The must-know terms and core concepts for Unit 5: Analytical Applications of Differentiation. Every vocabulary word, theorem, and idea you need to master.
The first derivative reveals increasing/decreasing behavior and relative extrema
The sign of f′ tells you where f is increasing (f′ > 0) or decreasing (f′ < 0). Extrema can occur only at critical points (where f′ = 0 or is undefined) or at endpoints. The First Derivative Test classifies a critical point by how f′ changes sign — from + to − is a relative maximum, from − to + a relative minimum. The Mean Value Theorem underlies all of this, guaranteeing a point where the instantaneous rate equals the average rate.
MVTCritical PointsFirst Derivative Test
Key Concept 2
The second derivative reveals concavity, inflection points, and classifies extrema
The sign of f″ gives concavity: concave up where f″ > 0, concave down where f″ < 0. An inflection point is where concavity changes (f″ changes sign). The Second Derivative Test classifies a critical point where f′(c) = 0: f″(c) > 0 means a relative minimum, f″(c) < 0 a relative maximum — and if f″(c) = 0 the test is inconclusive, so you fall back on the First Derivative Test.
ConcavityInflectionSecond Derivative Test
Key Concept 3
Connecting f, f′, f″ drives curve sketching and optimization
Reading the graphs of f, f′, and f″ against one another lets you translate features: zeros of f′ (with sign change) are extrema of f; extrema of f′ are inflection points of f. The Candidates Test finds absolute extrema on a closed interval by checking all critical points and endpoints. Optimization applies this: write the quantity to optimize in one variable using a constraint, find critical points, and test to identify the maximum or minimum.
Connecting GraphsCandidates TestOptimization
Mean Value Theorem
Some c in (a,b) has f′(c) = [f(b)−f(a)]/(b−a).
Theorems
Extreme Value Theorem
Continuous f on [a,b] attains an absolute max and min.
Theorems
Critical point
x where f′(x) = 0 or is undefined, in the domain of f.
Extrema
Relative (local) extremum
A highest/lowest value in a neighborhood of a point.
Extrema
Absolute (global) extremum
The highest/lowest value over the whole interval or domain.
Extrema
Increasing / decreasing
f rises where f′ > 0 and falls where f′ < 0.
First Derivative
First Derivative Test
Sign change of f′ classifies a critical point.
First Derivative
Candidates Test
Check critical points and endpoints for absolute extrema.