§ 02
The 2025 FRQs, decoded
The most recent real exam. Each card tells you which units a question draws on and what each part is really testing — then try the real thing from the official PDF, 15 minutes per question.
FRQ 1CalculatorUnit 8Unit 5Unit 1
An invasive plant spreads through a fruit grove: \(C(t) = 7.6\arctan(0.2t)\) acres after \(t\) weeks — a rate/accumulation story wrapped around an inverse-trig model.
AAverage value of \(C\) on \([0,4]\) — the \(\frac{1}{b-a}\int_a^b\) formula (topic 8.1). "Show the setup" = write the integral before the calculator touches it.
BFind where the instantaneous rate equals the average rate — Mean Value Theorem territory (5.1), solved numerically with the calculator.
CWrite and evaluate a limit at infinity of \(C'(t) = \tfrac{38}{25+t^2}\) — end behavior, straight from topic 1.15.
DMaximize \(A(t) = C(t) - \int_4^t 0.1\ln x\,dx\) — set \(A'(t)=0\) using FTC (6.4), then justify with the candidates test (5.5).
Where points are lost: forgetting the \(\frac{1}{b-a}\) in part A, and "justifying" the max in D without comparing endpoint values.
FRQ 2CalculatorUnit 8Unit 5
A region \(R\) trapped between \(f(x)=x^2-2x\) and \(g(x)=x+\sin(\pi x)\) — the classic area-and-volume factory.
AArea between curves (8.4): \(\int (g-f)\,dx\) with calculator-found intersections.
BVolume by cross-sections (8.7): rectangles of height \(x\) on base \(g-f\) → \(\int x\,(g(x)-f(x))\,dx\).
CWasher method about \(y=-2\) (8.11–8.12): radii shift by the line — \(R = g+2\), \(r = f+2\). Setup only, don't evaluate.
DParallel tangents: solve \(f'(x) = g'(x)\) — derivatives and equation-solving (Units 2–3).
Where points are lost: forgetting the +2 shift in the washer radii, and dropping the \(\pi\) outside the washer integral.
FRQ 3No calculatorUnit 6Unit 2Unit 1
A student's reading rate \(R(t)\) in words per minute, given only as a table at \(t = 0, 2, 8, 10\) — the table-skills question.
AEstimate \(R'(1)\) with a difference quotient from the table (2.3) — with units (words/min per min).
B"Must there be a \(c\) with \(R(c) = 155\)?" — IVT justification (1.16): cite continuity, show 155 is between table values.
CTrapezoidal sum with the three unequal subintervals (6.2) to approximate \(\int_0^{10} R(t)\,dt\).
DIntegrate a polynomial rate model exactly (6.7) — total words read = \(\int_0^{10} W(t)\,dt\).
Where points are lost: using equal-width trapezoids (the intervals are 2, 6, 2!), and IVT answers that never say "continuous."
FRQ 4No calculatorUnit 6Unit 5
The signature AP problem: the graph of \(f\) (semicircles + segment) defines \(g(x)=\int_6^x f(t)\,dt\); every part reads \(g\) through \(f\).
A\(g'(8)\) — FTC Part 1 (6.4): \(g' = f\), just read the graph.
BInflection points of \(g\) (5.6/5.9): where \(g'' = f'\) changes sign = where \(f\) has a local max/min.
C\(g(12)\) and \(g(0)\) by geometry (6.6): semicircle areas, watching signs and integral direction (\(g(0)=-\int_0^6 f\)).
DAbsolute minimum of \(g\) (5.5): candidates test on critical points of \(g\) (zeros of \(f\)) and endpoints.
Where points are lost: the sign flip when \(x < 6\) (lower limit is 6!), and calling a zero of \(f\) an inflection of \(g\).
FRQ 5No calculatorUnit 4Unit 8
Two particles on the \(x\)-axis — but one is given by its position \(x_H(t)=e^t-4t^2\) and the other by its velocity \(v_J(t)=2t\,(t^2-1)^3\). Knowing which is which is half the problem.
AVelocity of \(H\) = derivative of position (4.2).
BOpposite directions: sign analysis of \(v_H\) and \(v_J\) on intervals (4.2/5.3).
CSpeed increasing? — compare signs of \(v_J(2)\) and \(v_J'(2)\): same sign = speeding up (4.2).
DPosition from velocity (8.2): \(x_J(2) = 7 + \int_0^2 v_J(t)\,dt\) — a u-substitution integral (6.9).
Where points are lost: answering "speed increasing" from \(v' > 0\) alone (you need \(v\) and \(v'\) to agree in sign), and forgetting the initial position 7 in D.
FRQ 6No calculatorUnit 3Unit 4Unit 5
An implicit curve \(y^3 - y^2 - y + \tfrac14 x^2 = 0\) — the implicit-differentiation gauntlet, ending with related rates.
AShow \(\frac{dy}{dx} = \frac{-x}{2(3y^2-2y-1)}\) — implicit differentiation (3.2); "show that" means every step visible.
BTangent-line approximation (4.6) of \(y\) near \((2,-1)\) at \(x = 1.6\).
CVertical tangent (5.12): denominator \(= 0\) with \(x \neq 0\) → solve \(3y^2 - 2y - 1 = 0\) for the valid \(y\).
DRelated rates (4.4–4.5) on a second curve: differentiate \(2xy + \ln y = 8\) with respect to \(t\), plug the point and \(\frac{dx}{dt}=3\).
Where points are lost: in "show that" parts, skipping the step where \(\frac{dy}{dx}\) terms are collected; in D, forgetting the chain rule on \(\ln y\).