For question 4 I know that in the u direction there is at most 5, non-zero basis functions N_(i,4) to N_(i-4,4) and in the v direction there is at most 4 non-zero basis functions N_(j,3) to N_(j-3,3) but from here i'm lost as to how to figure out the number of non-zero terms as well as figure out what those values are.
For question 5 i'm not even sure where to start.
Thanks for any help!
A term can only be non-zero if both basis functions $N_{i,4}(u)$ and $N_{j,3}(u)$ are non-zero. So, the number of non-zero terms is $5 \times 4 = 20$.
For question #5, you'll have to tell us what your teacher regards as the the "necessary properties". If it's just continuity of derivatives, you can check this by straightforward calculation.