Section II is six free-response questions worth 9 points each, half of the exam score, in 90 minutes. Part A (Questions 1 and 2, 30 minutes) requires a graphing calculator; Part B (Questions 3 through 6, 60 minutes) allows none. The questions are not officially named task types, but the same archetypes recur, and BC adds two of its own: series and parametric, polar, and vector motion. Every question is scored by a rubric that pays for setups, justifications, and interpretations as much as for final numbers. For the section-by-section format of the whole exam, see the exam guide.
Questions
6
9 points each
Time
90 min
30 + 60 by part
Weighting
50%
of the exam score
Calculator
Part A only
required on Questions 1-2
The recurring archetypes
Seven setups that keep coming back
1
Contextual rate and accumulation
Units 4, 6, 8 · Real-world context · Either part
A rate of change in a real setting: water flowing into a tank, people joining a line, sand arriving on a beach. The question asks for the accumulated change over an interval, an average value, and what a definite integral of the rate means in the story's own units.
Write the integral before computing anything: the setup ∫ab r(t) dt earns its points even when the arithmetic never finishes.
Interpret with units: "the amount of water, in gallons, that enters the tank from t = 0 to t = 8" is worth a point; a bare number is not.
Net change needs a starting value: the amount at time b is the amount at time a plus the integral of the rate, not the integral alone.
2
Particle motion
Units 4, 6, 8 · Straight-line motion
A particle moves along a line with velocity v(t). Position, displacement, total distance, and whether the speed is increasing all come from reading v and its derivative correctly, and every claim needs a sign-based justification.
Speed increases when v and a share a sign. Justify with the signs of both velocity and acceleration, never with two speed values.
Displacement integrates v(t); total distance integrates |v(t)|. They differ whenever the particle turns around.
A direction change is a sign change of v, not merely a zero of v; check that v actually crosses.
3
Area and volume
Unit 8 · Region bounded by curves
A region bounded by curves: its area, the volume of the solid formed by revolving it about a line, or the volume of a solid whose cross sections are built on it. Nearly every year this is a full question.
Find the intersection points first, and on the calculator part store them to full precision instead of retyping rounded values.
Area is top minus bottom (or right minus left) across the whole interval; check whether the curves trade places partway.
Washers subtract squared radii: π times the integral of R2 - r2, never of (R - r)2.
4
Table and graph analysis
Units 2, 5, 6 · Values, not formulas
The function arrives as a table of values or as the graph of its derivative, so every estimate and every conclusion must come from what is given: Riemann or trapezoidal sums from the table, average rates of change, and reasoning about f from the graph of f′.
A Riemann sum from a table uses the table's subintervals, which are rarely equal width; show the sum term by term.
State the hypotheses before MVT or IVT: f is continuous on [a, b], and differentiable on (a, b) when the Mean Value Theorem is the tool.
On a graph of f′, heights are slopes of f and areas are changes in f: f increases where f′ is positive, and extrema of f sit where f′ changes sign.
5
Differential equations
Unit 7 · Slope fields and separation
A differential equation with a slope field or an initial condition. Sketch or match solution curves, verify that a proposed function satisfies the equation, and solve by separation of variables for the particular solution.
Separate completely before integrating: every y on one side, every x on the other, then write + C on one side immediately.
Use the initial condition to find C before solving for y, and keep the branch (sign or domain) the initial condition selects.
Verifying a solution means substituting into both sides of the equation, not re-solving it from scratch.
6
Series
Unit 10 · BC only
A Taylor or Maclaurin polynomial built from given derivative values or from a known series, questions of interval and radius of convergence, and an error bound: alternating series error or the Lagrange error bound.
The nth coefficient is f(n)(a) / n!, read straight from the derivative values the problem supplies; do not differentiate what you were already handed.
Endpoints of the interval of convergence are tested separately, one at a time, with the convergence test named.
The Lagrange error bound needs a bound on the next derivative over the interval; say where that bound comes from, not just the number.
7
Parametric, polar, and vector motion
Unit 9 · BC only
Motion in the plane: a parametric or vector-valued position, the slope dy/dx along a parametric curve, speed and total distance traveled, and the area enclosed by a polar curve.
Slope on a parametric curve is (dy/dt) / (dx/dt), evaluated at the named value of t; it is not dy/dt alone.
Speed is the square root of (dx/dt)2 + (dy/dt)2, and distance traveled integrates the speed.
Polar area is one half the integral of r2 dθ; the θ bounds come from where curves intersect or where r = 0, and those must be found, not guessed.
The section does not quarantine BC content: a motion question can go parametric or vector-valued, and series work can appear inside an approximation or accumulation context. Expect the archetypes above to mix rather than arrive one per question.
Earning the points
What the rubric rewards
Show the setup. Each question is scored out of 9 by a rubric that pays for the pieces: the correct integral or derivative expression earns its points even when the final number is wrong or missing. Write the expression first, always.
Interpret in context, with units. When a question asks what a value means, the answer is a sentence naming the quantity, its units, and the interval, read back into the situation. A number with no units earns no interpretation point.
Justify from the hypotheses. Before applying the Intermediate Value Theorem, the Mean Value Theorem, or the Extreme Value Theorem, state that the conditions hold: "f is continuous on [a, b]" is the line the rubric looks for.
Three decimal places. Where a calculator value is expected, report answers accurate to three places after the decimal point, and store intermediate values in the calculator instead of retyping rounded ones.
Never leave a bare number where reasoning is asked. "Justify your answer" and "explain" attach the point to the reasoning; the number alone earns nothing there, even when it is right.
Stay in radian mode on the calculator questions. Degree mode quietly wrecks every trigonometric evaluation, and no rubric forgives it.
The fastest way to stop losing FRQ points
Most lost points trace to a small set of documented misconceptions: subtracting radii before squaring in a washer, forgetting to test a series at its endpoints, taking dy/dt alone as a parametric slope, or applying a theorem whose hypotheses were never checked. Diagnose which ones are costing you, then drill exactly those in Unit 4, Unit 7, Unit 8, Unit 9, and Unit 10.
College Board publishes decades of past free response questions with scoring guidelines, sample student responses, and chief reader commentary. Scoring your own attempts against the real rubrics is the single best way to learn where the points actually sit. Find them on the AP Calculus BC exam page, along with the Course and Exam Description behind the whole framework. For how Section II fits into the full exam day, see the exam guide.
Common questions
FAQ
How many free response questions are on AP Calculus BC?
Six, each scored out of 9 points by rubric, for 50 percent of the exam score. Part A is Questions 1 and 2 in 30 minutes with a required graphing calculator; Part B is Questions 3 through 6 in 60 minutes with no calculator.
Which AP Calculus BC FRQs allow a calculator?
Part A, which is Questions 1 and 2, requires a graphing calculator and runs 30 minutes. Part B, which is Questions 3 through 6, allows no calculator and runs 60 minutes. Keep the calculator in radian mode.
What do the AP Calculus BC FRQs cover that AB's do not?
Two archetypes are BC only: series, meaning Taylor and Maclaurin polynomials, convergence, and the Lagrange error bound, and parametric, polar, and vector motion. The rest of the section draws on the same archetypes as AB, and BC content can mix into any of them.
How precise do calculator answers need to be?
Unless a question says otherwise, report decimal answers accurate to three places after the decimal point, and avoid rounding intermediate values; store them in the calculator instead of retyping rounded copies.
Do I lose points for a right answer with no work?
You can. The 9 points on each question sit on a rubric that pays for setups, justifications, and interpretations as well as answers. Where work or reasoning is requested, an unsupported answer may earn nothing.
