Mean value theorem
If f is continuous on [a, b] and differentiable inside, there is a point c between a and b where f'(c) equals the average slope.
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Analogy
If your average speed on a trip was 60 km/h, at some moment your speedometer showed exactly 60.
Example
Average slope: (9 - 1)/(3 - 1) = 4.Solve f'(c) = 2c = 4.So c = 2.The point c = 2 lies inside [1, 3].
In practice
It underlies many proofs, such as why a function with zero derivative must be constant.
Practice
Flashcards
- State the mean value theorem.Some c in (a, b) has f'(c) = (f(b) - f(a))/(b - a).
Questions people ask
What does the mean value theorem say in plain words?
Somewhere between two points on a smooth curve, the tangent line is parallel to the straight line joining the two ends. In driving terms, if your average speed was 60 km/h, you must have travelled at exactly 60 km/h at some instant.
Why does the theorem need differentiability?
If the graph has a sharp corner, the tangent may never match the average slope. The function absolute x on [-1, 2] has average slope 1/3, but its slope is only -1 or 1, never 1/3. The corner at 0 breaks the theorem.