Why did Prospro give Ferdinand a task to try and prove to him that he can take care off Miranda?
Although Prospero has picked Ferdinand out as a husband for Miranda, he still wants to test whether the young man is worthy of her love. As Ferdinand carries logs—normally Caliban's work—Prospero secretly observes him. Ferdinand does the work willingly, and when Miranda offers to help, he refuses to let her.
Ferdinand expresses his willingness to be a "slave" for Miranda. Prospero looks with approval at the love blossoming between the two. Miranda, who has been isolated all her life, is open about her love, not knowing it might be safer to hide it. However, with Ferdinand, her love is in no danger of being taken advantage of.
Willingness to labor is a sign of love and worth in this novel. Ariel, for example, shows his love for Prospero by willingly working for him. However, unsurprisingly, these characters also want their rewards in the end, be it freedom or the hand of a beloved in marriage.
Prospero's task for Ferdinand can be interpreted in several ways.
1. Prospero gives Ferdinand the task to test his love for Miranda. Ferdinand is the first "real" man Miranda has ever met; Prospero is playing the "protective father" role to ensure that Ferdinand's love is real and not fickle.
2. Ferdinand is a nobleman--King Alonso's son. It was the king who conspired with Antonio to have Prospero removed. It was the king's army who physically removed Prospero and Miranda and carried them to the island where they currently reside. Because of these factors, Prospero is getting revenge through Ferdinand.
3. "The Tempest" is a play about power and the effects of power. Prospero is proving how powerful he is by forcing Ferdinand to perform the task.