Advanced flashcards covering the Intermediate Value Theorem, Mean Value Theorem, differentiability vs. continuity, L'Hopital's Rule, and complex limits.
20 cards
Front
Intermediate Value Theorem (IVT)
Back
If f is continuous on [a, b] and N is any number between f(a) and f(b), then there exists a c in (a, b) such that f(c) = N. Does not work on discontinuous functions.
Front
Mean Value Theorem (MVT)
Back
If f is continuous on [a, b] and differentiable on (a, b), then there exists a c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a). Guarantees a point where instantaneous rate equals average rate.
Front
Extreme Value Theorem (EVT)
Back
If f is continuous on a closed interval [a, b], then f has both an absolute minimum and an absolute maximum on that interval. Requires both continuity and a closed interval.
Front
Differentiability Implies Continuity
Back
If a function f is differentiable at x = a, then f is continuous at x = a. The converse is NOT true (e.g., absolute value function is continuous at x=0 but not differentiable there due to a cusp/corner).
Front
Forms of Non-Differentiability
Back
A function is not differentiable at x = a if there is: 1. A cusp or corner (sharp turn). 2. A vertical tangent line. 3. A discontinuity (jump, removable, infinite).
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