--- title: "AP Calculus AB/BC No-Calculator FRQ Practice" description: "59 No-Calculator FRQs written in the AP format, each scored part by part." canonical: "https://fiveable.me/ap-calc/frqs/no-calculator" type: "frq-type-practice" subject: "AP Calculus AB/BC" --- # AP Calculus AB/BC No-Calculator FRQ Practice ## What the No-Calculator FRQ asks No-calculator FRQs are questions 3 to 6 on the AP Calculus AB and BC exams, in Part B of the FRQ section. Each has four parts with numbers built to be worked by hand, and you justify answers with the right theorem or test. - **60 min**: for the 4 questions in Part B - **9 points**: across parts A to D - **50%**: of your score, for all six FRQs | Part | What it asks | Points | | --- | --- | --- | | A: Evaluate | A value, rate, or estimate worked by hand, often with units | 2 | | B: Apply | A derivative or integral applied to the function or context | 2 | | C: Extend | Another calculus task on the same setup, such as an area or an approximation | 2 | | D: Conclude | A final result, often backed by a reason or a named theorem | 3 | ## How No-Calculator FRQ practice works 1. **Read the full question.** The setup, any graph or table, and parts A to D, laid out the way the exam shows them. 2. **Write on a 15-minute timer.** About the time each question gets on exam day. Pause it or turn it off, and your response saves as you go. 3. **Submit for a score out of 9.** Your response is scored against the scoring guidelines written for that question. Your first score is included. ## All 59 questions ### Unit 1: Limits and Continuity (3) - [Piecewise function continuity and function rates of change](/ap-calc/frq-practice?id=6abc47e5ab79ec2ed63de4f3&type=frqs-no-graphing-calculator) - [Limits, continuity, and rates of change of functions](/ap-calc/frq-practice?id=6abd64c72dc3fbef5b3814a6&type=frqs-no-graphing-calculator) - [Limits, average rates of change, and continuity theorem](/ap-calc/frq-practice?id=6abd65302dc3fbef5b3814b1&type=frqs-no-graphing-calculator) ### Unit 2: Differentiation: Definition and Fundamental Properties (2) - [Average rate of change and tangent lines](/ap-calc/frq-practice?id=6abc2dcf962b3568ec95d8d5&type=frqs-no-graphing-calculator) - [Polynomial rates of change and tangent lines](/ap-calc/frq-practice?id=6abbf44a4209d9fbdc0d37ba&type=frqs-no-graphing-calculator) ### Unit 3: Differentiation: Composite, Implicit, and Inverse Functions (3) - [Derivative calculations and tangent line slopes](/ap-calc/frq-practice?id=69781d78fdc816ae7dfbc248&type=frqs-no-graphing-calculator) - [Derivatives of composite and inverse functions](/ap-calc/frq-practice?id=6abc48ab44825425e3a57e5f&type=frqs-no-graphing-calculator) - [Inverse trigonometric derivatives and chain rule applications](/ap-calc/frq-practice?id=6abd6ec8d7c548680aa34b4f&type=frqs-no-graphing-calculator) ### Unit 4: Contextual Applications of Differentiation (2) - [Water drainage rates and tangent line approximation](/ap-calc/frq-practice?id=6abc34fc868334c2f215d563&type=frqs-no-graphing-calculator) - [Metal rod temperature and heating rates](/ap-calc/frq-practice?id=6abc354d868334c2f215d56f&type=frqs-no-graphing-calculator) ### Unit 5: Analytical Applications of Differentiation (3) - [Cubic function extrema and mean value theorem](/ap-calc/frq-practice?id=698a1f701ee6abd3ecdb1154&type=frqs-no-graphing-calculator) - [Cubic function extrema and mean value theorem](/ap-calc/frq-practice?id=698a1fa91ee6abd3ecdb1156&type=frqs-no-graphing-calculator) - [Derivative graphs, concavity, inflection points, mean value theorem](/ap-calc/frq-practice?id=6abea172efde7205a0bbffef&type=frqs-no-graphing-calculator) ### Unit 6: Integration and Accumulation of Change (2) - [Definite integrals, Riemann sums, trapezoidal approximation](/ap-calc/frq-practice?id=6978048c01261f3ca34513f6&type=frqs-no-graphing-calculator) - [Water inflow rate and tank volume accumulation](/ap-calc/frq-practice?id=698a24501ee6abd3ecdb1188&type=frqs-no-graphing-calculator) ### Unit 7: Differential Equations (5) - [Logistic growth model for deer population](/ap-calc/frq-practice?id=698a24871ee6abd3ecdb118c&type=frqs-no-graphing-calculator) - [Logistic population growth with carrying capacity](/ap-calc/frq-practice?id=698a259e1ee6abd3ecdb1193&type=frqs-no-graphing-calculator) - [Logistic population growth differential equation](/ap-calc/frq-practice?id=69d56612ef45aff030f3fc69&type=frqs-no-graphing-calculator) - [Medication concentration decay over