Step By Step Calculus » 4.0 - History and Applications

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History and Applications
Leibniz, in 1694, first used the word “function” as a mathematical term to describe a quantity related to a curve, such as a curve’s slope at a specific point. All the functions considered by Leibniz were differentiable functions. Later, in the mid-18th century, Euler used the word “function” to describe an expression or formula involving various arguments, e.g. f(x)= x^2+\cos(x).f(x)= x^2+\cos(x).
Towards the end of the 19th century, mathematicians began a program of developing the foundations of mathematics, starting with the theory of sets. It is Dirichlet and Lobachevsky who are credited for independently giving the modern “formal" definition of a function as a relation in which every element in a given set is assigned a single image element in a second set.
Functions are often used to represent physical quantities such as velocity, temperature, pressure, or voltage. For example, the instantaneous velocity of a rocket as a function of the amount of fuel burnt, the atmospheric pressure as a function of height, a population as a function of time, etc.