Calculus

Calculus is a branch of mathematics invented in the 17th century by I. Newton and G. W. Leibniz amid controversies of continental proportions. First, there was an acrimonious question of precedence which took a long way to settle. It is now accepted that the two founding fathers made their discoveries independently, with Newton being the first to come up with some basic ideas, while Leibniz was the first to publish his results. To Leibniz, we also owe the common notations nowadays used in Calculus and the term Calculus itself.

Second, the fundamental notion of an infinitely, or indefinitely, small quantity, the infinitesimal, in other words, was assumed to have such strange properties, like sometimes being zero and sometimes being non-zero, that the best contemporary minds refused to accept the emergent methods. The forefathers have been driven by their intuition and the final results of their methods were indisputably correct. However, the methods themselves were suspect.

As an example, let’s find the derivative y’ of y = x². Assuming o to be small, but a non-zero, quantity,

[(x + h)² – x²] / h = (x² + 2hx + h² – x²) / h
= (2hx + h²) / h
= 2x + h.

By setting h = 0 we conclude that y’ = 2x. Which is a meaningful and, in fact, correct result. But, according to Berkeley, it has been derived by both letting h to be zero and non-zero in the same breath. This he justly found utterly objectionable.

So what is Calculus?

Calculus is the branch of mathematics that defines and deals with limits, derivatives and integrals of functions. It is often divided into two parts: Differential Calculus (dealing with derivatives, i.e., rates of change and tangents) and Integral Calculus (dealing with integrals, i.e., areas and volumes). Differential Equations that use methods from both grew into a separate discipline (actually more than just one), as did Calculus of Variations.

To understand what is meant by infinitesimal, consider the formula for the area of a circle: A=πr². The following demonstration is adapted from one given by Professor Steve Strogatz of Cornell, who points out that despite this formula’s simplicity, it is impossible to derive without the utility of infinitesimals.

To start, we recognize that the circumference of a circle divided by its diameter (or twice the radius) is approximately 3.14, a ratio denoted as pi (π). With this information, we can write the formula for a circle’s circumference: C=2πr. To determine a circle’s area, we can start by cutting the circle into eight pie wedges and rearranging them to look like this:

Rearranging eight pie wedges.

We see the short, straight edge is equal to the original circle’s radius (r), and the long, wavy side is equal to half the circle’s circumference (πr). If we repeat this with 16 pieces, it looks like this:

Rearranging 16 pie wedges.

Again, we see the short, straight edge is equal to the original circle’s radius (r), and the long, wavy side is equal to half the circle’s circumference (πr), but the angle between the sides is closer to a right angle and the long side is less wavy. No matter how much we increase the number of pieces we cut the circle into, the short and long sides keep the same respective lengths, the angle between the sides gets progressively closer to a right angle, and the long side gets progressively less wavy.

Now, let’s imagine we cut the pie into an infinite number of slices. In the language of mathematics, the slices are described as “infinitesimally thick,” since the number of slices “is taken to the limit of infinity.” At this limit, the sides still have lengths r and πr, but the angle between them is actually a right angle and the waviness of the long side has disappeared, meaning we now have a rectangle.Rearranging an infinite number of pie wedges.

Calculating the area is now just the length × width: πr × r=πr². This case-in-point example illustrates the power of examining variables, such as the area of a circle, as a collection of infinitesimals.

The study of calculus has two halves. The first half, called differential calculus, focuses on examining individual infinitesimals and what happens within that infinitely small piece. The second half, called integral calculus, focuses on adding an infinite number of infinitesimals together (as in the example above). That integrals and derivatives are the opposites of each other, is roughly what is referred to as the Fundamental Theorem of Calculus. To explore how this is, let’s draw on an everyday example:

A ball is thrown straight into the air from an initial height of 3 feet and with an initial velocity of 19.6 feet per second (ft/sec).

If we graph the ball’s vertical position over time, we get a familiar shape known as a parabola.

