Here are the verb moods identified in each of the sentences:
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If the weather improves, we will go to the beach tomorrow. - Indicative Mood (the statement is based on a condition that will happen in the future).
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Please close the door quietly. - Imperative Mood (the sentence is giving a command or request).
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I have a dentist appointment on Monday. - Indicative Mood (the statement is a declaration of fact).
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If I were a bird, I would fly across the ocean. - Subjunctive Mood (the sentence expresses a hypothetical or unreal situation).