What is the definition of a cusp? - Mathematics Stack Exchange
A cusp is a point where you have a vertical tangent, but with the following property: on one side the derivative is +∞ + ∞, on the other side the derivative is −∞ − ∞. The paradigm example was stated above: y =x2 3 y = x 2 3. The limit of the derivative as …
How to tell if a function has a cusp without a graph?
Jul 21, 2016 · 1. I need to be able to identify if a function is indifferentiable at any point. The common way to do that is to actually determine the derivative and inspect it for singularities. This is generally easy with elementary functions. In your example: f(x) =x2 3 f (x) = x 2 3. f′(x) = 2 3x−1 3 = 2 3 x−−√3 for x ≠ 0 f ′ (x) = 2 3 x − ...
cusp vs. corner? or both? - Mathematics Stack Exchange
Sep 14, 2017 · Cusp: where the slope of the tangent line changed from -infinity to +infinity (or the other way around) Corner: left-sided and right-sided derivatives are different. And I saw a problem which was asking if there is a corner or a cusp given a graph. The graph looked like: f(x)=-x, if x<0. =sqrt(x), if x>=0.
Notion of cusp in algebraic geometry - Mathematics Stack Exchange
Nov 20, 2020 · Cusps are defined in a rather roundabout manner in the exercises of Chapter I Section 5. Basically, a curve has a cusp at the origin if its Taylor expansion looks like y2 − x3 + (higher-order terms), which is precisely your curve in the chart z = 1. – Tabes Bridges. Nov 20, 2020 at 17:36. 1.
Definition of cusp or corner - Mathematics Stack Exchange
Feb 5, 2018 · This leads us to: Definition 1 (used by some authors): a cusp is precisely a point where a curve is equivalent to y3 = x2 y 3 = x 2 via a smooth local change of variables. Definition 2 (more general): A piecewise smooth curve in, say, a euclidean space: f: R → Rn f: R → R n. has a cusp at a point p = f(x) p = f (x) if there is ε> 0 ε> 0 ...
what does cusp form mean? - Mathematics Stack Exchange
Nov 28, 2013 · A cusp form is a modular form which vanishes at the cusps. As for visualization, well... below, you'll find a rendering of the real part of the simplest cusp form of all, the modular discriminant Δ(q) Δ (q). Beautiful, but singularly unhelpful as far as intuition goes.
How "sharp" does a cusp have to be in order for the equation to …
Dec 16, 2014 · A cusp is where the one-sided derivatives tend to opposite infinities. Either the left derivative is ...
Should a cusp or corner have infinite many tangents?
Oct 11, 2021 · As we know a corner is where you have two distinct tangent lines and a cusp is where you have one tangent line which is vertical. I realized that a cusp and corner can have infinitely many tangent lines. See the image below of what I mean. I draw vertical lines at the corner and cusp. The first left graph is y =|x| y = | x | and the the right ...
Why non-regular curves have cusp (s) - Mathematics Stack Exchange
Sep 13, 2015 · 2. Not quite. If γ′(t) ≠ 0, that does imply there is no kink there. If γ′(0) = 0, then that doesn't necessarily mean there is a kink there. Maybe there is, maybe there isn't. And no, a regular curve does not have a cusp. A cusp necessitates that the point tracing it slows down to a stop for a moment, which would make γ′(t) = 0, which ...
geometry - What is the definition of a cusp of a plane curve and …
Oct 8, 2021 · For a cusp, you want the two (local) branches of the curve to have the same tangent line at the singular point (so compute left- and right-hand limits of the derivative). I don't understand your sentence "So I think the cusps are indeed singular points, since this curve is everywhere differentiable."