# Differentiable Function

Awesome Fractals by Silvia Cordedda -- For the ones not familiar with fractals, a fractal is an image built with math, a repetition of the same geometric module over and over, with different dimensions, according to a mathematical function.

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Theory and Applications of Differentiable Functions of Several Variables by S. M. Nikolskii Download

graph of $f(x)$ and $f'(x)$ Can there be an injective function whose derivative is equivalent to its inverse function?

This activity engages the students in studying several continuous, non-continuous, differentiable, and non-differentiable functions to decide what conditions must be met to guaranteed the conclusions of the mean value theorem.

Dirichlet’s function is nowhere continuous and nowhere differentiable. It is also nowhere Riemann integrable since its upper integral and lower integral do not equal anywhere.

DERIVATIVE ~ m = ∆y / ∆x ~ where the symbol ∆ (Delta) is an abbreviation for 'change in' [*not* just 'change']

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Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by German mathematician Emmy Noether in 1915 and published in 1918.[1] The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action…

Before jumping on to the applications of Rolle’s theorem let us study its definition. Rolle’s theorem simply states that if a function f is differentiable in the open interval (a, b) and continuous in the closed interval [a, b] and if it also attains equal value at two distinct points, i.e., f(a) = f(b), then there is at least one point c between a and b where the first derivative of the function ( the slope of the tangent line to the graph of the function) is zero.

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