Using auxiliary (helping) verbs to express politeness and certainty

AUXILIARY VERB USE EXAMPLE
can ability I can understand his English fairly well.
informal permission You can use my copy of the agenda.
informal polite request Can I borrow your copy of the agenda?
impossibility (with negative) That can't be true!
could past ability I could understand the last speaker better.
polite request Could I borrow your copy of the agenda?
possibility I could understand him better if he spoke more slowly.
suggestion I need some new ideas for this project. You could have a brainstorming session with the whole team.
less than 50% certainty Where is John? He could be stuck in traffic.
impossibility (with negative) That couldn't be true!
to have to necessity I have to go to a meeting today. (Future) I had to go to a meeting yesterday. (Past)
lack of necessity (with negative) I don't have to go to a meeting today. (Future) I didn't have to go to a meeting yesterday. (Past)
have got to necessity I have got to go to that meeting today. (Future)
had better advisability with threat of bad result You had better be on time, or we will start without you.
may polite request May I interrupt you for a moment?
permission (formal) You may interrupt me if it is important.
less than 50% certainty Robert may be stuck in traffic, so let's begin the meeting without him.
might less than 50% certainty Robert might be stuck in traffic, so let's go ahead and begin the meeting without him.
must strong necessity I must go to a planning session today.
prohibition (negative) You must not withhold information during the meeting.
95% certainty John isn't here. He must be sick (present) John must have been sick yesterday. (Past)
ought to advisability We ought to invite an IT specialist to the meeting next time.
90% certainty The meeting ought to end on time.
shall polite question to make a suggestion Shall I take the minutes?
should advisability

We should meet again next week.

We should have met more often last month. (past)

90% certainty The meeting should finish on time.
to be supposed to expectation The meeting is supposed to begin at 9:00.
to be strong expectation

You are to be there at 9:00. (future) You were to be here at 9:00. (Past)

to be able to ability My colleague is always able to help.
to be going to 90% certainty He is going to be here at 9:00.
intention I'm going to ask Mr. Smith to join us. (Future) I was going to ask Mr. Smith to join us but I couldn't reach him. (Past)
used to repeated action in the past We used to meet from 3:00 to 5:00 every Friday.
will 100% certainty I will be here at 9:00. (only future)
willingness Who would like to take the minutes? I'll do it.
polite request Will you please take the minutes?
would polite request Would you please send me another copy of the agenda? Would you mind if I left the meeting a few minutes early?
preference I would rather meet in the conference room than his office.
repeated action in the past At my previous job meetings would always start at 3:00 p.m.