Unit 5 · Analytical Applications of Differentiation
The Complete AP Calculus BC Guide · AP Calculus BCUnit 5 · Analytical Applications of Differentiation
- The Complete AP Calculus BC Guide
- AP Calculus BC
- 10 sections
Unit 5: Analytical Applications of Differentiation accounts for 10–15% of AP Calculus BC multiple-choice content. Section I has 42 multiple-choice questions in 100 minutes and contributes 50% of the score. For Section II's 6 free-response questions in 90 minutes (50%), be ready to carry the same unit skills and representations into a complete solution. College Board assigns this unit 10–15% of AP Calculus BC multiple-choice content; use that range to plan review time, not to predict a fixed number of questions or a free-response allocation.
- How AP Calculus BC assesses this 10–15% of the multiple-choice section · Section I: 42 MCQs in 100 min, 50% · Section II: 6 FRQs in 90 min, 50% · show the model with first-derivative sign chart, aligned graphs of f f' and f'', objective and constraint diagram
- Key skills Apply MVT and EVT, Analyze monotonicity and concavity, Optimize under constraints
- How to study for Unit 5 This page turns first-derivative sign chart, aligned graphs of f f' and f'', objective and constraint diagram into one route: build the candidate set and sign chart before naming a conclusion.
- The organizing decision use derivative signs and theorem hypotheses to justify behavior extrema and approximation
What AP Calculus BC Unit 5 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Apply MVT and EVT
first-derivative sign chart; Critical numbers are candidates rather than automatic extremaAPCALCBC-U5-S2Analyze monotonicity and concavity
aligned graphs of f f' and f''; A derivative sign change classifies local extremaAPCALCBC-U5-S3Optimize under constraints
objective and constraint diagram; Closed-interval absolute extrema require endpoints and all interior candidatesUnit 5: Analytical Applications of Differentiation accounts for 10–15% of AP Calculus BC multiple-choice content.
Official unit name and weighting: College Board course and exam description.
Use derivative evidence to justify global and local behavior
Connect the published share to the unit model
College Board assigns this unit 10–15% of AP Calculus BC multiple-choice content; use that range to plan review time, not to predict a fixed number of questions or a free-response allocation.
Unit map language to know: critical point, local extremum, absolute extremum, increasing interval, decreasing interval, concavity, inflection point, Mean Value Theorem, first derivative test, and second derivative test; each term is used on this page and defined again in Appendix B.
Name the target before reading a derivative graph: sign controls increase, change in sign controls extrema, and change in the derivative controls concavity.
The decision that organizes this unit
Define the system and choose the route before calculating
use derivative signs and theorem hypotheses to justify behavior extrema and approximation
build the candidate set and sign chart before naming a conclusion
Mechanism route and repair branches
Relationships to preserve
- Critical numbers are candidates rather than automatic extrema
- A derivative sign change classifies local extrema
- Closed-interval absolute extrema require endpoints and all interior candidates
Representations to read
- first-derivative sign chart
- aligned graphs of f f' and f''
- objective and constraint diagram
Branches to reject
- calling every stationary point an extremum
- using a theorem without its continuity or differentiability conditions
- ignoring endpoints in an absolute-extrema problem
| Key concept | Why it's hard | What scores |
|---|---|---|
| Extrema | Zeros alone do not classify behavior | Shows a sign change or compares all closed-interval candidates |
| Mean Value Theorem | The secant slope is not automatically attained | States continuity, differentiability, and solves f′(c)=secant slope |
| Concavity | The sign of f′ and change in f′ answer different questions | Uses f″ or monotonicity of f′ |
How AP Calculus BC assesses Analytical Applications of Differentiation
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Apply MVT and EVT | first-derivative sign chart; Critical numbers are candidates rather than automatic extrema | calling every stationary point an extremum |
| Analyze monotonicity and concavity | aligned graphs of f f' and f''; A derivative sign change classifies local extrema | using a theorem without its continuity or differentiability conditions |
| Optimize under constraints | objective and constraint diagram; Closed-interval absolute extrema require endpoints and all interior candidates | ignoring endpoints in an absolute-extrema problem |
Resolve the Analytical Applications of Differentiation evidence conflict
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Absolute extrema on a closed interval
A differentiable function is defined on [−2,3], and a table gives f and f′ at the endpoints and critical points.
Which procedure correctly locates the absolute extrema of f on [−2,3]?
Select the best answer.
A. Compare f at −2, 3, and every valid interior critical point; choose the largest and smallest f-values.
✅ Correct! 😊 This is the closed-interval method: endpoints and all valid interior critical candidates are tested using function values.
B. Compare only the interior inputs where f′=0.
❌ Not quite 😭 This omits endpoints and any domain points where f′ does not exist, so the absolute extrema could be missed.
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C. Choose the inputs having the largest and smallest values of f′.
❌ Not quite 😭 Derivative values describe rates of change, not the heights that determine absolute extrema.
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D. Every critical point is automatically an absolute maximum or minimum.
