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This is an audio version of the Wikipedia Article:\nhttps://en.wikipedia.org/wiki/Mean_value_theorem\n\n\n00:01:22 1 History
00:02:01 2 Formal statement
00:04:57 3 Proof
00:05:38 4 A simple application
00:09:51 5 Cauchy's mean value theorem
00:10:10 5.1 Proof of Cauchy's mean value theorem
00:11:50 6 Generalization for determinants
00:14:06 7 Mean value theorem in several variables
00:15:03 8 Mean value theorem for vector-valued functions
00:15:06 9 Mean value theorems for definite integrals
00:15:29 9.1 First mean value theorem for definite integrals
00:18:11 9.2 Proof of the first mean value theorem for definite integrals
00:18:34 9.3 Second mean value theorem for definite integrals
00:20:46 9.4 Mean value theorem for integration fails for vector-valued functions
00:28:48 10 A probabilistic analogue of the mean value theorem
00:31:06 11 Generalization in complex analysis
00:44:16 12 See also
00:44:26 13 Notes
00:46:19 14 External links
00:52:03 Second mean value theorem for definite integrals
00:54:49 Mean value theorem for integration fails for vector-valued functions
00:57:43 A probabilistic analogue of the mean value theorem
00:59:02 Generalization in complex analysis
01:00:35 See also
01:00:53 Notes
01:01:02 External links
\n\n\nListening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.\n\nLearning by listening is a great way to:\n- increases imagination and understanding\n- improves your listening skills\n- improves your own spoken accent\n- learn while on the move\n- reduce eye strain\n\nNow learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.\n\nListen on Google Assistant through Extra Audio:\nhttps://assistant.google.com/services/invoke/uid/0000001a130b3f91\nOther Wikipedia audio articles at:\nhttps://www.youtube.com/results?search_query=wikipedia+tts\nUpload your own Wikipedia articles through:\nhttps://github.com/nodef/wikipedia-tts\nSpeaking Rate: 0.8900721231279423\nVoice name: en-AU-Wavenet-D\n\n\n"I cannot teach anybody anything, I can only make them think."\n- Socrates\n\n\nSUMMARY\n=======\nIn mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.
This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
More precisely, if



f


{\displaystyle f}
is a continuous function on the closed interval



[
a
,
b
]


{\displaystyle [a,b]}
, and differentiable on the open interval



(
a
,
b
)


{\displaystyle (a,b)}
, then there exists a point



c


{\displaystyle c}
in



(
a
,
b
)


{\displaystyle (a,b)}
such that:





f
′

(
c
)
=



f
(
b
)
−
f
(
a
)


b
−
a



.


{\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}.}
It is one of the most important results in real analysis.