Transmission problem
 

My daughter has a 2001 Camry with 198,000 miles on it. It goes in forward gears just fine, but has trouble backing up. It will reverse normally if on a flat surface, but you have to increase the RPM's to get the car to move if you are reversing on even a slight hill. She says it gets worse when the car has been driven for awhile. Does this sound like a major problem or is there something I can try first?

Could be major. Lack of reverse can lead to lack of forward gears in the future as it indicates a worn trans.

Check fluid level and condition. Should be red or reddish, not brown or black. You can change ALL the fluid and see what happens. There are posts at this site on how to change all the fluid, not just what is in the trans pan.

If not trans fluid level or condition my guess guess? is the reverse brake is not working. When reverse is engaged, the brake stops planetary gears from rotating allowing the trans output shaft to rotate in the opposite direction.

The problem is not going to get any better and could get worse. Suggest telling your daughter to assume the car no longer has reverse and to park the car accordingly so it is not needed. Otherwise she could soon be stuck without reverse.

A clutch brake repair requires taking the trans apart. It might be possible to fix the trans without taking it out of the car but most trans shops want to sell a replacement. If possible find a shop willing to do some diagnostic beyond "you need a new trans."