AP Calculus BC Exam Guide and Review for May 2027
AP Mathematics · May 2027
AP Calculus BC Exam Guide and Review for May 2027
- 53-page complete guide
- 10 course units
- 3 hours 10 minutes
- 42 MCQ · 6 FRQ
AP Calculus BC includes the AP Calculus AB ideas and extends them through additional integration techniques, parametric and polar curves, vector-valued functions, and infinite sequences and series. The guide moves from a unit map to newly written problems, visible calculations, checked answers, formula recall, and coordinate plots.
The exam gives equal score weight to multiple choice and free response. Calculator access changes by part, and free-response credit depends on mathematical setup, notation, units, and a conclusion that answers the question rather than a calculator display alone.
- Plan by range. Use the published multiple-choice ranges to distribute practice without turning a range into a promised question count.
- Write the evidence. State theorem conditions, preserve signs and bounds, and show the expression that supports each numerical result.
- Check another way. Confirm a symbolic answer with a graph, table, derivative, integral, limiting case, or unit analysis whenever possible.
What AP Calculus BC covers
The ten units appear in course order. Every percentage is a share of multiple-choice content, not a fixed question count and not a separate free-response weight. Open a unit to practice its main decisions, representations, and common errors.
Three hours and 10 minutes across four timed parts
The complete exam has 42 multiple-choice questions in 100 minutes and 6 free-response questions in 90 minutes. Each section contributes 50% of the score. Students answer multiple-choice questions and view free-response prompts in Bluebook, then handwrite free-response answers in paper exam booklets.
| Section or part | Questions | Time | Score weight | Calculator and response |
|---|---|---|---|---|
| Multiple choice | 42 questions | 100 minutes | 50% | Answer in Bluebook |
| Part A | 29 | 62 minutes | 35% | Calculator not permitted |
| Part B | 13 | 38 minutes | 15% | Graphing calculator required |
| Free response | 6 questions | 90 minutes | 50% | View in Bluebook; handwrite answers |
| Part A | 2 | 30 minutes | 16.7% | Graphing calculator required |
| Part B | 4 | 60 minutes | 33.3% | Calculator not permitted |
| Complete exam | 48 questions | 3 hours 10 minutes | 100% | Hybrid digital |
Check the official AP Calculus BC exam page and the current course and exam description for later changes. These details are stated as of 2026-08-29.
Prove a zero exists without claiming uniqueness
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Let p(x)=x³+x−3.
Which argument proves that p has at least one zero between x=1 and x=2?
Select the best answer.
A. Because p is continuous, p(1)=−1, and p(2)=7, the Intermediate Value Theorem gives some c in (1,2) with p(c)=0.
✅ Correct! 😊 This verifies continuity, brackets 0 with the endpoint values, and states the precise existence conclusion supported by the Intermediate Value Theorem.
B. Because p(1) and p(2) have different signs, p has exactly one zero in (1,2).
❌ Not quite 😭 The sign change supports existence when continuity is stated, but it does not establish uniqueness.
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C. Because p is differentiable, the Mean Value Theorem gives c in (1,2) with p(c)=0.
❌ Not quite 😭 The Mean Value Theorem concerns a derivative matching a secant slope; it does not directly guarantee an intermediate function value.
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D. Because p(1)≠p(2), p must cross the x-axis between 1 and 2.
❌ Not quite 😭 Unequal endpoint values need not bracket 0, and the argument omits the required continuity condition.
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Full explanation
- Step 1A polynomial is continuous on the closed interval [1,2].
- Step 2Compute p(1)=−1 and p(2)=7, so 0 lies between the endpoint outputs.
- Step 3The Intermediate Value Theorem therefore guarantees at least one c in (1,2) for which p(c)=0.
Define f(x)=(x²−1)/(x−1) for x≠1 and f(1)=k.
For which value of k is f continuous at x=1?
Select the best answer.
A. k=0
❌ Not quite 😭 This assigns the canceled numerator value, but continuity requires the limiting value of the quotient.
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B. k=1
❌ Not quite 😭 This uses the input value itself rather than the output approached by the surrounding function.
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C. k=2
✅ Correct! 😊 For x≠1, the quotient equals x+1, whose limit at 1 is 2; setting f(1)=2 makes the three continuity conditions agree.
D. No value of k works
❌ Not quite 😭 The discontinuity is removable because the finite two-sided limit exists, so redefining the point can restore continuity.
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Full explanation
- Step 1Factor x²−1=(x−1)(x+1).
- Step 2For x≠1, f(x)=x+1, so lim as x approaches 1 of f(x) is 2.
