Check both hypotheses
For , continuity on [a,b] and differentiability on (a,b) guarantee an interior point where the tangent slope equals the secant slope.
Calculus / English
Mean Value Theorem
The Organic Chemistry Tutor · YouTube · 19:39
YouTube · The Organic Chemistry TutorOriginal video
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Reviewed learning material · Video analysis · English01 02 03
Read the full overview
Apply the mean value theorem carefully, distinguishing endpoint behavior from interior differentiability.
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Chapters
0:00Check both hypotheses6:00Solve for the matching slope9:00Endpoint versus interior cornerLearning script
Generated from the video's visuals and explanation; not verbatim speech.
For , continuity on [a,b] and differentiability on (a,b) guarantee an interior point where the tangent slope equals the secant slope.
For ²− on [1,5], the secant slope is ()/4=2. Solving gives , inside the interval.
The cusp of at the endpoint 0 does not violate differentiability on (0,1). By contrast || has a corner at inside (0,2), so the theorem does not apply there.
Knowledge cards
Solve for the matching slope
For ²− on [1,5], the secant slope is ()/4=2. Solving gives , inside the interval.
Endpoint versus interior corner
The cusp of at the endpoint 0 does not violate differentiability on (0,1). By contrast || has a corner at inside (0,2), so the theorem does not apply there.
Explore the knowledge in this video
- Mean value theorem ExplanationAt 0:00
Why this connection?
For , continuity on [a,b] and differentiability on (a,b) guarantee an interior point where the tangent slope equals the secant slope. For ²− on [1,5], the secant slope is ()/4=2. Solving gives , inside the interval. The cusp of at the endpoint 0 does not violate differentiability on (0,1). By contrast || has a corner at inside (0,2), so the theorem does not apply there.
Similar explanations
YouTube7:37Another explanation of Mean value theorem.
Mean value theorem
Connect the next step
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YouTube2:39Calculus·English
Product rule | Derivative rules | AP Calculus AB | Khan AcademyFoundationalReviewed notes
Prerequisite in Calculus: Derivatives → Mean value theorem.
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A different approach
YouTube7:37Lecture for Mean value theorem, compared with the current whiteboard.
Mean value theorem