It would have to be quite a flexible petrol engine to be able to 'constantly' be changing up below 2000 rpm.
I consider driving 'economically' to be when I manage to change up before 3000 rpm with a petrol engine (at least not a large V8 or V6), If making 'standard progress' it would be between 3 and 4000, and if getting a move on, I will extend it to the edge of the red-line in each gear.
But these are fairly grand generalisations.
Normally, just drive the car in the rev range where the engine 'feels' sweetest and happiest. Yes, labouring it at low revs is bad for it. Constantly running at high revs isn't good either, but you need to give it a good cardiovascular workout reasonably regularly to minimise the risk of things gumming up, like EGR valves.
Be gentle with the engine until it has completely warmed up, use correct-spec good quality oil and filter, changed at least once a year or 10,000 miles, whichever comes soonest - that's your best chance at preserving the engine.
Oh and get some reasonably brisk longer journeys done on A roads or motorways - at least 40 miles - if your car normally just does city traffic trips.
Edited by HandCart on 09/05/2016 at 11:14
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