time](/ap-calc/frq-practice?id=6abc421446d5d9853837bc1c&type=frqs-no-graphing-calculator) - [Salt concentration approaching equilibrium in a tank](/ap-calc/frq-practice?id=6abc425146d5d9853837bc26&type=frqs-no-graphing-calculator) ### Unit 8: Applications of Integration (14) - [Water flow rates and tank accumulation over time](/ap-calc/frq-practice?id=6abc2dc1fd23875525569657&type=frqs-no-graphing-calculator) - [Vase volume and water accumulation rates](/ap-calc/frq-practice?id=6abd602152be57f42940e456&type=frqs-no-graphing-calculator) - [Parabolic base, square cross-sections, glass volume accumulation](/ap-calc/frq-practice?id=6abd6ffa87ef3825c0cbcb12&type=frqs-no-graphing-calculator) - [Average function value and enclosed area calculation](/ap-calc/frq-practice?id=697804a401261f3ca34513f7&type=frqs-no-graphing-calculator) - [Velocity functions, position, and bounded area](/ap-calc/frq-practice?id=698a205d1ee6abd3ecdb115e&type=frqs-no-graphing-calculator) - [Particle motion, velocity function, position integration](/ap-calc/frq-practice?id=698a20d11ee6abd3ecdb1162&type=frqs-no-graphing-calculator) - [Particle motion: velocity, displacement, distance traveled](/ap-calc/frq-practice?id=698a210a1ee6abd3ecdb1164&type=frqs-no-graphing-calculator) - [Particle motion, velocity functions, area under curves](/ap-calc/frq-practice?id=698a215a1ee6abd3ecdb1169&type=frqs-no-graphing-calculator) - [Particle motion, displacement, and average velocity](/ap-calc/frq-practice?id=698a22101ee6abd3ecdb1171&type=frqs-no-graphing-calculator) - [Particle motion, velocity function, position displacement](/ap-calc/frq-practice?id=698a233f1ee6abd3ecdb117b&type=frqs-no-graphing-calculator) - [Piecewise function: area, average value, displacement](/ap-calc/frq-practice?id=698a24341ee6abd3ecdb1186&type=frqs-no-graphing-calculator) - [Particle motion, velocity function, displacement integral](/ap-calc/frq-practice?id=698a24631ee6abd3ecdb1189&type=frqs-no-graphing-calculator) - [Area between curves, particle motion, average value](/ap-calc/frq-practice?id=698a24dc1ee6abd3ecdb1190&type=frqs-no-graphing-calculator) - [Reservoir water inflow, outflow, and accumulation](/ap-calc/frq-practice?id=6abdba0ffbde87256bdd554f&type=frqs-no-graphing-calculator) ### Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC only) (18) - [Parametric curves: derivatives, tangent slopes, arc length](/ap-calc/frq-practice?id=697804c501261f3ca34513f9&type=frqs-no-graphing-calculator) - [Particle velocity, acceleration, and position over time](/ap-calc/frq-practice?id=6abd7248839c13d8e177b215&type=frqs-no-graphing-calculator) - [Particle motion with vector-valued acceleration](/ap-calc/frq-practice?id=6abea376084050db83b34e85&type=frqs-no-graphing-calculator) - [Parametric curves, derivatives, arc length](/ap-calc/frq-practice?id=698a201d1ee6abd3ecdb1159&type=frqs-no-graphing-calculator) - [Parametric curve motion and arc length](/ap-calc/frq-practice?id=698a210c1ee6abd3ecdb1165&type=frqs-no-graphing-calculator) - [Parametric motion, arc length, polar areas](/ap-calc/frq-practice?id=698a21431ee6abd3ecdb1166&type=frqs-no-graphing-calculator) - [Parametric curves, derivatives, velocity, arc length](/ap-calc/frq-practice?id=698a21851ee6abd3ecdb116b&type=frqs-no-graphing-calculator) - [Parametric curves, particle motion, arc length](/ap-calc/frq-practice?id=698a21b21ee6abd3ecdb116d&type=frqs-no-graphing-calculator) - [Parametric and polar curves, derivatives, arc length](/ap-calc/frq-practice?id=698a21e61ee6abd3ecdb1170&type=frqs-no-graphing-calculator) - [Parametric motion: tangent slope, speed, arc length](/ap-calc/frq-practice?id=698a22111ee6abd3ecdb1172&type=frqs-no-graphing-calculator) - [Parametric curves, derivatives, arc length](/ap-calc/frq-practice?id=698a22901ee6abd3ecdb1176&type=frqs-no-graphing-calculator) - [Parametric spiral motion and tangent lines](/ap-calc/frq-practice?id=698a235a1ee6abd3ecdb117d&type=frqs-no-graphing-calculator) - [Parametric motion, arc length, velocity analysis](/ap-calc/frq-practice?id=698a23b71ee6abd3ecdb1182&type=frqs-no-graphing-calculator) - [Parametric and polar curves, derivatives, arc length](/ap-calc/frq-practice?id=698a23eb1ee6abd3ecdb1184&type=frqs-no-graphing-calculator) - [Parametric motion, arc length, polar curves](/ap-calc/frq-practice?id=698a24b91ee6abd3ecdb118e&type=frqs-no-graphing-calculator) - [Parametric particle motion, velocity, and distance traveled](/ap-calc/frq-practice?id=6abc41c15e8636432f6f123a&type=frqs-no-graphing-calculator) - [Polar curve intersections, enclosed