I don't know if I'm actually using the right terminology here, to be clear I'm going to state explicitly what I'm trying to figure out to see if I can be pointed in the right direction:

Let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between 2 abelian categories $\mathcal{A}$ and $\mathcal{B}$, let $X^\bullet$ be the complex $0 \rightarrow X^0 \rightarrow X^1 \rightarrow X^2 \rightarrow X^3 \rightarrow \cdots$ of objects in $\mathcal{A}$, let $T^\bullet$ be a complex obtained by removing the FIRST TERM (or maybe the first $q$ terms from the left?), I mean let $T^\bullet$ be $X^0 \rightarrow X^1 \rightarrow X^2 \rightarrow X^3 \rightarrow \cdots$, the complex obtained by removing the $0$ from $X^\bullet$, NO 0's to the left, so $T$ becomes "unbounded". Let's say I have a way of obtaining the derived functor $RF(T^\bullet)$ using the machinery developed for unbounded complexes. Is there any way I can compute some of the terms in $RF(T^\bullet)$ from knowing this?

oneterm! :-) – Mariano Suárez-Alvarez♦ Sep 14 '12 at 20:16