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12 votes

Distance Between Curves in First-Semester Calculus

How about this? Say the two curves are $A$ and $B$, with the min distance achieved at point $a \in A$ and $b \in B$. Let me assume both curves are smooth, or at least have 1st derivatives everywhere. ...
Joseph O'Rourke's user avatar
5 votes

Distance Between Curves in First-Semester Calculus

[T]o justify that it should occur when the slopes of the two curves are parallel: If $ab$ is a minimum, then the curve through $b$ has no points in the interior of the circle centered at $a$ with ...
user1815's user avatar
  • 5,595
5 votes
Accepted

Ideas for teaching a bit of linear optimization to advanced undergraduates?

Four things come to mind. 1) You could focus on how the linearity of the objective function and the convexity of the feasible region yields that local optima are global, and how the linear nature of ...
John Coleman's user avatar
  • 1,536
3 votes

Distance Between Curves in First-Semester Calculus

I would be more careful with the phrasing of the question. Here are some points to explain why. The first point is an answer under certain conditions. When looking for the distance between a point ...
Ferenc Beleznay's user avatar
3 votes

Ideas for teaching a bit of linear optimization to advanced undergraduates?

I have not tried anything similar myself, but if I had to, I would start by looking at Gilbert Strang's book "Introduction to applied mathematics" in chapter 8 "Optimization" (8.1 is "Introduction to ...
Dirk's user avatar
  • 2,991
3 votes

Good examples of non-convex optimization

Maybe consider a sinusoid(ish) curve superimposed on a general trend. A place where you can see this is in business (e.g. volumes of production) where there is some general trend over time but there ...
guest's user avatar
  • 111
2 votes

Ideas for teaching a bit of linear optimization to advanced undergraduates?

One of my classes actually has linear optimization on their corriculum (Switzerland, Berufsmatur). I teach according to the book "Mathematik in der Berufsschule". It's focused on students who want to ...
Richard Conrardy's user avatar
2 votes

Ideas for teaching a bit of linear optimization to advanced undergraduates?

Just an idea: If your students know some statistics, including least squares regression, you could introduce optimization via some machine learnings methods, such as the lasso. The lasso can be ...
kjetil b halvorsen's user avatar
2 votes

Distance Between Curves in First-Semester Calculus

Let $h: \mathbb{R} \to \mathbb{R}$ be a function. Let $(a,b)$ be an arbitrary point in $\mathbb{R}^2$. We want to minimize the distance from the point $(a,b)$ to the graph of $h$. This is a single ...
Steven Gubkin's user avatar
1 vote

Favorite linear programming (not integer) examples?

[Chris, convert to comment, if you deem. /Not cookie mousing.] I think you are best off for an intro to do an example that is extremely simple and toy-like and might even be simply solved by high ...
guest troll's user avatar
1 vote

Textbooks for Non-Convex Optimization

Approaches to global optimization problems are broadly divided into deterministic methods (with some convergence guarantee, but requiring some assumptions on the objective function and constraints) ...
Brian Borchers's user avatar
1 vote

Distance Between Curves in First-Semester Calculus

I like the geometric approaches of Joseph O’Rourke and my other answer, but here is an algebraic-calculus approach that is Calculus-I level except for the number of variables. It also assumes ...
user1815's user avatar
  • 5,595
1 vote

Distance Between Curves in First-Semester Calculus

So, you want to find the minimum distance between two curves in the most general form using calculus and optimization. The curves you are interested at this particular example are: \begin{equation} xy=...
Subhendu Chakraborty's user avatar

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