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 .
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 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 but that's just a theory.)
well, we know that gives the slope for the tangent line.
this means that the slope for the normal line is given by 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.