Applications of Derivatives: every key term you need (+ practice quiz)
31 flashcard terms for AP Calculus BC Unit 3, written to match the course framework. Read them here, drill them as flashcards, or take the 27-question quiz. Free, no account needed.
If f is continuous on [a,b] and differentiable on (a,b), some c in (a,b) has f'(c) = [f(b)-f(a)]/(b-a). It converts an average rate into an instantaneous one.
Rolle's Theorem
The special case of the MVT with f(a)=f(b), guaranteeing a point where f'(c)=0. It underlies the claim that between two roots lies a critical point.
Critical Point Definition
An interior point where f'=0 or f' fails to exist. Endpoints are candidates for absolute extrema but are not critical points.
First Derivative Test
A sign change of f' from + to - at c gives a local max; - to + gives a local min; no sign change gives neither.
Second Derivative Test Limits
If f'(c)=0 and f''(c)<0 it is a local max; f''(c)>0 a local min; f''(c)=0 is inconclusive and forces you back to the first derivative test.
Inflection Point
A point where concavity actually changes sign. f''=0 is necessary but not sufficient — x⁴ has f''(0)=0 with no inflection.
Candidates Test
For absolute extrema on a closed interval, evaluate f at every critical point and at both endpoints, then compare the outputs directly.
Write a relation among the quantities, differentiate with respect to t, then substitute the instantaneous values — never substitute before differentiating.
Optimization Workflow
Build the objective function, use a constraint to reduce to one variable, state the domain, then apply a derivative test and justify the answer.
Motion Sign Analysis
An object speeds up when v and a share a sign and slows down when they differ. Speed is |v|, so speed can increase while velocity decreases.
Total Distance vs Displacement
Displacement is the integral of velocity; total distance is the integral of speed, computed by splitting at each time the velocity changes sign.
L'Hôpital in Applications
Rates that approach 0/0 as a parameter shrinks — such as average versus instantaneous cost — can be resolved by differentiating numerator and denominator.
Concavity and Rate Language
f'' > 0 means the rate is increasing: the quantity grows at an accelerating pace. This is how AP prompts phrase concavity in context.
Justification Standard
AP requires you to cite the sign behavior that supports your claim, for example 'f' changes from positive to negative at x=3', not merely to name the answer.
Global Behavior of Rational Functions
Combine asymptote analysis with critical points to sketch a full curve; extrema may exist even when the function is unbounded.
Increasing Function Test
f is increasing on any interval where f'>0. A single point with f'=0 does not break monotonicity, as x³ shows at the origin.
Marginal Analysis
In economics, marginal cost is C'(x), and profit is maximized where marginal revenue equals marginal cost with revenue growth falling below cost growth after.
Newton-Style Reasoning with MVT
If |f'| ≤ M on an interval, then |f(b)-f(a)| ≤ M|b-a|. Bounding the derivative bounds how far the function can travel.
Absolute Extrema on Open Intervals
On an open or unbounded interval, extrema may fail to exist; check limits at the endpoints of the domain before claiming a maximum.
Second Derivative and Graph Reading
On a graph of f', the extrema of f' are the inflection points of f, and where f' crosses zero are the local extrema of f.