![]() Having them in this explicit form isn’t critical, but we will assume $y$ is locally a differentiable function of $x$. Let the two curves be given by $y=f(x)$, $y=g(x)$. ![]() ![]() It also assumes differentials are covered in Calculus I. 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. The concept of 'slope' is very vague when you go dimensions higher than 2D. Think of a 3D line and a parabola when both are not in the same plane. To generalize this by stating that 'at minimum distance the slope is parallel', this might work for 2D but not in higher dimensions or between a curve and surface. See, even though you missed one solution, mathematics guided us to both the solutions. The corresponding pints on curves in cartesian co-ordinate are $$ and $$ on respective curves. Substitute $t=1$ in second equation to get $s=0$. Which is not possible as $s$ is a real valued parameter. (This, of course, has the solution of $\sqrt=0$. no tricks allowed that take advantage of the symmetry/simplicity of these specific curves):įind the minimum distance between the curves $xy=1$ and $y=-x$. And perhaps that's the true nature of this type of problem and so it just can't be done in Calculus 1.Īlternatively, feel free to solve the following problem using only Calculus 1 optimization techniques that would generalize to slightly more difficult curves (i.e. The justifications I've been able to find so far rely on Vector and/or Multivariable Calculus. Minimum distance between curves occurs when the slopes are equal? Other types of optimization problems that commonly come up in calculus are. In this video, we'll go over an example where we find the dimensions of a corral (animal pen) that maximizes its area, subject to a constraint on its perimeter. Is there a Calculus-1-level justification that the Optimization, or finding the maximums or minimums of a function, is one of the first applications of the derivative you'll learn in college calculus. This fact is visually intuitive, but I'd like to establish it more algebraically. However I can't seem to justify that it should occur when the slopes of the two curves are parallel. I'd like to extend this idea and be able to compute the minimum distance between two (smooth and non-intersecting) curves. In the optimization section of Calculus 1 a common problem is to find the minimum distance between a curve and a point.
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