On my VISA application I wrote down 36 days as my duration of stay, but was only issue a 30-day VISA

It can't be considered as their mistake. Now, you can take the visa with 30 days stay duration to enter China, when you stay in China for 24 days, you can go to the local Exit-Entry Administrative Bureau to extend the visa (in fact, you'll be issued a new visa), it takes about one week for processing. You can't go to extend your visa when you already stay in China for 28 - 30 days. If it is on weekend or can't get a new visa before the present visa expires, you will overstay in China.