Question

Calculus help please?

Find the point on the parabola y^2 = 2x that is closest to the point (1, 4).

Answer 1

so I have to take the derivative of the distance formula right?
which would be
http://www.wolframalpha.com/input/?i=root%28+%28x-1%29%5E2+%2B+%28+%282x%29%5E%281%2F2%29+-+4%29%5E2+%29

Answer 2

but I was supposed to take the derivative by hand... but I had a brain fart... HELP?!

Answer 3

y^2=2x => x=(1/2)y^2

any point on the parabola has the form (x, y), that is (0.5y^2, y)

We may as well minimise the square of the distance, as it is easier to calculate. We can do this because distance is positive, suppose the distance be d, then:

d^2 = (0.5y^2 - 1)^2 + (y - 4)^2

which is the formula for distance. Now expand the right-hand side.

d^2 = 0.25y^4 -y^2 + 1 + y^2 - 8y +16
d^2 = 0.25y^4 - 8y + 17

minimise by setting the derivative equal to zero.

y^3 - 8 = 0

So y =2

substitute this back into your formula for distance

d = sqrt[ (0.5y^2 - 1)^2 + (y - 4)^2 ]

d=sqrt[ 1 + 4 ] = sqrt(5).

so sqrt(5) is the final result xD

Answer 4

I hope I'm not wrong :P I suck at calculus xDD

Answer 5

thanks :)
can't be worse than me ;) got a 62% on this test :P

Answer 6

xDD I had 70% marks in my multi variable calculus & 75% in calculus 1

Answer 7

erm @annas how did ya get from
d^2 = 0.25y^4 - 8y + 17
to y^3 - 8 = 0
wasn't there a d^2 on the left? :3
so would you derive implicitly or something?

Answer 8

hmm I've no idea lol I just remember my instructor told me this ^_^ lol
told you I'm pretty bad at calculus >.>

Answer 9

mk then :P

Answer 10

lol wait I've called the expert ^_^

Answer 11

nevermind, it still works ;)

Answer 12

okie dokie ^_^

Answer 13

@annas
part b: verify that the tangent to the parabola y^2 = 2x at your answer is perpendicular to the line connecting your answer to the point (1, 4)

Answer 14

@jigglypuff314
I dont remember how to verify but try the link I think they explained well.
http://www.dummies.com/how-to/content/how-to-find-the-tangent-lines-of-a-parabola-that-p.html