Question

How quickly does the distance between the parabola y=x^2 and y=x^2+1 change as you increase x?

Answer 1

by "distance" do you mean the perpendicular distance between the tangent lines at a given value for \(x\)?

Answer 2

I think that distace is constant and equal to 1

Answer 3

oops...distance...

Answer 4

it is constant if you interpret the distance as the vertical distance between the two curves at any given \(x\)

Answer 5

I used this formula:
\[d=\sup|(f(x)-g(x)|\]

Answer 6

or does @Kainui mean the "shortest" distance between the two curves as we increase \(x\)?

Answer 7

is the max rate of increasing =1?

Answer 8

oops not 1, the formula is:
\[\lim _{x \rightarrow +\infty}\frac{ 4 }{ \sqrt{4+(1/x ^{2})} }=\]

Answer 9

\[=2\]

Answer 10

I think we first need Kainui to clarify what he/she means by "distance" here

Answer 11

By distance I mean shortest length between one parabola to the other.