- Step 3Continuity requires f(1) to equal that limit; therefore k=2.
Worked solution from the guide: Prove a zero exists without claiming uniqueness
Question. Let p(x)=x³+x−3. Prove that p has at least one zero between x=1 and x=2. The polynomial and constants were selected for this guide; a decimal root is not required.
- Step 1State that p is continuous on [1,2] because every polynomial is continuous on every real interval.
- Step 2Evaluate the left endpoint: p(1)=1+1−3=−1.
- Step 3Evaluate the right endpoint: p(2)=8+2−3=7, so p(1)<0<p(2).
- Step 4Invoke the Intermediate Value Theorem to conclude that some c in (1,2) satisfies p(c)=0.
Fifteen terms to use precisely
These definitions come from the guide’s appendix. Connect each term to a formula, graph, table, interval, or written justification so it does mathematical work in a response.
- Limit
- Output approached as the input nears a target.
- Continuity
- Agreement among function value, limit, and their equality.
- Intermediate Value Theorem
- Continuity guarantee for outputs between endpoint values.
- Derivative
- Instantaneous rate of change or tangent slope.
- Chain rule
- Derivative rule multiplying nested-layer derivatives.
- Critical point
- Domain input where the derivative equals zero or does not exist.
- Fundamental Theorem of Calculus
- Link between differentiation and definite integration.
- Integration by parts
- Product-based integration method using uv minus a residual integral.
- Differential equation
- Equation relating a function to one or more derivatives.
- Euler method
- Stepwise tangent-line approximation method.
- Polar coordinate
- Location described by radius and angle.
- Sequence
- Ordered list of terms, often indexed starting at n=0 or n=1.
- Series
- Sum of terms from a sequence.
- Absolute convergence
- Convergence after replacing terms by magnitudes.
- Taylor polynomial
- Finite derivative-based polynomial about a center.
What to expect and how to use this guide
Does AP Calculus BC provide a formula sheet?
No. College Board specifies calculator access for some parts but does not provide a formula sheet for this exam. Memorize the convergence tests and standard series expansions you plan to use.
How is calculator access divided?
A calculator is not permitted for the 29-question multiple-choice Part A and is required for the 13-question Part B. It is required for the 2-question free-response Part A and not permitted for the 4-question Part B.
Are the practice problems copied from AP exams?
No. Every practice setting, function, constant, table, and graph in this guide was newly written for learning.
What do the unit percentages mean?
They are College Board ranges for shares of multiple-choice content. They are not fixed question counts and do not assign free-response weight by unit.
Will AP Calculus BC results include an AP Calculus AB subscore?
College Board reports a Calculus AB subscore for AP Calculus BC.
Are official multiple-choice questions included in the publicly available materials?
No. The publicly available materials used for this guide contain no officially released AP Calculus BC multiple-choice questions. The multiple-choice practice in this guide was independently written for instruction.
Use four passes to make calculus reasoning exam-ready
Pass 1 — identify the mathematical job. Decide whether the prompt asks for a limit, rate, accumulation, model, geometric quantity, convergence decision, approximation, or justification. Mark the domain, interval, orientation, units, and calculator rule before choosing a method.
Pass 2 — reproduce complete examples. Work from Question to Calculation to Answer without hiding endpoint evaluations, derivative factors, integral bounds, convergence conditions, or units. Then use the four-column rubric check to name the model element, likely error, and evidence needed for credit.
Pass 3 — connect representations. Turn a solved expression into a labeled graph or table, and turn a graph or table back into a defensible expression. Check signs, slopes, accumulated change, endpoints, asymptotes, orientation, and limiting behavior in a second representation.
Pass 4 — practice under both tool settings. Alternate a 62-minute calculator-free multiple-choice block with a 38-minute calculator-required block, then practice the 30-minute calculator-required and 60-minute calculator-free free-response parts. Show the setup even when technology supplies a decimal.
Use the ten published multiple-choice ranges to distribute study time, but keep every unit active in mixed review. Units 6 and 10 each carry a 15–20% range, yet a range never guarantees a fixed item count or a free-response topic.
Build recall deliberately. The exam does not provide a formula sheet. Reconstruct derivative and integral rules, convergence tests, standard Taylor and Maclaurin expansions, polar and parametric formulas, and error bounds from memory; then check them against the appendix and apply them in a complete problem.
AskSia is not affiliated with or endorsed by College Board®. AP is a registered trademark of College Board®. Exam facts and unit percentages are based on official College Board information as of 2026-08-29. The practice settings, functions, values, calculations, and graphs in this guide were independently written for instruction.