area, and slope](/ap-calc/frq-practice?id=6abd4b4b993c54ba70ba4afe&type=frqs-no-graphing-calculator) - [Polar curves, radial rates, and tangent slopes](/ap-calc/frq-practice?id=6abea16921f0a8759b65f2c0&type=frqs-no-graphing-calculator) ### Unit 10: Infinite Sequences and Series (BC only) (7) - [Maclaurin polynomial approximation of logarithmic function](/ap-calc/frq-practice?id=698a214d1ee6abd3ecdb1167&type=frqs-no-graphing-calculator) - [Alternating series convergence and error bounds](/ap-calc/frq-practice?id=698a224f1ee6abd3ecdb1174&type=frqs-no-graphing-calculator) - [Alternating series convergence and remainder estimation](/ap-calc/frq-practice?id=698a24851ee6abd3ecdb118b&type=frqs-no-graphing-calculator) - [Convergence tests for infinite series and integrals](/ap-calc/frq-practice?id=6abd4aa5ac289158fbc4733d&type=frqs-no-graphing-calculator) - [Convergence tests for infinite series](/ap-calc/frq-practice?id=6abd4af2ac289158fbc47347&type=frqs-no-graphing-calculator) - [Taylor polynomial approximation and Lagrange error bounds](/ap-calc/frq-practice?id=6abd45e3918039f9563296e7&type=frqs-no-graphing-calculator) - [Geometric series integration and approximation errors](/ap-calc/frq-practice?id=6abd4a43ac289158fbc47332&type=frqs-no-graphing-calculator) ## Sample question **Piecewise function continuity and function rates of change** The function f is defined by f(x) = (x² - 5x + 6)/(x - 3), x < 3; 2x - 5, x ≥ 3 The function g is continuous on the closed interval 0 ≤ x ≤ 8. Selected values of g(x) are given in the table shown. Data Tables: Headers: x | 0 | 2 | 4 | 6 | 8 g(x) | -3 | -1 | 2 | 5 | 4 Part A (3 points): Prompt: Find lim_(x → 3⁻)f(x) and lim_(x → 3⁺)f(x). Is f continuous at x = 3? Justify your answer using the definition of continuity. Verb: find Topic: 1.11 Defining Continuity at a Point Part B (1 point): Prompt: Use the average rate of change of g over the interval 2 ≤ x ≤ 4 to approximate the instantaneous rate of change of g at x = 3. Show the work that leads to your answer. Verb: approximate Topic: 2.1 Defining Average and Instantaneous Rates of Change at a Point Part C (2 points): Prompt: Must there be a value c, for 0 < c < 2, such that g(c) = -2? Must there be a value k, for 6 < k < 8, such that g(k) = 4.5? Justify your answers. Verb: determine Topic: 1.16 Working with the Intermediate Value Theorem (IVT) Part D (3 points): Prompt: A new function h is defined by h(x) = (x² - 5x + 6)/(x - 3), x < 3; ax² - 4, x ≥ 3 where a is a constant. Find the value of a for which h is continuous at x = 3. Justify your answer using the definition of continuity. Verb: find Topic: 1.11 Defining Continuity at a Point ## Questions **How many responses can I get scored?** You can read every question without a plan. Your first scored response is included. A plan unlocks unlimited scoring. **How is the no-calculator FRQ different from the calculator FRQ?** Calculator questions lean on integrals you can’t do by hand and on solving messy equations. No-calculator questions use friendly numbers and reward exact algebra and clear justifications. **How do I type math in my answer?** Type it in plain text, use the superscript button for exponents, and pick symbols like ∫ and √ from the special characters menu. You can also switch to handwrite and add a photo of your work. **Are these for Calculus AB or BC?** Both. The AB and BC exams use the same FRQ format, and some of these questions cover BC-only topics like series and parametric equations. **How are the 9 points split?** Across parts A to D, most often 2, 2, 2, and 3. The split changes by question, and on the exam each part is scored on its own. ## Related - [No-Calculator FRQ exam guide](/ap-calc/ap-calculus-ab-bc-exams/ap-calc-frq/study-guide/ap-calc-frq) - [AP Calculus AB/BC Calculator FRQ practice](/ap-calc/frqs/calculator) - [AP Calculus AB/BC FRQs](/ap-calc/frqs) - Any question by id: the `get_frq` tool on the [Fiveable MCP server](/mcp/docs) ## About This Document Canonical Fiveable pages are available as Markdown at the same path plus `.md`. - [llms.txt](https://fiveable.me/llms.txt): index of Fiveable's sections and URL patterns - [llms-full.txt](https://fiveable.me/llms-full.txt): complete subject and unit listing - [MCP server](https://fiveable.me/mcp): call Fiveable as tools instead of fetching pages (`https://fiveable.me/api/mcp`) - [MCP server for AP teachers](https://fiveable.me/mcp/teachers): a teacher's classes, assignments and AP-rubric grading (`https://fiveable.me/api/mcp/teacher`)