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What we are taught at school is that

"the" before vowel pronounce as a long "thee" [ði:]

It sounds that the lecturer is saying [ðə]

we can then say that the gradient of this line is equal to the amount of that function that increases in this interval divided by the length of the interval

So, which one is the correct pronunciation of "the", [ði:] or [ðə]?

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  • I guess from your previous question that the lecturer is speaking from his head (not a prepared text) and perhaps had not decided what word he would say next next, or changed his mind as he was saying "the". I wouldn't worry about it. People speak with all sorts of inconsistencies. Aug 10 '19 at 10:14
  • @WeatherVane I agree with you about "people speak with all sorts of inconsistencies" and for this specific case, "the" should pronounce as a long "thee" [ði:], right?
    – fu DL
    Aug 10 '19 at 10:16
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    I intended to say Sometime a long "thee" is said, instead of the usual short vowel, when a pause for emphasis is made. That was a general observation (unrelated to the speaker's use of a short vowel) to say that rules are just guidelines. We don't consult rules when we speak naturally. Aug 10 '19 at 10:32
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    /ði/ before vowels is a genuine property of (most speakers') English, and so is almost never taught to native speakers, because it doesn't need to be, In this it is almost exactly like "an" before vowels. ("almost" because of pesky words like "historical").
    – Colin Fine
    Aug 10 '19 at 22:37
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    Does this answer your question? What is the correct pronunciation of "the"? And this one: correct pronunciation of "the university"
    – Void
    Jun 26 '21 at 10:33

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