Mathematics · Ch 9 — Differential Calculus – Limits and Continuity
Introduction
Introduction
Calculus: The Mathematics of Change
Calculus is fundamentally the mathematics of change, and its reach extends across virtually every branch of science and social science. Whenever an observer wants to know not just the value of a changing quantity but how fast it is changing — the rate of change — calculus is the tool required, because most changing quantities in nature do not change at a constant rate.
Different disciplines encounter this same underlying idea from different angles: a mathematician studying a curve is interested in measuring how sharply it bends away from a straight line at a point on it; a physicist studies velocity (the rate of change of displacement); a chemist studies the rate of a chemical reaction that turns one or more reactants into one or more products; a biologist studies the rate of growth of an animal or plant population, or the rate at which blood flows through a vein or artery (and which part of the vessel carries the fastest or slowest flow); an economist studies marginal demand, marginal revenue and marginal profit; a geologist studies the rate at which an intruded body of molten rock cools by conducting heat into the surrounding rock; an engineer studies the rate at which water flows into or out of a reservoir; an urban geographer studies the rate at which population density in a city changes as the city expands; a meteorologist studies the rate of change of atmospheric pressure with height; and a psychologist studying a "learning curve" is interested in the rate at which performance improves with practice. Even something as ordinary as the eye's pupil widening in a darkened room and contracting in a bright one is, at heart, governed by a limiting process.
Velocity, density, current, power and a temperature gradient in physics; rate of reaction and compressibility in chemistry; rate of growth and blood velocity in biology; marginal cost and marginal profit in economics; rate of heat flow in geology; rate of improvement of performance in psychology — every one of these "rates" is, mathematically, the same concept: the derivative. This is exactly why calculus is powerful — a single abstract idea, once its properties are worked out once, applies unchanged to every one of these situations, instead of each science needing to invent its own separate machinery. One of the greatest creations of the ancient world was Euclidean geometry, and that monumental achievement was not matched in importance again until the discovery of calculus, almost two thousand years later.
A brief history. Calculus was developed independently, in the last quarter of the seventeenth century, by Sir Isaac Newton in England and Gottfried Wilhelm Leibnitz in Germany. Newton had absorbed the mathematics of his time (Euclid's Elements, Descartes's geometry, and the work of Galileo and Fermat) and by 1665 began studying rates of change of continuously varying quantities — what he called "fluxions" — which is essentially what we today call differential calculus. Leibnitz, publishing independently through the journal Acta Eruditorum which he co-founded, is credited with introducing much of the notation still used today; the near-simultaneous discovery led to a long and bitter priority dispute between the two camps. It took roughly another century and a half before the logical foundations of calculus were made fully rigorous: Augustin-Louis Cauchy, in his 1829 lecture notes, gave the first reasonably precise definition of a limit and defined the derivative as the limit of a difference quotient; a little later the German mathematician Karl Weierstrass supplied the precise "epsilon–delta" formulation of limit, continuity and differentiability that underlies the modern treatment (and that the informal definitions in this chapter are a stepping-stone towards).
What is calculus? In one line, calculus is the mathematics of the rate of change of quantities. It is also the mathematics of tangent lines and slopes, of areas and volumes, of arc lengths, centroids and curvatures — a family of concepts that has let scientists, engineers and economists model real-life situations that pre-calculus mathematics simply could not reach.
What makes calculus different from pre-calculus mathematics. Pre-calculus mathematics is essentially static — a constant velocity, the slope of a straight line, the area of a rectangle, a line tangent to a circle — while calculus is dynamic: the velocity of an accelerating body, the slope of a curve at a single point, the area under a curved boundary, a line tangent to an arbitrary smooth curve. What makes each of these dynamic problems solvable is the same three-stage strategy: start from an ordinary pre-calculus quantity, pass it through a limiting process, and arrive at a genuinely new calculus quantity (a derivative, or later an integral). This chapter is devoted entirely to building that middle stage — the limit — since it is the foundation on which the derivative (taken up in the next chapter) is built. Because differentiation has not yet been introduced, every limit encountered in this chapter must be evaluated using purely algebraic and geometric reasoning: direct substitution, factoring and rationalising, the algebra-of-limits theorems, the standard trigonometric/exponential/logarithmic limit results, and the Sandwich (Squeeze) Theorem — never a differentiation-based shortcut.