Pls find attached work.
Hi everyone. I've got a homework problem with Differential Calculus that drives me crazy for a couple of days so far and would like to ask you guys for a few suggestions
"Given two parabolas, (C1): y = x^2 and (C2): y = - (4-x)^2. Find two points, each point on one parabola so that the distance between them is smallest."
What I got so far: Well, I called the point on (C1) (x1, y1), the other point (x2, y2). After that, I set up the distance equation and replace both y(s) with their respective x(s) as follow:
After that I got stuck ... since I don't know how to differentiate an equation with two variables. Here I need to find the minimum of d, which is the distance between the two points.
I wonder if my approach to the problem is wrong or what. Any suggestion is highly appreciated. Thank you. (Also sorry if I confused you guys because I don't know how to type subscript and superscript in the forum yet)
OK I understand what you wrote in the latest attachment. However, in this case, the two parabolas do not appear on straight line. One is x^2 and has its vertex at (0,0) and concave upward, while the other has a vertex at (4,0) and looks downward... so somehow I don't think the distance between the two points is |y2 - y1| ...
I just don't get it at this point. Please bear with me as I'm just in my first semester calculus. Thank you.
there is one more way to go about this. chose a point on the first parabola and fix it. Now try to solve the following one-variable-calculus problem.
Find a point on the second parabola such that the distance between theses two points is minimum.
Note that is a constant now. you have to find in terms of .
[QUOTE=sparkyboy;647734]OK I understand what you wrote in the latest attachment. However, in this case, the two parabolas do not appear on straight line. One is x^2 and has its vertex at (0,0) and concave upward, while the other has a vertex at (4,0) and looks downward... so somehow I don't think the distance between the two points is |y2 - y1| ...
I just don't get it at this point. Please bear with me as I'm just in my first semester calculus. Thank you.[/QUOTE
Pls find attached graph.
Umm ... I still don't understand, would you please tell me why? Thank youYES, "first find the shortest distance from a point to a curve". this expression of shortest distance will be completely the function of x_1. minimize this expression too! and that will be the answer. (do you see why?)
Thank you for the graph. However, I'm still confused. What if the first point is (-2, 4) and the second point is (3, -1)? Then the distance cannot be a straight line right? I mean I still don't get why the distance is just simply |y2 - y1|. Really sorry about my dumbness but please bear with me. Thank you.