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You said you are shooting your daughter's soccer and gymnastics so I am approaching my answer as a sports shooter. Probably 90 percent of what I shoot involves a ball, a bat, a stick or a puck.
With that being said, the best lens to shoot sports is the Nikon 70-200 f/2.8. VRI or VR II is up to you. I have owned both, but went back to the VRI and shoot it on the VRI on my D3s and D300 and have never had any issues (some say there are vignetting problems with the VRI on the full frame bodies, but it won't impact you at all on DX.)
The full stop difference between the f/2.8 and f/4 version of the lens is HUGE. She may be playing daytime soccer right now, but as she moves up, it will be under the lights -- and I promise you the lights for high school soccer are miserable. Gymnastics is worse. I can't imagine hand-cuffing myself at f/4. And while I never use the VR for shooting sports -- I use it quite a bit when I need a little help with a really slow shutter speed in low light situations. I can hand-hold it with VR on at 1/10 and it gives you a lot of flexibility.
There is a reason the 70-200 f/2.8 is in the bag of every serious sports shooter. It is the gold standard for a reason.