Learning Rumble 101
tagged a link to LoLKing's Rumble guides.
i'll just give you some personal tips while fighting Jayce in lane.
Starting Doran's Shield + pot or cloth + 5 pots would be highly advised.
you run on CDs with a 'heat bar'. try to keep above 50% (in da danger zone) for bonus effects on your spells.
DONT BE SCARED TO GO OVERHEAT! (100% heat bar). you silence yourself. but you deal more damage from your basic attacks.
try to hit around 55% heat. then go in with
Ult + E (75% heat) + W (95%) + Q (OVERHEAT!) -> E.
ult -> E OR W. if you can land your harpoon. definitely harpoon first. then shield. or vice versa. -> Q -> 2nd E.
chances are if done correctly you'll dish out overkill damage. if you can lock them in your ult with harpoons and chase them with flamespitter while basic attacking them overheated.
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but while laning versus Jayce.
start with harpoon's for last hitting. then shield for damage mitigation. then feel free to max out flamespitter for DPS.
if you're getting zoned. call for jungle support. you can use the previous inscribed combo pre-6 too. just minus the ult obviously.
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your choice on what item to rush.
i'd just keep the cloth armor there for later (ninja tabi's or seekers)
i'd rather build into Rylai's or Liadry's
Liadry's will give your flamespitter respectable damage. combined with your ult. Jayce is gonna NEED to land a few empowered shots to poke you down.
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and rule of thumb vs skillshot champs. hide behind minions. side steppin FTW, and yada yada.
getting used to 'reading a player' takes time.
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Jayce walks off to the side of the minion wave. chances are. He's probably setting an empowered shot.
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you're sitting at your caster minions. Jayce is pushing your melee minions hard. he's setting up a surprise empowered shot RIGHT AFTER YOUR "MINION SHIELD" IS DESTROYED. simply walk a bit further behind your 2nd meatshield minions.
^ of he does empower shots at you. that means a pushed lane. pushed lanes mean's farming under tower.
WHICH ALSO MEANS. jungle support.
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otherwise. just do your best not to die. while farming with harpoons and flamespitter. SHIELD TO MITIGATE DAMAGE!
and eventually you'll hit HUGE power spikes from Liandry's + Rylais item combo.
from there you may as well build into Hourglass. HOURGLASS > DEATHCAP. cuz active baiting too stronk. and you're melee. so it works out even better.
definitely check out the guides recommended items and their more in-depth descriptions.