# Point on a line closest to the origin

• Nov 18th 2007, 05:21 PM
angel.white
Point on a line closest to the origin
On my last calc test, they asked f(x)=10x+5, what point in the function is closest to the origin?

I solved it by finding the normal line which intersects the origin, then finding where that line intersects f(x). (We haven't gotten the tests back yet, so I don't know how if that worked)

Anyway, this didn't involve any calculus, so I'm concerned that the problem was written wrong, and was supposed to be something like $10x^{3}+5$.

This question, I am not sure how to solve, so can anyone help me figure out how I would find the point on the function $f(x)=10x^{3}+5$ which is closest to the origin?

And if this method does not require normal lines, could you also explain to me how I would find the normal line for such a function? (I have a theory that it would be $-\frac{1}{30x^{2}}$ but that's just a theory.)
• Nov 18th 2007, 05:32 PM
Jhevon
Quote:

Originally Posted by angel.white
On my last calc test, they asked f(x)=10x+5, what point in the function is closest to the origin?

I solved it by finding the normal line which intersects the origin, then finding where that line intersects f(x). (We haven't gotten the tests back yet, so I don't know how if that worked)

Anyway, this didn't involve any calculus, so I'm concerned that the problem was written wrong, and was supposed to be something like $10x^{3}+5$.

This question, I am not sure how to solve, so can anyone help me figure out how I would find the point on the function $f(x)=10x^{3}+5$ which is closest to the origin?

And if this method does not require normal lines, could you also explain to me how I would find the normal line for such a function? (I have a theory that it would be $-\frac{1}{30x^{2}}$ but that's just a theory.)

this is an optimization problem. what you are really being asked to do is to minimize the distance function between the two points (0,0) and (x, 10x + 5)
• Nov 18th 2007, 06:51 PM
angel.white
Quote:

Originally Posted by Jhevon
this is an optimization problem. what you are really being asked to do is to minimize the distance function between the two points (0,0) and (x, 10x + 5)

My solution I used on the test was that the normal line which intersects (0,0) is $y=-\frac{1}{10}x$ Hmm, well I got the same answer either way, so hopefully my instructor won't take off points for not using calculus to solve it :/

Anyway, how would I find the normal line for:
$f(x)=10x^{3}+5$?
• Nov 18th 2007, 06:57 PM
Jhevon
Quote:

Originally Posted by angel.white
Hmm, perhaps you can help me figure out where I am going wrong.

My solution I used on the test was that the normal line which intersects (0,0) is $y=-\frac{1}{10}x$ Hmm, well I got the same answer either way, so hopefully my instructor won't take off points for not using calculus to solve it :/

Anyway, how would I find the normal line for:
$f(x)=10x^{3}+5$?

well, we know that $f'(x) = 30x^2$ gives the slope for the tangent line.

this means that the slope for the normal line is given by $- \frac 1{30x^2}$ as your theory was (the normal line is a line perpendicular to the tangent line, and so their slopes are the negative inverses of each other).

so now the challenge is to find the line with that slope going through the origin to our curve. which shouldn't be too hard i think
• Nov 18th 2007, 08:12 PM
angel.white
Quote:

Originally Posted by Jhevon
well, we know that $f'(x) = 30x^2$ gives the slope for the tangent line.

this means that the slope for the normal line is given by $- \frac 1{30x^2}$ as your theory was (the normal line is a line perpendicular to the tangent line, and so their slopes are the negative inverses of each other).

so now the challenge is to find the line with that slope going through the origin to our curve. which shouldn't be too hard i think

Hmm, I'm actually not sure how to do that, it has a vertical asymptote at x=0, so moving it up and down won't help, and I can't figure out how to move it left/right without changing the slope.

However, your suggestion earlier will obviously work to figure out the closest distance between the origin and the line, so if he pops this question on a quiz, I'll be fine, won't need the normal line to solve it.
• Nov 18th 2007, 08:27 PM
Jhevon
Quote:

Originally Posted by angel.white
Hmm, I'm actually not sure how to do that, it has a vertical asymptote at x=0, so moving it up and down won't help, and I can't figure out how to move it left/right without changing the slope.

However, your suggestion earlier will obviously work to figure out the closest distance between the origin and the line, so if he pops this question on a quiz, I'll be fine, won't need the normal line to solve it.

yeah, go the optimization route. i think i see a way to solve it using the normal line, but i end up with a quartic, which is too much trouble to go through for something like this