Fundamental theorems
Rolle, the mean value theorem, Cauchy, L'Hôpital and the fundamental theorem of calculus: the results that turn local information (derivatives) into global guarantees — error bounds, limits, areas.
5 topics
Topics
Rolle's theorem
If a smooth function takes the same value at both ends of an interval, somewhere in between its derivative is zero: a ball thrown up and caught at the same height was momentarily still.
Mean value theorem
Somewhere on the instantaneous rate of change equals the average rate: . It is the bridge from derivatives to inequalities — and therefore to every error bound in numerical analysis.
Cauchy's mean value theorem
The mean value theorem for two functions at once — for a parametric curve . It is the lemma from which L'Hôpital's rule follows.
L'Hôpital's rule
For indeterminate forms or , when the right-hand limit exists. It settles races between growth rates, such as versus .
Fundamental theorem of calculus
Differentiation and integration are inverse operations. Accumulating a rate of change recovers the total change: , and the derivative of an accumulated quantity is the rate.