So, in the last few days I've done these directly in front of a cop/sheriff deputy.
Passed by a cop going 60+ in a 45.
Passed by a cop going 80+ in a 65.
This stuff depends on the area. Where I live, if you actually do 55 on the interstates marked 55, you will create just as much of a road hazard as the crazy people who drive way faster than the average flow of traffic. Doing about 70 is normal in these areas; in the 65, it's typical to do somewhere between 70-80 mph. If you're in city limits, you can often even get a ticket for obvious speeding (such as 65 in a 55) thrown out simply by bothering to show up at the courthouse, as long as you weren't doing anything truly reckless.
Of course, speeding in front of a cop--daring the police--is stupid. You have to read the situation. Sometimes the cops, who are locals too, drive just as badly as the rest of us. The ones in traffic are usually OK as long as you're not the idiot who decides to go around them. The ones on the shoulder, however...watch out; they might be in a bad mood.
If you are on the interstate, I'd say keep up with traffic, but do not be the fastest car on the interstate unless you are very sure a) no cops are around and b) you don't cut people off, change lanes without a turn signal, ride bumpers, or make other reckless maneuvers; especially if you're going fast,
you want to be very predictable for your fellow drivers. You do not want them guessing as to what you will be doing in any given moment.
Passed a car on a double yellow in front of a deputy (he yelled at me but didn't give me a ticket or warning)
Blew through a stop sign in front of a deputy.
Went through a red light with a cop following me.
Made a left turn from a right turn lane in front of a cop.
And almost rear ended a cop at the airport.
These here are all damn stupid. Fast is one thing, especially if that's the typical driving custom in your area. Fast and reckless is a horrible combination. You never, ever want to be unpredictable to your fellow drivers, because unexpected maneuvers cause accidents. You're also clearly pushing--and exceeding--the boundaries of your reaction times if you are unable to react appropriately to situations like you're describing, and that also causes accidents, whether or not it happened to this time around. You should be watching a good, long stretch of road ahead of you and thinking several maneuvers ahead (including anticipating other drivers' moves, which you can often catch from subtle mannerisms and even tiny moves the car makes before it actually does whatever the driver is thinking about doing); the smart driver plays "chess" rather than simply making a mad, uncoordinated dash to one's destination.