【1】I need a knife to cut the watermelon.【目的状语】
=To cut the watermelon , I need a knife.
=I need a knife so that I can cut the watermelon.
备注:不定式to cut the watermelon作目的,不加介词with
【2】I need a knife to cut the watermelon with.【后置定语】
=I need a knife with which I can cut the watermelon
=I need a knife with which to cut the watermelon
备注:不定式to cut the watermelon with作后置定语,必须加上介词with, 而且作后置定语不定式(to cut the watermelon with)与被修饰名词(a knife)属于“动宾关系”即,a knife为不定式(to cut the watermelon with)逻辑宾语. 即(to cut the watermelon with a knife)即(用一把小刀来切这个西瓜)