[Introductory Calculus] Applied Optimization; Find the closest point to (3,0) on y=(x

[zimmertr]

Hello, I've been assigned two calculus problems and have completed both of them. I'm pretty sure the first one is correct but I'm iffy on the second one. I would really appreciate it someone here could check my work on the second problem, and maybe even on the first problem if they have the time.

The problem in question is: "Use calculus to find the point (x,y) on the parabola traced out by y=x2that is closest to the point (3,0)."

Here is a copy of my work: https://www.dropbox.com/s/8jp4ad5qdeo4wtv/Miniproject.pdf?dl=0

 

[ksdhart]

In the first problem, most of your work seems good, but you seem to have made a small error when taking the derivative. The derivative is correct up until this point:

\(\displaystyle L'\left(x\right)=\frac{2x-200}{2\sqrt{x^2-200x+13600}}+\frac{2x}{2\sqrt{x^2+6400}}\)

Then you try to cancel out the 2 in both denominators, but you never divide the numerators by 2. Try fixing that slip-up and see if the answer you get feels a bit more certain for you.

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As for the second problem, you can tell your answer is wrong immediately because the point (3,1) is not on the parabola y=x². That said, where you went wrong is when you substituted in a value for y. In your work you say, "In order to minimize the length, we’ll substitute the square of the length in to the function as y and solve for x." But I don't know why you're doing that. You already have a value for y, as given by the parabola. That's the value you should be substituting in for y. Like so:

\(\displaystyle L=\sqrt{\left(x-3\right)^2+y^2}\)
\(\displaystyle L=\sqrt{\left(x-3\right)^2+\left(x^2\right)^2}\)
\(\displaystyle L^2=S=\left(x-3\right)^2+x^4\)

You should be able to continue from there.

 

[pka]

The problem in question is: "Use calculus to find the point \(\displaystyle (x,y)\) on the parabola traced out by \(\displaystyle y=x^2\) that is closest to the point \(\displaystyle (3,0)\).

L=\sqrt{\left(x-3\right)^2+y^2}[/tex]
\(\displaystyle L=\sqrt{\left(x-3\right)^2+\left(x^2\right)^2}\)

I am confused by what both of you are doing. You need to minimize \(\displaystyle L\).
To do that solve \(\displaystyle 2(x-3)+4x^3=0\). That is the numerator of the derivative of \(\displaystyle D_x(L)\).