At every point along this curve, the ball is changing velocity, so there’s no timespan where the ball is travelling at a constant rate. We can, however, find the average velocity over any timespan. For example, to find the average velocity from 0.1 seconds to 0.4 seconds, we find the position of the ball at those two times and draw a line between them. This line will rise some amount compared with its width (how far it “runs”). This ratio, often referred to as slope, is quantified as rise ÷ run. On a position versus time graph, a slope represents a velocity. The line rises from 4.8 feet to 8.3 feet for a rise of 3.5 feet. Likewise, the line runs from 0.1 seconds to 0.4 seconds for a run of 0.3 seconds. The slope of this line is the ball’s average velocity throughout this leg of the journey: rise ÷ run = 3.5 feet ÷ 0.3 seconds = 11.7 feet per second (ft/sec).

The progress of the vertical position of a ball over time when it is thrown straight up from a height of 3 feet and a velocity of 19.6 feet per second. The average velocity from 0.1 seconds to 0.4 seconds is 11.7 ft/sec.

At 0.1 seconds, we see the curve is a bit steeper than the average we calculated, meaning the ball was moving a bit faster than 11.7 ft/sec. Likewise, at 0.4 seconds, the curve is a bit more level, meaning the ball was moving a bit slower than 11.7 ft/sec. That the velocity progressed from faster to slower means there had to be an instant at which the ball was actually traveling at 11.7 ft/sec. How might we determine the precise time of this instant?

Let’s back up and observe that the span of 0.1 seconds to 0.4 seconds isn’t the only timespan over which the ball had an average velocity of 11.7 ft/sec. So long as we maintain the line’s slope, we can move it any place over this curve and the average velocity over the timespan between the two places the line intersects the curve will still be 11.7 ft/sec. If we move the line farther toward the edge of the parabola, the timespan decreases. When the timespan reaches zero, the points land on the same spot and the line is said to be tangent to (just barely resting against) the parabola. The timespan is described as having been “taken to the limit of zero.”

At the instant of 0.25 seconds, the ball's velocity is 11.7 feet per second.

Here’s where the notion of infinitesimals enters into play. Until this point, we’ve talked about velocity over a finite span of time, but now we’re talking about a velocity at an instant; a timespan of infinitesimal length. Notice how we can’t take the slope between two points that are infinitesimally far apart; we’d have rise ÷ run = 0 feet ÷ 0 seconds, which doesn’t make any sense. To find the slope at any point along the curve, we instead find the slope of the tangent line. The results of six points are plotted below:

Taking the slope of a tangent line at six points to get a derivative.

This graph is what’s known as the original graph’s derivative. In the language of mathematics and physics, it’s said that “the derivative of an object’s position with respect to time is that object’s velocity.”

This process works in reverse, too. The opposite of a derivative is an integral. Thus, “the integral of an object’s velocity with respect to time is that object’s position.” We found derivatives by calculating slopes; we find integrals by calculating areas. On a velocity versus time graph, an area represents a length. The matter of finding areas under a graph is relatively simple when dealing with triangles and trapezoids, but when graphs are curves instead of straight lines, it is necessary to divide an area into an infinite number of rectangles with infinitesimal thickness (similar to how we added an infinite number of infinitesimal pie wedges to get a circle’s area).

Taking the cumulative area under the function at six points to get an integral. Areas below the x-axis (shown in red) are negative, so they decrease the total area.

You might have noticed that this integral graph doesn’t quite give us the same vertical-position graph that we started with. This is because it is just one of many vertical-position graphs that all have the same derivative. A few similar curves are shown below:

Some examples of position curves that all have the same derivative. The desired curve is identified by the initial condition, which is shown as a dotted red circle.

To determine which of these curves will give us the original graph of position, we must also use some knowledge about the position of the ball at a certain time. Examples of this include the height from which it was thrown (the vertical position of the ball at time zero), or the time at which it hit the ground (the time where the vertical position was zero). This is referred to as an initial condition because we’re usually concerned with predicting what happens after, though it’s a bit of a misnomer, since an initial condition can also come from the middle or end of a graph.

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