❌ Not quite 😭 Critical points are candidates only; their function values must be compared with one another and with endpoint values.
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Full explanation
- Step 1List both endpoints and every interior point where f′=0 or f′ does not exist.
- Step 2Discard any input outside the function domain or the closed interval.
- Step 3Compare f, not f′, at every remaining candidate.
- Step 4The greatest and least values identify the absolute maximum and minimum and their locations.
Practice 2: Applying the Mean Value Theorem
Let f(x)=x² on [1,3].
At which value c in (1,3) does f′(c) equal the average rate of change of f on [1,3]?
Select the best answer.
A. c=1
❌ Not quite 😭 The left endpoint is not in the required open interval, even though it is used to compute the secant slope.
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B. c=2
✅ Correct! 😊 The secant slope is 4 and f′(x)=2x, so f′(2)=4 at an interior point.
C. c=3
❌ Not quite 😭 The right endpoint is excluded from the theorem’s conclusion and also gives derivative 6 rather than 4.
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D. No such c exists
❌ Not quite 😭 The polynomial is continuous on [1,3] and differentiable on (1,3), so the theorem applies and the calculation finds c=2.
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Full explanation
- Step 1The polynomial is continuous on [1,3] and differentiable on (1,3).
- Step 2The endpoint secant slope is (9−1)/(3−1)=4.
- Step 3Solve f′(c)=2c=4 to obtain c=2, which lies in (1,3).
Key terms for Unit 5: Analytical Applications of Differentiation
Models, uses, and boundaries
- Identify Critical-Number Candidates
- A critical number lies in the function domain where the derivative is zero or undefined Choose this formula when the prompt asks you to identify critical-number candidates and the function, domain, interval, or representation matches the symbols shown. A Identify Critical-Number Candidates solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Classify Extrema from Derivative Signs
- Derivative sign changes classify local maxima and minima Choose this formula when the prompt asks you to classify extrema from derivative signs and the function, domain, interval, or representation matches the symbols shown. A Classify Extrema from Derivative Signs solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Optimize on a Closed Interval
- Closed-interval absolute extrema occur among endpoints and interior critical candidates Choose this formula when the prompt asks you to optimize on a closed interval and the function, domain, interval, or representation matches the symbols shown. A Optimize on a Closed Interval solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Apply the Mean Value Theorem
- The Mean Value Theorem matches an interior tangent slope to the endpoint secant slope Choose this formula when the prompt asks you to apply the mean value theorem and the function, domain, interval, or representation matches the symbols shown. A Apply the Mean Value Theorem solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
AP Calculus BC Unit 5 FAQ
How much of AP Calculus BC does Unit 5 carry?
Unit 5: Analytical Applications of Differentiation accounts for 10–15% of AP Calculus BC multiple-choice content.
What is the first move on a Analytical Applications of Differentiation problem?
build the candidate set and sign chart before naming a conclusion
Which relationships should I preserve?
Critical numbers are candidates rather than automatic extrema A derivative sign change classifies local extrema Closed-interval absolute extrema require endpoints and all interior candidates
Which representations should I practice?
Practice moving among first-derivative sign chart, aligned graphs of f f' and f'', objective and constraint diagram.
What error should I check before submitting an answer?
Check for calling every stationary point an extremum; using a theorem without its continuity or differentiability conditions; ignoring endpoints in an absolute-extrema problem.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Run the full closed-interval candidate test
- Match an instantaneous rate to the interval’s average rate
- Read extrema, concavity, and signed change from f′
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related AP Calculus BC unit guides
AP Calculus BC Exam Guide & Review
The whole exam and its official unit sequence.01Limits and Continuity
5–10% of the multiple-choice section02Differentiation: Definition and Fundamental Properties
5–10% of the multiple-choice section03Differentiation: Composite, Implicit, and Inverse Functions
5–10% of the multiple-choice section04Contextual Applications of Differentiation
5–10% of the multiple-choice section06Integration and Accumulation of Change
15–20% of the multiple-choice section07Differential Equations
5–10% of the multiple-choice section08Applications of Integration
5–10% of the multiple-choice section09Parametric Equations, Polar Coordinates, and Vector-Valued Functions
10–15% of the multiple-choice section10Infinite Sequences and Series
15–20% of the multiple-choice sectionHow to study AP Calculus BC Unit 5
Start with the organizing decision
Before solving, restate the decision in operational terms: use derivative signs and theorem hypotheses to justify behavior extrema and approximation. Your first written move should be to build the candidate set and sign chart before naming a conclusion.
Practice the same idea in several representations
Rotate through first-derivative sign chart, aligned graphs of f f' and f'', objective and constraint diagram. Use each representation to practice Apply MVT and EVT, Analyze monotonicity and concavity, Optimize under constraints, and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for calling every stationary point an extremum; using a theorem without its continuity or differentiability conditions; ignoring endpoints in an absolute-extrema problem. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official College Board course and exam description for the May 2027 administration.
