Taylor series

Inspect the first three terms of the Taylor series of any function.

Tags: Taylor-Series, calculus, derivative, differentiation

Taylor series

Look at the first second or third order Taylor Series Approximation of a function.

Tags: taylor-series, calculus, approximation, function

Caustic curve as focus of parabola

The red curve is f(x) (for -2≤x≤2).  The blue curve is the terms up to the square term of the Taylor series.  A is the vertex of this parabola, B is its focus. The orange curve is the caustic - envelope of the reflection of vertical rays reflected in a mirror aligned with f(x). Observe that the focus of the parabola traces the caustic.

Tags: caustic, Taylor-series, parabola