An arcanist must choose and prepare her spells ahead of time by getting 8 hours of sleep and spending 1 hour studying her spellbook. While studying, the arcanist decides what spells to prepare and refreshes her available spell slots for the day.
I would interpret this to mean no to your question.
Here is another way to think of it, an arcanist prepares spells known, not spells in spell slots. The spell slots represent potential spell power, just like a sorcerer.
In fact, the easiest way to think of a arcanist is as a sorcerer who can change his spells known each day. Because that's what he does. If he has unused spell slots from the day before, that doesn't mean he can use them to cast spells he knew yesterday because it's not a prepared slot. It's prepared spells known.
Claxon is describing, in ordinary (as opposed to RAW) terms, how the arcanist works.
You're preparing the list of spells that you will have access to, and using those spells using sorcerer game mechanics --- in ordinary terms, "preparing spells known" or "being a sorcerer that can use a spellbook to swap spells known."
It would seem, for the moment, to fall into that category of situations that is "technically allowed, but obviously abusive, and don't be surprised if you get your head bopped by your GM's CRB for trying it."
The same thing would go for spontaneous casters retraining metamagic feats to achieve a similar (if far less efficient in effect!) result.
Thank you sandslice for understanding what I meant.
HatlessAtlas, that section applies to how arcanist prepare their "spells known".
An arcanist has a spell book. It can contain any number of spells. Effectively, each day an arcanist can go through his spell book and rememorize any spells he's learned before (by writing them in his spellbook). Thus preparing his "spells known". He then has a number of spell slots that he can cast out of. As long as the spell known is of an equal or lower spell slot the arcanist can cast it from that slot but does not "forget" the spell as a wizard does.
Example, a 7th level arcanist has three 3rd level spell slots and can prepare 2 3rd level spells known. This means he could cast 1 or 1 spell and 2 of the other or 3 of 1 and non of the other without having to choose how many times a particular spell is prepared.
The wizard is referred to as "fire and forget" because when the spell slot is used up he "forgets" how to cast that spell. He can't cast in again in a different slot unless he has unprepared slots left open, and takes the time to prepare ("learn") the spell again.
Why do I say all this? Because, if you "forget" your spells known you no longer have access to any of those spells. They aren't actually tied to the spell slot in any way. Those spell slots had the potential to become any spell that could fit in them. You can carry over your prepared spells known each day if you want, but if you trade it out for a new one then you don't know it any more and can't cast that spell out of your spell slots.
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I actually think that leaving a spell memorized with a metamagic feat then swapping the metamagic feat and memorizing a spell with a different feat is OK. This can't be done before 11th level, which is about when things start getting a little. . . outlandish. . . even for martials.
But remember the big limiting factor on arcanists: number of spells memorized. Take that 11th level arcanist. He can memorize a Persistent fireball one day and an Empowered lightning bolt the next, but he has filled both of his 5th level slots. So if he decides that today he will need to cast teleport, or break enchantment, or overland flight he has to replace one of those. I don't think that's out of line with the intention of the exploit.
edit: And in your original example, Widen is a 3 level bump so your arcanist couldn't memorize Widened fireball until he gets a single 6th level slot at Arcanist level 12 (barring trait shenanigans).