# symmetry-translation-tangent-zoom.mp4

Chapter section: Generator of Translations

Caption (verbatim):

A function value is translated by the function's slope

Image description (verbatim):

Tangent approximation under local zoom contact sheet

## Before the animation

Chapter lines 574–574:

We have a function representation. Now we want to find an infinitesimal generator that acts on functions to produce a translation. We can do so by considering what happens to a function's value under "tiny" displacements. In that case, the new value $f(x-a)$ is close to the old value $f(x)$, and correction is given, in the infinitesimal limit, by the slope at $x$. This is the same idea that any curve becomes flat when zoomed in sufficiently:

## After the animation

Chapter lines 582–582:

That is the visual version of the first-order **Taylor expansion** that relates a function's derivatives to its value under translation:

Chapter lines 584–592:

```math
f(x-a)
=
f(x)
-
a\frac{df}{dx}
+
O(a^2).
```