of:

• methodology for calculating these "infinitely close approximations", and
• analysis for determining when such approximations are appropriate, in the sense that:
• these approximations can be calculated, and
• the result of this calculation will be equal to the desired value.

(Caveat: the actual branch of mathematics called "analysis" is closely related to my second bullet point, but the bullet point is not actually intended to be a true definition of that branch of mathematics.)

Thus, the first concept taught in calculus is that of a limit, which is the fundamental building block for the "infinitely close approximation" methodology. The next concept is typically that of convergence of limits, and examples of functions for which the limit at a particular point does not equal the value of the function itself at that point. This corresponds, broadly speaking, to my second bullet point, of analyzing when limits can be computed and when they are equal to the originally-desired value.

As mentioned elsewhere, calculus is typically divided into "differential" calculus and "integral" calculus. Each branch uses the concepts above to find values that can be represented in standard coordinate geometry but can't typically be computed directly using algebraic or geometric methods:

• Differential calculus concerns the problem of finding the local "slope" (called the "derivative") of a function at a given point. "Slope" is an algebraic/geometric concept, but cannot typically be computed directly except in simple cases (e.g. the slope of $y = x^2$ at $x = 0$ is $0$). The slope can, however, be approximated by taking two points near the point of interest and calculating the slope of the line connecting them; this approximation can be improved by bringing the points closer to the point of interest. The "infinitely close approximation" is the limit of this sequence of approximate slopes.
• Integral calculus concerns the problem of finding the area under the curve (called the "integral") of a function. (It appears that this is the branch of calculus that you've heard of, since you mention the area under a graph.) Again, for most functions, this value cannot be calculated directly. But it can be approximated, for instance by picking some number of points along the graph and creating a sequence of line segments connecting them, then calculating the sum of the areas of the trapezoids formed by these line segments, the $x$ axis, and the vertical lines connecting the selected points to the $x$ axis. As the number of points selected is increased, this approximation improves; the "infinitely close approximation" is, again, the limit of this sequence of approximate areas.

In both cases, methodology for calculating these limits is developed; it is often possible (and in some cases, surprisingly easy!) to take a function definition and describe a new pair of functions representing the derivative and integral of the original function at every point. And in both cases there are degenerate functions that make it impossible to calculate either the derivative or the integral (or both) at a certain point or set of points (or even the entire domain of the function), which is why the analysis of when such methods are appropriate is important.

The connection between these two branches is known as the "fundamental theorem of calculus"; it is a pair of theorems that essentially state that integration and differentiation are inverses in the sense that, given the integral of a function, one can find the original function using differentiation, and (modulo a constant) vice-versa.

The subject that you asked about is usually referred to by the term "calculus" in English. In German, this subject is called Analysis. This name is not used outside of the mathematics world. So it is not the translation of the English word "analysis"!

I also had difficulties with figuring out what all this was about, until I saw it referred to as the "Analysis einer Veränderlichen", which then can indeed be translated as "the analysis of a changing/changeable (value)".

So my personal definition is: Calculus is about figuring out how the value of a function changes when its argument changes. Or more broadly:

How a function behaves, depending on its inputs.

You may have a function like $f(x) = 1/x$, and want to analyze what happens when $x$ approaches zero. Doing this (without actually evaluating the function for each possible $x$ and comparing the values) is one part of the analysis - particularly, computing the derivative $f'(x) = -1/x^2$.

For people familiar with the basic concept of limits, I like to describe it as the geometric application of indeterminate forms.

A derivative is the slope of a function computed at every point. For a line, we can take any two points and calculate the slope as $\frac{\Delta y}{\Delta x}$. For a curve, we do exactly the same, but we have to take the limit as the two points converge, leaving us with the indeterminate form $\frac{0}{0}$. Differential calculus is effectively the study of this limit.

An intergral is the area under a curve. For a constant function $f(x)=c$ from $0$ to $a$, this is just $c \times a$. For a curve, we can approximate it with a bunch of skinny rectangles with height $f(x)$ and width $\Delta x$. It's an approximation as long as the rectangles have some nonzero width, so we make it exact by taking the limit as $\Delta x \to 0$, and equivalently the number of rectangles goes to infinity. This leaves us with the indeterminate form $0 \times \infty$. Integral calculus is effectively the study of this limit.

Differential calculus is not about infinitesimals or limits. if we think of a smooth function as a curve, that curve has a steepness for every value of the independent variable. (For roads the corresponding term is "grade.") For good reason the value of that steepness is taken to be the slope of the line tangent to that point. The new function that gives the steepness shouldn't be called the "derivative" of the original but we're stuck with the name. The limit is only relevant when we want to actually derive that function from the original function. Similarly, over any interval of the independent variable of a function, there is an area between the function and the axis of the independent variable. The new function that gives that area is called the "integral" of the original function. Once again this is not a descriptive name for what the new function IS and once again the limit is used to derive the integral of a function. We should never confuse a device used to obtain something (a limit in this case) with the thing obtained, in this case two functions (the "derivative" and the "integral") that describe properties of the original function. Abstraction and formality can be egregiously confusing. bs

• Oops, the lower case "bs" at the end shouldn't be there. It is no doubt the work of the monster who has lived under my bed for the last 86 years. Apr 26, 2017 at 0:46

There are several topics of calculus: limits, differentiation, integration, power series, approximation of functions... all are bound together by the concept of INFINITY. That is, they are the result if an infinite process. So I would say that calculus is the study of a particular set of infinite processes that have useful problem solving ability.

The essence of Calculus is the consideration of quantities that are very small such that they are almost zero but not quite. Doing so allows for the calculation of things that occur in an instant, whether an instant of time, or an instant of any dimension. These quantities usually are infinitely small and when they are infinitely large, they are used as their inverse which makes them infinitely small. That is the reason for calculus as conceived by Newton and Leibniz is called infinitesimal calculus. Using infinitesimals explains why Achilles passes the turtle and why the border of a rugged coast has a finite length which otherwise would be infinite. Infinitesimals allow to approximate the probability of an event that changes continuously and gives an exact number which otherwise would be zero. Making a quantity infinitely small forces a sequence when quantities go from being large to being small, therefore the concept of a rate of change is implicit in the conception of calculus. This is applied in derivatives that calculate instantaneous rates of change through differential calculus. Infinitesimals are the basis of calculating the areas under curves that are changing continuously which is the basis of integral calculus. These concepts can be extended to any discipline that deals with events that do not take a discrete value. The numbers of apples in a box is discrete and can be counted using algebra. The exact temperature outside can take infinite values. When we say that is 70 degrees we are using the differential of the temperature reached with the use of calculus.

It's too simple. The derivative of a function gives its rate of change at a given point. The integral of a function gives the area under the curve at a given point. The fundamental theorem merely says that the rate of change of the area under the curve is the value of the function at a given point.

In other words, the derivative of the integral of a function is the function itself. Voila!

• Downvotes by pedantic teachers long on formalism but short on concepts? Apr 26, 2017 at 4:03