Day 25 · Strategy Guide

FRQ Strategy

The free-response section is 50% of your score and the most coachable part of the exam. This is how the points are actually awarded — and how to collect them.

How the section is built Where students leak points Justification language Using the calculator well Practice

How the section is built

Section II is 6 questions, each worth 9 points, 90 minutes total. The 9 points on a question are split across its parts (a), (b), (c), (d) — usually 1 to 4 points each — and every point is tied to an identifiable piece of work, not to the final number alone.

Part A — 2 questions, 30 minutes, graphing calculator required.
Part B — 4 questions, 60 minutes, no calculator. You may keep working on the Part A questions here, but without the calculator.
Budget about 15 minutes per question. Do the calculator questions first, while you have the tool.
Parts are often independent — part (d) rarely needs part (c). Never leave a part blank; a blank is a guaranteed zero, a reasonable attempt often is not.

What a rubric line looks like

A part might be scored “1 point: definite integral 1 point: answer”. If you write only

12.408

from the calculator, the setup point is gone — you needed

\int_2^7 R(t)\,dt

on the page. Two points become one.

Where students leak points

Most lost points on the FRQ are not from not knowing the calculus. They come from four habits:

Writing the answer without the setup — the integral, the derivative expression, or the equation being solved is usually its own point.
Dropping units — on rate, accumulation, and average-value answers, units are frequently a separate point. “ $-3.8$ ” loses to “ $-3.8$ liters per minute”.
Rounding early — carry at least 3 decimals through the middle of a problem; round only the final answer, to 3 decimal places unless told otherwise.
Erasing work — cross out neatly instead. Readers score the best complete attempt; erased correct work earns nothing.

Calculator syntax is not work

“fnInt(v(t),t,0,6)” or “nDeriv(f,x,3)” written as your steps earns nothing. Translate to real notation:

\int_0^6 v(t)\,dt

f'(3)

Do this

On every rate/accumulation part, answer in a full sentence with units and context: “The tank contains

s(8) = 63.104

gallons at time

t = 8

hours.”

Justification language

“Justify your answer” has a precise meaning: name the theorem or the sign fact, state that its hypotheses hold, then state the conclusion. Describing the picture (“the graph goes up”) earns nothing — you must reference

f'

f''

, or a named theorem. Fill in these templates:

Increasing / decreasing: “ $f$ is increasing on $(a, b)$ because $f'(x) > 0$ there.”
Local extremum: “ $f$ has a local maximum at $x = c$ because $f'$ changes from positive to negative at $c$ .”
Concavity / inflection: “The graph is concave up where $f'' > 0$ ; $x = c$ is a point of inflection because $f''$ changes sign at $c$ .”
A value is attained (IVT): “ $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$ , so by the IVT $f(x) = N$ for some $x$ in $(a, b)$ .”
A derivative value is guaranteed (MVT): “ $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , so by the MVT $f'(c) = \dfrac{f(b) - f(a)}{b - a}$ for some $c$ .”
Absolute extremum: “Comparing $f$ at the critical numbers and the endpoints, the largest value is … , so the absolute maximum is … .”

Common mistake

Two of the most common zero-credit justifications: “

f

increases because the curve rises” (no mention of

f'

), and “increasing” with no interval stated.

Using the calculator well

On Part A you are expected to use exactly four built-in operations. The calculator does the arithmetic; you still write the mathematical setup by hand.

Graph a function in a window you choose.
Solve an equation numerically, e.g. $f(x) = 0$ .
Evaluate a numerical derivative $f'(a)$ at a point.
Evaluate a numerical definite integral $\int_a^b f(x)\,dx$ .

Note

Everything else — algebra, exact antiderivatives, limits, sign charts, solving by hand — must be shown as written work. Store exact values in the calculator; do not retype rounded numbers between steps.

“Find when the particle is farthest right”

Write

x'(t) = v(t) = 0

by hand. Use the calculator to solve it and to evaluate

x

at each candidate time and the endpoints. The equation and the comparison are the points — the raw numbers are not.

Practice

Today's task is two full FRQs under timed conditions — 15 minutes each, then score yourself against the rubric before looking at a full walkthrough.

Work from the site's FRQ bank — 7 archetypes (tables, accumulation, particle motion, graph analysis, area/volume, differential equations, implicit), each with its own step-by-step approach and worked examples: open /frq.
Each FRQ page has a timed test mode at /frq/[type]/test.
For the official scoring style, read the year-by-year walkthroughs at /frq-answers.
Keep an error log: for every point you missed, write the one sentence that would have earned it.