If a car travels 180 km in 2 hours, then at some moment its speedometer must have read exactly 90 km/h. This is not obvious — the car could have been faster or slower at every individual instant — but it is guaranteed by the Mean Value Theorem. The MVT formalises the intuition that an instantaneous rate of change must equal the average rate of change at least once over any interval on which the function is smooth. That structural guarantee is the engine of the rest of calculus: it is what lets us translate derivative information ( means increasing, means decreasing) into reliable information about the function itself. In quantitative finance, the MVT underpins error bounds on discrete approximations (finite-difference Greeks, log-return approximations) and confirms that if two functions have identical derivatives everywhere, they can differ only by an additive constant — the uniqueness of antiderivatives.
Learning objectives. By the end of this note you should be able to:
- state Rolle's Theorem with its three hypotheses and verify each in an example;
- derive the Mean Value Theorem from Rolle's Theorem, understanding the geometric meaning;
- use the MVT to bound the change in a function value from a bound on its derivative;
- prove the increasing/decreasing test: on implies increasing on ;
- state and use the antiderivative uniqueness corollary: everywhere implies .
In this lesson
- Rolle's Theorem
- The Mean Value Theorem
- Consequences for monotonicity
Before you start
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