AP Calculus AB · Unit 3
Extreme Value Theorem: every key term you need
20 flashcard terms for AP Calculus AB Unit 3, written to match the course framework. Study them here, then drill them as interactive flashcards — free, no account needed.
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Extreme Value Theorem Continuous function on closed interval [a,b] attains max and min; critical points found where f'(x)=0 or undefined.
Rolle's Theorem If f continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃c in (a,b) where f'(c)=0; derivative equals zero somewhere.
Mean Value Theorem If f continuous on [a,b], differentiable on (a,b), then ∃c in (a,b) where f'(c)=[f(b)-f(a)]/(b-a); instantaneous rate equals average rate.
First Derivative Test Sign of f'(x): + to - means local max, - to + means local min; doesn't change sign means neither.
Critical Points Points where f'(x)=0 or f'(x) undefined; candidates for local extrema; must be in domain of f.
Absolute vs Local Extrema Absolute: largest/smallest on entire domain; local: largest/smallest in neighborhood; absolute extrema occur at critical points or endpoints.
Monotonicity f increasing where f'(x)>0, decreasing where f'(x)<0; analyzing sign of derivative determines intervals of increase/decrease.
Second Derivative Test If f'(c)=0: f''(c)>0 means local min, f''(c)<0 means local max; inconclusive if f''(c)=0.
Inflection Points Points where concavity changes; f''(x)=0 or undefined (and changes sign); graph changes from concave up to down or vice versa.
Concavity Analysis f concave up where f''(x)>0, concave down where f''(x)<0; second derivative test for concavity.
Curve Sketching Find domain, intercepts, asymptotes, critical points, extrema, inflection points, intervals of increase/decrease/concavity; sketch graph.
Optimization Problems Maximize/minimize quantity subject to constraint; write objective function, use constraint to simplify, find critical points.
Closed Interval Extrema Evaluate f at critical points and endpoints; absolute max/min occurs at one of these points.
Implicit Differentiation for Extrema When curve defined implicitly, find dy/dx via implicit differentiation; find critical points where dy/dx=0 or undefined.
Absolute Extrema Location Global max/min on closed interval: evaluate at critical points and endpoints; on open interval: approach limits at boundaries.
Business Optimization Revenue, cost, profit functions; marginal cost = derivative of cost; optimize production levels, pricing; real-world applications.
Related Rates Revisited Optimization in motion: volumes, areas changing with time; differentiate constraint equation, substitute known rates, solve.
Antiderivatives Preview Function F where F'(x) = f(x); integration reverses differentiation; fundamental connection to integrals.
Slope Fields (Direction Fields) Visual representation of dy/dx at points (x,y); shows direction of solutions to differential equations.
Optimization with Constraints Lagrange multipliers concept; use constraint to eliminate variable or solve system of equations from optimization conditions.
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