Explain how far the title of the play The Tempest is appropriate.

According to the Oxford English Dictionary (OED), the word "tempest" has multiple meanings, but one of the older (and currently obsolete) definitions is "calamity, misfortune, trouble" (OED, "tempest, n."). A deeper explanation of the play's title can be found through the use of carefully dissected definitions.
My answer, then, will be two-fold: the first part will be how students can use proper dictionary definitions of words to create analysis for student writing, and the second part will be an example of how the word "tempest" could fit into my use of the OED's definition.
Students seeking an answer to the question of how Shakespeare's The Tempest is appropriately titled should start with a dictionary definition of the word tempest. Personally, I use the Oxford English Dictionary, and I try to pinpoint when and where the definitions were used and which definitions are now obsolete or rarely used. In the case of the definition I used above, the two citations the OED offers are from the 1300s and the 1400s, meaning that Shakespeare most likely was aware of this definition of the word tempest when he was writing this play.
Of course, the definition a student chooses does not exclusively have to be the obsolete or rarely used version of the word. In fact, a critical analysis can still be created with the more familiar definition of the word tempest. The OED provides that definition, which is a "violent storm of wind, usually accompanied by a downfall of rain . . . or by thunder" (OED, "tempest, n.").
These definitions can form the basis for a carefully wrought piece of analysis by a student. Let's start with the second definition I provided from the OED, revolving around an actual physical rainstorm. The keywords of the definition are violent and downfall of rain and thunder. This is where key analysis of this play can happen for students:
How do the characters reflect a tempest with their actions (violent actions, namely)?
How do the characters reflect a tempest with their words (words that clap like thunder, words that are violent)?
How do the elements of a tempest reflect the characters?
For the first OED definition I provided, analysis could move in a different direction, such as:
What kinds of calamities, misfortunes, or troubles do the characters suffer through?
What kinds of calamities or troubles do the characters subject each other to?
All of the above questions are fantastic questions to begin a student analysis of one of Shakespeare's most brilliant plays. Remember, students: most analysis is simply answering questions that start with how or what happened when.

The Tempest is an ideal title for this play, one of Shakespeare's greatest and last. Not only is an actual tempest portrayed in the opening scene—conjured up by Prospero and serving to capture his foes (and a couple of friends) on the island—but all the characters in the play are experiencing their own personal "tempests" as well.
Miranda and Ferdinand are feeling the tempestuous effects of first love. Caliban is full of tempestuous rage at his slavery and lot in life under Prospero's mastery. Ariel longs for the tempest of the air of freedom, long promised by Prospero but not yet delivered. Sebastian and Alonso toss from the tempest of their conspiracies and guilt. And most importantly, Prospero, at the end of his sorcery and even his life, longs for life's tempest to subside. His soliloquy at the end of the play is considered by most scholars to be Shakespeare's own farewell to the constant tempest of his own genius.