AP Calc AB FRQ: All six question types with worked examples

The AP Calculus AB free-response section is six questions in 90 minutes and worth 50 percent of your total score. Two questions require a graphing calculator and four don't, and the six rotate through a stable set of types year after year: rate-in rate-out, particle motion, area and volume, differential equation, analytical function, and graph or table interpretation. Recognizing which type you're looking at within thirty seconds is one of the biggest score levers on the exam.

FRQ scoring is granular. Each part awards points for specific pieces of the response: setup, correct application of a definition, correct notation, correct final answer with units. A student who arrives at the correct answer with no visible setup can lose the setup points; a student who sets up correctly but slips on the arithmetic usually keeps them.

Section II has six FRQs in 90 minutes, split into two parts. Part A is Questions 1 and 2 with a graphing calculator, 30 minutes. Part B is Questions 3 through 6 without a calculator, 60 minutes. Once you've moved to Part B you may still go back to the Part A questions, but you are not allowed to use your calculator on them.

Each FRQ is worth 9 points. Every part is independently scored, so a wrong answer to part a doesn't cascade into zero on parts b, c, and d as long as your later work is internally consistent; readers apply what they call "read on" scoring, giving credit for correct application of an incorrect earlier answer.

AP Calc readers use conventions that don't always feel intuitive from a math class. Knowing them upfront is worth several points across the section.

Units earn points. If the problem is in context (gallons per minute, meters per second), your final answer needs units. Missing units can cost a point where the rubric asks for them.

Definite integrals with limits and integrand written out earn setup points before you compute. Writing the integral correctly is often worth 1 of 2 or 3 points on that part; leaving it out to jump straight to a decimal answer forfeits the setup mark.

Calculator use requires the setup shown. On calculator-active questions, write the integral or derivative expression, then "= [decimal]" from the calculator. Just writing the decimal earns no setup credit.

Interpretation in context earns points on rate and particle motion problems. When asked what a value means, a numerical answer without a sentence like "the amount of water in the tank is decreasing at a rate of 12 gallons per minute at t = 5" often loses a point.

Decimal places matter. Three decimal places on calculator-active questions unless otherwise specified. Rounding to two can cost the final answer point.

The rate-in rate-out problem shows up almost every year. You're given two rates (water flowing in and out of a tank, cars entering and leaving a parking lot) as functions of time, plus an initial condition. Typical sub-parts ask for the net rate at a specific time, the total change over an interval, and the maximum or minimum quantity.

The standard trap: mixing up rate and quantity. Net rate at time t is R_in(t) minus R_out(t). Total change from a to b is the definite integral of that net rate. Quantity at time t is the initial quantity plus the definite integral of the net rate from 0 to t.

For the max or min sub-part, use the sign of the net rate: quantity is maximized when net rate changes from positive to negative; minimized when it changes from negative to positive. Check endpoints too.

Area and volume problems live in Part B (no calculator) most years, which means the functions are chosen so the integrals are manageable by hand. You're usually given two curves and asked for the area between them, then the volume of the solid formed by rotating that region, then sometimes the volume of a solid with a specified cross-section.

For area, always sketch the region and identify which curve is on top over the interval. Set up the integral of (top minus bottom) dx or (right minus left) dy depending on orientation. Solve.

For volume of revolution, use disks or washers: cross sections taken perpendicular to the axis of rotation. This is the method the AP Calculus course covers, and it's what the rubric expects. Write out the integral expression before evaluating.

For cross-sectional volume, the integrand is the area of one cross-section as a function of the position variable. Square cross-sections give side-squared; equilateral triangle gives (side-squared) times root three over four; semicircular gives pi over 8 times side-squared.

The analytical problem is the one that most rewards clean calculus and precise notation. It usually asks you to use the definition of the derivative, apply the Mean Value Theorem or Intermediate Value Theorem, or identify local and global extrema using the first and second derivative tests.

When invoking a theorem, cite it by name and check its conditions. The Mean Value Theorem requires the function to be continuous on the closed interval and differentiable on the open interval; state both before applying. Skipping the condition check often costs a point even if your conclusion is correct.

For extrema, be explicit about your reasoning. "f'(x) = 0 at x = 2 and x = 5. f' changes from positive to negative at x = 2, so f has a local maximum at x = 2. f' changes from negative to positive at x = 5, so f has a local minimum at x = 5." The rubric wants the sign analysis, not just the answer.