AP Calculus AB Source-First Exam Guide 2027
The Complete AP Calculus AB Guide · AP Calculus ABAP Calculus AB Source-First Exam Guide 2027
- The Complete AP Calculus AB Guide
- AP Calculus AB
- 8 sections
The complete 70-page book follows all eight official units from limits and continuity through differential equations and applications of integration. Every worked route starts with stated givens, displays the computation, and checks the result against a graph, table, sign, interval, or unit.
- How AP Calculus AB is assessed 42 multiple-choice questions in 100 minutes and 6 free-response questions in 90 minutes, with calculator access split by part
- Key terms Limit, Continuity, Derivative, Critical point
- How to study for AP Calculus AB Size effort by weight first: Unit 5 (15–20%), Unit 6 (15–20%), Unit 1 (10–15%) carry the most Section I score, then work each unit as mechanism routes, error repairs and FRQ task language.
- AP Calculus AB FAQ When is the AP Calculus AB exam in 2027?
What AP Calculus AB covers
Eight official units appear in course order. Each percentage below comes from the College Board's published ranges and is stated with its multiple-choice denominator; it is a range, not a promised question count or a free-response allocation.
How AP Calculus AB is assessed
Every figure below is source-verified against the official College Board specification. Confirm the current structure, timing and tools on the official specification before test day.
| Multiple choice | 42 questions · 100 minutes · 50% of the exam score |
|---|---|
| Free response | 6 questions · 90 minutes · 50% of the exam score |
| Calculator split | Calculator required on 13 MCQs and 2 FRQs; not allowed on 29 MCQs and 4 FRQs |
Official sources (as of 2026-08-29): AP Calculus AB Exam · Course and Exam Description · 2027 AP Exam Dates. Structure, timing, and calculator rules can change; confirm them at the official source before test day.
An Intermediate Value Theorem guarantee with visible hypotheses
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 − 1.
Which statement correctly proves that p has at least one zero in the interval (1, 2)?
Choose the best answer.
A. Because p is a polynomial, it is continuous on [1, 2]; p(1) = −1 and p(2) = 5, so the Intermediate Value Theorem guarantees some c in (1, 2) with p(c) = 0.
✅ Correct! 😊 This states continuity, shows that 0 lies between the endpoint values, and gives the correct interior conclusion. It does not claim uniqueness.
B. Because p(1) and p(2) have opposite signs, p has exactly one zero in (1, 2).
❌ Not quite 😭 The sign change helps establish existence, but continuity is not stated and the theorem does not by itself prove exactly one zero.
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C. Because p is differentiable, p has a point c in (1, 2) where p′(c) = 0, so p(c) = 0.
❌ Not quite 😭 This confuses a derivative conclusion with a function value. A point where p′ is zero need not be a zero of p.
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D. Because p(1) = −1 and p(2) = 5, one endpoint must be a zero of p.
❌ Not quite 😭 Neither endpoint value is zero. The relevant conclusion is that continuity forces an interior input with output 0.
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Full explanation
- Step 1A polynomial is continuous everywhere, so p is continuous on the closed interval [1, 2].
- Step 2Compute p(1) = −1 and p(2) = 5; therefore 0 lies between the two endpoint outputs.
- Step 3By the Intermediate Value Theorem, there is at least one c in (1, 2) such that p(c) = 0.
- Step 4No evidence given here establishes that this zero is unique.
The function g is continuous, and A(x) = ∫ from 0 to x² of g(t) dt.
What is A′(x)?
Choose the best answer.
A. g(x²)
❌ Not quite 😭 This applies the Fundamental Theorem but omits the derivative of the upper limit x², so the chain factor is missing.
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B. 2xg(x²)
✅ Correct! 😊 This evaluates the integrand at the upper limit and multiplies by the derivative 2x of that limit.
C. 2xg(x)
❌ Not quite 😭 The factor 2x is present, but the integrand must be evaluated at x² rather than at x.
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D. ∫ from 0 to x² of g′(t) dt
❌ Not quite 😭 Differentiating the integrand inside the same definite integral does not differentiate an accumulation function with a variable endpoint.
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Full explanation
- Step 1Treat the upper endpoint u(x) = x² as the inner function.
- Step 2The Fundamental Theorem gives the outer derivative g(u(x)).
- Step 3The chain rule contributes u′(x) = 2x, so A′(x) = 2xg(x²).
