Habluka's Weekly Challenge #2

[Roflcopter]

I'm kind of disappointed that no answers have been posted so far for at least the first question after the fast responses from last week, so here's a hint: use trapezoids

I guess these were a little difficult though since most people probably haven't taken calculus or linear algebra though, but at least try to work through them.

I've taken multivariate calculus and linear algebra but the first question is rather ambiuous and the second is just a definition.

For instance you say "there is a parametric equation (x1(t), y1(t)) where for each value of t, (x1(t), y1(t)) is the end point of a perpendicular line segment to the first equation with a starting point (x(t), y(t)) and a length d". I think what you're saying is that we're to take a straight line segment with start (x, y) and end (x1, y1) such that this is perpendicular to the original curve and has length d. But there are two such pairs of functions (x1, y1) that would satisfy this (one "under" and one "above").

I'm also not comfortable that I understand what you mean by trace "the movement of the line segment[s]".

Also see my earlier question where you don't state that we're working in R^2 but seem to assume it by giving us that V has a direction θ.


Anyway here is the argument I think you're looking for:

1. Put .

What you're asking us to compute if I understand correctly (I had to just provide links in a code block or something breaks them):

[SELECT ALL] Code:

http://latex.codecogs.com/png.latex?\lim_{\Delta%20\to%200}%20{\sum_{x%20\in%20I}^{%20}%20\frac{[y%28x+\Delta%29-y%28x%29]d+[y_1%28x+\Delta%29-y_1%28x%29]d}{2}}
http://latex.codecogs.com/png.latex?=%20\frac{d}{2}%20\lim_{\Delta%20\to%200}%20{%20\sum_{x%20\in%20I}^{%20}%20{y%28x+\Delta%29-y%28x%29+y_1%28x+\Delta%29-y_1%28x%29}}
http://latex.codecogs.com/png.latex?=%20\frac{d}{2}%20\lim_{\Delta%20\to%200}%20{%20\sum_{x%20\in%20I}^{%20}%20\frac{y%28x+\Delta%29-y%28x%29+y_1%28x+\Delta%29-y_1%28x%29}{\Delta}%20\Delta}
http://latex.codecogs.com/png.latex?=%20\frac{d}{2}%20\lim_{\Delta%20\to%200}%20{%20\sum_{x%20\in%20I}^{%20}%20[y%27%28x%29+y_1%27%28x%29]%20\Delta}
http://latex.codecogs.com/png.latex?=%20\frac{d}{2}%20%28y+y_1%29

And you can see where this is going...

2. See definition 1.1.