= Calculus
{wiki}
Calculus connects \b[local change] with \b[accumulated change]. A <derivative> asks how fast a <function> changes near one input; an <integral> adds its contributions across an <interval>. The <fundamental theorem of calculus> explains why these operations fit together: accumulating a continuous rate produces a quantity whose <derivative> is that rate.
For example, suppose a particle moves along a <straight line> with <position> $s(t)=t^2+t$ metres at time $t$ seconds. Its <velocity> is $s'(t)=2t+1$ metres per second. Between $t=0$ and $t=3$, adding the <velocity> over time gives
$$
\int_0^3(2t+1)\,dt=s(3)-s(0)=12\text{ metres}.
$$
Here the <velocity> stays positive, so the <displacement> also equals the distance travelled. If the particle reverses direction, the <integral> of its signed <velocity> still gives <displacement>, while the <integral> of its <speed> gives total distance.
This is useful because local rules are often easier to specify than whole trajectories. A <differential equation> can prescribe a particle's <velocity> or a population's growth rate; <integration> then reconstructs its evolution, subject to an <initial condition>. In the other direction, a <derivative> extracts local information from a known <function>, and <optimization> uses that information to locate or approach promising inputs. The same two operations therefore connect measurement, prediction and design.
The visual connection is developed in <derivative>, <integral> and <fundamental theorem of calculus>. The definitions depend on <limits>, which turn progressively finer finite measurements into exact mathematical statements. A graph with a corner need not have a <derivative> there, and the conditions permitting <integration> or interchange of <limits> matter; calculus is more than a collection of symbolic rules.
Historically, Isaac Newton described changing quantities and their rates in the language of fluents and fluxions; his manuscript \i[History of the Method of Fluxions] records that approach.https://www.newtonproject.ox.ac.uk/view/texts/normalized/NATP00356[Isaac Newton, History of the Method of Fluxions, Cambridge University Library MS Add. 3968]{ref} Gottfried Wilhelm Leibniz's \i[Nova methodus pro maximis et minimis] appeared in \i[Acta Eruditorum] in October 1684, pp. 467–473; the original article is preserved in the HKUST Library collection.https://digitalcollections.hkust.edu.hk/Detail/objects/b540295[Gottfried Wilhelm Leibniz, Nova methodus pro maximis et minimis, 1684]{ref} The modern <limit> definitions used here should not be read back into those seventeenth-century formulations.
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