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Rolle's theorem

Level UniversityDifficulty ★★★★★Theorem⌖ Open in the map

What is it?

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.

Statement

$f$ continuous on $[a,b]$ , differentiable on $(a,b)$ , $f(a) = f(b)$ $\implies \exists c \in (a,b):\ f'(c) = 0$ .

Idea of the proof

By Weierstrass, $f$ has a maximum and a minimum on $[a,b]$ . If both are at the endpoints, $f$ is constant and $f' \equiv 0$ . Otherwise one of them is at an interior point $c$ , and at an interior extremum the derivative vanishes (the difference quotients change sign).

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

Mean value theorem→Taylor's theorem and the remainder→Scientific computing★★★★★
Mean value theorem→Fundamental theorem of calculus→Cumulative distribution function→Monte Carlo methods★★★★★
Mean value theorem→Fundamental theorem of calculus→Symbolic computation (CAS)★★★★★
Mean value theorem→Fundamental theorem of calculus→Improper integrals→Integral test→Algorithm analysis and complexity★★★★★
Mean value theorem→Taylor's theorem and the remainder→Floating point (IEEE 754)★★★★★

ℒ AI and machine learning

Mean value theorem→Lipschitz continuity→Gradient descent★★★★★
Mean value theorem→Lipschitz continuity→Gradient descent→Learning rate★★★★★
Mean value theorem→Lipschitz continuity→Gradient descent→Backpropagation★★★★★
Mean value theorem→Lipschitz continuity→Gradient descent→Stochastic gradient descent (SGD)★★★★★
Mean value theorem→Lipschitz continuity→Gradient descent→Loss landscape★★★★★
Mean value theorem→Lipschitz continuity→Gradient descent→Reinforcement learning★★★★★
+7

3D Computer graphics

Mean value theorem→Fundamental theorem of calculus→Cumulative distribution function→Monte Carlo methods→The rendering equation★★★★★

⚙ Robotics and control

Mean value theorem→Fundamental theorem of calculus→Improper integrals→Laplace transform→Control theory★★★★★
Mean value theorem→Fundamental theorem of calculus→Improper integrals→Laplace transform→PID control★★★★★
Mean value theorem→Fundamental theorem of calculus→Improper integrals→Probability density function→Kalman filter★★★★★

⇄ Optimization and systems

Mean value theorem→Lipschitz continuity→Gradient descent→Recommender systems★★★★★
Mean value theorem→Fundamental theorem of calculus→Cumulative distribution function→Queueing theory and performance★★★★★

ψ Quantum computing and physics

Mean value theorem→Fundamental theorem of calculus→Improper integrals→Fourier transform→Uncertainty principle★★★★★
Mean value theorem→Taylor's theorem and the remainder→Taylor and Maclaurin series→Quantum computing★★★★★

⚛ Physics and simulation

Mean value theorem→Fundamental theorem of calculus→Physics engines★★★★★
Mean value theorem→Fundamental theorem of calculus→Physics engines→N-body gravitational simulation★★★★★
Mean value theorem→Fundamental theorem of calculus→Physics engines→Fluid dynamics and CFD★★★★★
Mean value theorem→Fundamental theorem of calculus→Physics engines→Fluid dynamics and CFD→Weather and climate modelling★★★★★

What depends on it

Mean value theorem