Tobias Holck Colding continues MIT's 18.100B Real Analysis course with a lecture on the deeper machinery built from the mean value theorem. He introduces the Cauchy mean value theorem, a more elaborate version involving two functions simultaneously, and shows how it underlies both versions of L'Hôpital's rule for evaluating limits of quotients. The lecture closes with Taylor expansion, developed rigorously as a consequence of these mean value results rather than presented as a formula to memorize. As with the rest of the course, the proofs are worked through on the board in full, building each theorem from the analytic definitions established earlier in the term. Running 78 minutes, this is lecture 17 of the Spring 2025 offering, continuing the course's rigorous treatment of single variable calculus from a real analysis perspective.