How to use the graph of the distance between any two points

[Juwane]

Homework Statement

Write an expression for the distance between the point P(1,2) and an arbitrary point $$( x, \sqrt{x} )$$ on the curve $$y = \sqrt{x}$$. Graph this distance versus x, and use the graph to find the x-coordinate of the point on the curve that is closest to the point P.

Homework Equations

N/A

The Attempt at a Solution

Well, here's the expression I wrote for the distance:

$$d = \sqrt{ (x - 1)^2 + ( 2 - \sqrt{x} )^2 }$$

I've graphed this on a graphic software. Now, how can I use this graph to answer the question? What do I have to look for on the graph?

 

[Mentallic]

Since this is the in the pre-calculus forum I'll assume you won't be using derivatives, so you'll be finding an approximate solution to the shortest distance.

When you graphed the distance versus x, at what (approx) x value is the distance the shortest? In other words, where is d the smallest?

 

[Juwane]

d is smallest when x is approx equal to 1.35296

The answer is correct as given at the back of the book.

Yes, I didn't want to use derivatives for this question; but if I were to use derivatives, I would have differentiated the function and equated it to zero, and then would've solved for x, right?

EDIT: Also note that in the graph there's only one minimum and no maximum extrema, so differentiating twice won't be necessary.

 

[Mentallic]

Yes that's correct

May I ask how you found that answer with such precision?

 

[Juwane]

Yes that's correct

May I ask how you found that answer with such precision?

I used a graphing software to create the graph, and then I zoomed in many times on the part of the graph where d was the smallest. I used http://www.walterzorn.com/grapher/grapher_app.htm".