The Calculus of Infinitesimals

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Calculus is often used as the complete name of a branch of mathematics dealing with rates of change and very small numbers. In fact, a calculus is a specific discipline or method for the analysis of some set of problems. The fact that the calculus of infinitesimals has come to be known in this way is a testament to its importance.

Functions

To understand calculus, it is critical to understand the derivative. Let us begin with the definition of a function.

$y=f(x)$

This expression defines a relation between two quantities x and y. It can be said that y is given in terms of x. Specifically, we have defined a function f that provides a mapping for values of x. The actual mapping of a given function varies; what follows is one example.

$f(x)=ax$

In this example, the value of the function is said to be equal to the product of its argument and a constant a. Take note of the fact that this is a constant multiple, as it means that the function is said to be linear--an important property. Evidently, $y=f(x)=ax\Rightarrow y=ax$ by the transitive property. Here, the relation between y and x becomes clear; y is greater than x by a multiple a. Observe the pattern that forms when evaluating f at various x.

$f(0)=a\cdot0=0$

$f(1)=a\cdot1=a$

$f(2)=a\cdot2=2a$

$f(3)=a\cdot3=3a$

As x increases by increments of 1, y increases by increments of a. This gives rise to the familiar slope-intercept form of a line, $y=mx+b$. m, here, is the slope of the line, the ratio between the growth of y and x, or the rate of change of the function mapping the two. Since b is a constant term, it has no impact on the rate of change of the function. It is this understanding that we will build upon. Consider two points in the form (x, y): (0, 0) & (3, 9). As x increases by 3, y increases by 9. We refer to these as delta x and delta y. We obtain the rate of change as follows:

$\Delta x = 3,\ \Delta y = 9$

$m = \frac{\Delta y}{\Delta x} = \frac{9}{3} = 3$

The implication of this is that a line drawn between the two points would have a slope of 3. We note two functions upon whose graphs these points fall: $y=3x$, and $y=x^2$. If graphed, the path of $y=3x$ would appear identical to a straight line drawn between the points, except that its line would continue infinitely in either direction. Recall that it is a linear function, and so every segment of its path has the same inclination as the whole. Compare this to $y=x^2$, a parabolic curve. Though both points fall upon its graph, the path it takes between them is not linear, but curved. It appears to follow a much less direct route. However, as each function has a slope of 3 over the interval [0, 3], we say that they share the same average rate of change over that interval. The linear function appears to move towards its destination faster in the start, but as the parabola steepens, it catches up, such that both functions reach (3, 9). If we continued observing this competition, the parabola would continue accelerating more and more rapidly, its value rapidly eclipsing the straight line.

Suppose we wished to know the slope of a function not over an interval but at a point. This is easily accomplished with a linear function--it is observed that the slope of a linear function is the same over any interval, and so it is reasonable to extend that to a single point. It is more difficult, however, with non-linear functions. Take $y=x^2$ on [0, 1]:

$f(x) = x^2,\ x = 1,\ \Delta x = 1$

$f(1) = 1,\ f(1+\Delta x) = 4,\ \Delta y = 4-1 = 3$

$m = \frac{\Delta y}{\Delta x} = 3$

Compare to the interval [0, 0.5]:

$\Delta x = 0.5$

$f(1+\Delta x) = 2.25,\ \Delta y = 1.25$

$m = \frac{\Delta y}{\Delta x} = 2.5$

Or to [0, 0.25]:

$\Delta x = 0.25$

$f(1+\Delta x) = 1.5625,\ \Delta y = 0.5625$

$m = 2.25$

As the interval becomes smaller, m decreases. One could use a computer to perform this calculation with increasingly small values, and would find that m gets very close to 2. As we can see, when moving from [0, 1] to [0, 0.5], m moved half of the way to 3. On the next interval, it moved half of the distance that remains, again. This would continue infinitely, with m never reaching 2. A new calculus is needed to precisely describe the slope of non-linear functions at a point. This value, the rate that the function will change over the next infinitesimally small interval, is known as its instantantaneous rate of change at a point.

Limits

The limit of a function is the value L it approaches as its argument approaches c.

$\lim_{x\to c} f(x) = L$

Limits enable us to make statements about the behaviour of functions near a point, even when the behaviour differs at that point. For example, 1/0 is undefined, but $\lim_{x\to 0} \frac{1}{x} = \infty$. This is fairly self-evident--division by an infinitely small number will produce an infinitely large one. The limit is formally defined as follows:

$\lim_{x\to c}f(x)=L\leftrightarrow\forall\epsilon\in\mathbb{R} : \epsilon>0\ \exists\delta\in\mathbb{R} : \newline|f(x)-L|<\epsilon\leftarrow 0 < |x-c| < \delta$