Analytical Applications of Differentiation

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 5. This unit is worth 8–11% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Optimization15 min · 3 objectivesTranslate a word problem into an objective function of one variable · Use critical points and endpoint analysis to find absolute extrema · Justify that a critical point is a maximum or minimum
• The Mean Value Theorem13 min · 3 objectivesState the hypotheses and conclusion of the Mean Value Theorem · Find the guaranteed value c for a given function and interval · Interpret the MVT as an average rate equaling an instantaneous rate

Formulas in Unit 5

Every term in Unit 5

All 8 terms we publish for Analytical Applications of Differentiation, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Mean Value Theorem hypotheses

Continuous on the closed interval and differentiable on the open one. Stating both is required before using the conclusion.

First derivative test

f′ changing positive to negative gives a local maximum, negative to positive a local minimum. No sign change means neither.

Second derivative test

At a critical point, f″ < 0 gives a local maximum and f″ > 0 a local minimum; f″ = 0 is inconclusive.

Point of inflection

Where concavity changes, so f″ must change SIGN. f″ = 0 alone is insufficient, as x⁴ at the origin shows.

Candidates test

On a closed interval, compare f at every critical point and both endpoints to find the absolute extrema.

Reading f′ to describe f

f increases where f′ is positive and is concave up where f′ is increasing. A minimum of f′ is an inflection point of f.

Justification language

Name the theorem or the sign change and say where. "Because f′ changes from positive to negative at x = 2" scores; "because it is the maximum" does not.

Optimization procedure

Express the quantity, eliminate a variable with the constraint, differentiate, find critical points, and justify the extremum.

What examiners penalize here

• On free response you must *justify* that your answer is a max or min — cite a sign change in f′ (first-derivative test), the sign of f′′ (second-derivative test), or the candidates comparison. An unjustified extremum loses the justification point.
• MVT justifications must state both hypotheses ("f is continuous on [a,b] and differentiable on (a,b)") before invoking the conclusion. Many free-response points hinge on that explicit verification, not on solving for c.

Practice Calculus BC

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 10 units of AP Calculus BC

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Fundamental Properties
3. Unit 3 · Differentiation: Composite, Implicit & Inverse Functions
4. Unit 4 · Contextual Applications of Differentiation
5. Unit 5 · Analytical Applications of Differentiation
6. Unit 6 · Integration & Accumulation of Change
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration
9. Unit 9 · Parametric, Polar & Vector-Valued Functions
10. Unit 10 · Infinite Sequences & Series