Worked solution from the guide: An Intermediate Value Theorem guarantee with visible hypotheses
A fresh polynomial changes sign between two stated inputs, and the task is to prove that a zero occurs between them.
- Step 1Establish continuity from the polynomial form.
- Step 2Evaluate both endpoint outputs and display the sign change.
- Step 3Apply the Intermediate Value Theorem on the named closed interval.
Key terms and definitions for AP Calculus AB
- Limit
- The value a function approaches as its input approaches a specified point or direction.
- Continuity
- The condition that a function value exists, its limit exists, and those two values are equal at the input.
- Derivative
- An instantaneous rate of change, equivalently the limiting slope of secant lines when that limit exists.
- Critical point
- An interior domain point where the derivative is zero or does not exist.
- Concavity
- The bending behavior determined by whether the derivative increases or decreases, or by the sign of the second derivative.
- Accumulation function
- A function defined by a definite integral whose variable limit records net accumulated change.
- Riemann sum
- A sum of rectangle areas used to approximate or define a definite integral.
- Slope field
- A grid of short segments showing the derivative prescribed by a differential equation at many points.
- Particular solution
- A solution to a differential equation whose constant is fixed by an initial condition.
- Washer
- A cross section perpendicular to an axis of rotation with area equal to pi times the difference of squared radii.
- Intermediate Value Theorem
- A continuity theorem guaranteeing that every output between two endpoint outputs occurs somewhere inside the interval.
- Fundamental Theorem of Calculus
- The link between accumulation by definite integration and differentiation of an accumulation function.
AP Calculus AB FAQ
When is the AP Calculus AB exam in 2027?
The regular exam is listed for May 10, 2027, Session 1. Start times vary by location under the 2027 policy, so students must use the time and reporting instructions supplied by their AP coordinator.
How many units and worked examples are in this AP Calculus AB guide?
The guide covers all 8 official units in College Board's current course outline and contains 24 complete worked examples, with three fresh examples per unit.
Does this guide promise a fixed number of questions per AP Calculus AB unit?
No. Published unit percentages are multiple-choice ranges, not fixed counts, and they do not create free-response unit weights. The guide uses them for prioritization without converting them into an unsupported layout promise.
Does this AP Calculus AB guide reproduce official questions?
No. Official questions, scoring guidelines, and commentary were mined only for public problem mechanics and loss patterns. Every problem, number, function, context, table, and figure in the guide is newly authored.
Can this guide predict an AP Calculus AB score from raw points?
No stable raw-to-1–5 conversion is promised. Administration-specific scaling can vary, so the guide focuses on complete computation, rubric-visible reasoning, and verified exam structure rather than an invented cutoff table.
What must a complete worked response contain?
Name the requested quantity. Display the rule or theorem and any required hypotheses, substitute the given values into a visible equation, carry units through rate or accumulation work, and end with a sentence or boxed expression that answers the original prompt. If direction, interval, or long-run behavior is requested, interpret it explicitly.
How should calculator output appear in a solution?
Write the mathematical setup before entering values. Preserve the integral, derivative, equation, or intersection condition that the calculator evaluates; then report a sensible approximation and attach units or context. Calculator access does not replace justification, and the exam instructions for required precision control how many digits should appear.
How can a wrong result be diagnosed efficiently?
Recheck the target and representation before recomputing. A wrong sign often comes from reversed bounds or lost direction; wrong units expose a mismatched derivative or integral; an implausible magnitude can reveal a missing width, chain factor, radius shift, or endpoint. Use the nearby graph or table as an independent scale check.
Why does each worked example show its computation?
The displayed setup, operation, and result make the mathematical justification auditable.
Are the practice numbers taken from released questions?
No. The examples use newly authored functions, contexts, constants, tables, and plots.
Does AP Calculus AB provide an exam-day formula sheet?
No. The 2026-27 CED identifies calculator access but no formula or reference sheet; formulas in the appendix are memory-required review material.
How to study for AP Calculus AB
Follow the book's computation-first route: identify the requested quantity, write the governing rule with its hypotheses, substitute visibly, and use the graph or table as an independent sign, scale, interval, and unit check.
Use the College Board's published ranges to prioritize review without predicting fixed item counts. Interleave a limit or theorem decision, a derivative interpretation, an accumulation setup, and a differential-model or geometry task so formula recall stays attached to meaning.
The complete 70-page book, included unchanged, supplies the full worked route and appendix. Because no exam-day formula or reference sheet is supplied, retrieve each formula from memory and state its condition before using it.
