Q: The RPMs rais between 3k and 4k without a change in throttle position, speed stays the same, and it does this at different speeds.

There are a number of things that could cause this to happen in a vehicle with an automatic transmission like yours. It sounds like what you are describing is that the transmission is slipping slightly while trying to maintain a specific gear at a given speed. It sounds like your vehicle may be a bit low on transmission fluid or you may potentially have a bad transmission control solenoid. As you know, the transmission works on hydraulic pressure built up from the fluid inside and uses this pressure to activate solenoids that cause the transmission to shift. When these are not working properly or when you have low transmission fluid, this does not happen which usually results in erratic shifting patterns, a jerking reaction or in some cases no shifting at all. I would suggest having a professional from YourMechanic come to your location to diagnose and inspect your vehicle.