AP Calculus AB/BC Unit 5 Cheat Sheet: Analytical Applications of Differentiation

The basics

What it covers: The Mean Value Theorem, critical points and extrema, the first and second derivative tests, concavity, curve sketching, and optimization.

Exam weight: About 15–18% of the AB exam (the largest AB unit) and 8–11% of the BC exam.

The big question: How do derivatives reveal the shape and extrema of a function?

Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.

Key topics at a glance

Mean Value Theorem

If f is continuous on [a,b] and differentiable on (a,b), some c has f′(c) = [f(b)−f(a)]/(b−a) — tangent parallel to the secant.

Critical Points

Where f′(x) = 0 or undefined (in the domain). Extrema occur only at critical points or endpoints.

Increasing / Decreasing

f′ > 0 → increasing; f′ < 0 → decreasing. Use a sign chart of f′.

First Derivative Test

At a critical point: f′ + → − gives a relative max; − → + gives a relative min.

Candidates Test

Absolute extrema on [a,b]: evaluate f at all critical points and both endpoints; pick largest/smallest.

Concavity & Inflection

f″ > 0 → concave up; f″ < 0 → concave down. Inflection where f″ changes sign.

Second Derivative Test

At f′(c)=0: f″(c) > 0 → min; f″(c) < 0 → max; f″(c)=0 → inconclusive.

Optimization

Write the objective in one variable via a constraint, find critical points, and test to identify the max/min.

The key terms and theorems you must know

Mean Value Theorem — f′(c) = [f(b)−f(a)]/(b−a) for some c.
Extreme Value Theorem — continuous f on [a,b] attains abs. max and min.
Critical point — f′ = 0 or undefined, in the domain.
Increasing / decreasing — sign of f′.
First Derivative Test — sign change of f′ classifies extrema.
Candidates Test — check critical points and endpoints for absolute extrema.
Concavity — sign of f″ (up if +, down if −).
Inflection point — where f″ changes sign.
Second Derivative Test — f″(c) sign classifies extrema; 0 is inconclusive.
Optimization — extremize an objective built from a constraint.

Key themes to remember

f′ controls slope; f″ controls bend. Sign charts turn derivatives into shape.
Extrema live at critical points or endpoints — never anywhere else.
A sign change is what matters. f′=0 alone isn’t an extremum; f″=0 alone isn’t an inflection.
Connect f, f′, f″. Reading one graph tells you features of the others.
Optimization is calculus + a constraint. Reduce to one variable, then test.

Common exam traps

f′(c)=0 does not guarantee an extremum. You need a sign change (or the 2nd-derivative test).
f″(c)=0 does not guarantee an inflection point. Concavity must actually change.
Don’t forget endpoints in the Candidates Test for absolute extrema.
Check MVT hypotheses first. Both continuity and differentiability are required.
The Second Derivative Test can fail (f″=0) — fall back on the First Derivative Test.
Justify with the derivative’s sign, not just a number, on FRQs.
Relative ≠ absolute. A local max may not be the global max.