@nicktheturtle
NickTheTurtle
commented
We can easily find the inverse function as #g(x)="arcosh"(x)-a/x#.
The functions #f(x)# and #g(x)# are closest when each of the two is closest to the line of symmetry #y=x#. The point at which they are closest to #y=x# is when the functions have a slope of #1#.
@nicktheturtle
NickTheTurtle
commented
Just a quick clarification: even though it seems that this goes against Einstein's theory of relativity, the speed of light is still #3*10^8\ "m"/"s"# when traveling through the medium. You can think of it as light being constantly absorbed by atoms and then remitted (which takes time), causing the overall speed to be reduced to #2*10^8\ "m"/"s"#. But in fact, when light travels from atom to atom, it still travels at #3*10^8\ "m"/"s"#.
on
If the index of refraction of a lens is 1.5, how fast does light travel in the lens?
@nicktheturtle
NickTheTurtle
commented
Can't this problem be solved by just finding the partial derivative of #int_x^y(3/4-t-t^2)dt# with respect to #x# and then #y# and then set the two equal to #0#?