Question

For `f(t)= (lnt/t, t^2/e^t)` what is the distance between `f(2)` and `f(4)`?

Answer 1

`(4(e^2-4))/e^4`#

Explanation:

We have:

`f(t)= (lnt/t, t^2/e^t)`

Put `t=2`:

`f(2)= (ln2/2, 4/e^2)`

Put `t=4`:

`f(4)= (ln4/4, 16/e^4)`
`\ \ \ \ \ \ \= (ln2^2/4, 16/e^4)`
`\ \ \ \ \ \ \= (2ln2/4, 16/e^4)`
`\ \ \ \ \ \ \= (ln2/2, 16/e^4)`

By Pythagoras:

`d^2 = Delta x^2 + Delta y^2`
`\ \ \ = (0)^2 + (4/e^2-16/e^4)^2`

`:. d = 4/e^2-16/e^4`
`\ \ \ \ \ \ \ = (4(e^2-4))/e^4`