Dot product of the tangent vector, right? So above ##(p-X(t_0))## is the vector between point and curve and ##X'(t_0)## is the tangent vector. But I don't see why should ##p\dot X'(t_0)-X(t_0)X'(t_0)=0##
Isn't that the dot product is zero? That or that the slopes of the tangent lines are inverse reciprocals of one another. But I don't see how the latter definition can be applied...
Given a parametrized curve ##X(t):I\to\mathbb{R}^2## I am trying to show given a fixed point ##p##, and the closest point on ##X## to ##p##, ##X(t_0)##, the line between the point and the curve is perpendicular to the curve. My only idea so far is to show that...
Assume that ##\{f_n\}## is a sequence of monotonically increasing functions on ##\mathbb{R}## with ## 0\leq f_n(x) \leq 1 \forall x, n##. Show that there is a subsequence ##n_k## and a function ##f(x) = \underset{k\to\infty}{\lim}f_{n_k}(x)## for every ##x\in \mathbb{R}##.
(1) Show that some...
Homework Statement
Warning: I realize the title is misleading... the function itself isn't what's constant.
Mod note: Edited to fix the LaTeX
If ##f## is a continuous at 0 such that ##\lim_{x \to 0}\frac{f(x)-f(g(x))}{g(x)}=M##, where ##g(x)\to 0## as ##x\to 0## does this generally mean that...