[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-311438-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-311438-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","ap-calculus-ab-section-ii-part-a-free-response-questions","AP® Calculus AB - Section II Part A - Free-Response Questions","","AP® Calculus AB Section II Part A provides two free-response questions to assess students’ calculus skills within a 30-minute exam window. The instructions require a radian-mode graphing calculator, scratch work for planning only, and solutions written in the free-response booklet to be scored. Students approximate rates and definite integrals, interpret integral meaning, solve a systems-style setup involving intersecting graphs, and justify answers using correct mathematical notation and supporting reasoning.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/ap-calculus-ab-section-ii-part-a-free-response-questions/311438/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/ap-calculus-ab-section-ii-part-a-free-response-questions/311438.png","ImageObject",300,407,{"name":42,"@type":43},"Angel","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-29","2026-09-20",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",18,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How long is Section II Part A, and how many free-response questions does it contain?","Question",{"text":63,"@type":64},"Section II, Part A lasts 30 minutes and includes 2 free-response questions.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What calculator settings are required for this exam section?",{"text":68,"@type":64},"A graphing calculator is required, and it must be in radian mode.",{"name":70,"@type":61,"acceptedAnswer":71},"What counts for grading when using scratch paper or the free-response booklet?",{"text":72,"@type":64},"Only work written in the free-response booklet is scored; work on scratch paper is not graded.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},311438,1789978181,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":96,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":124,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":144,"read_time":145,"preview_page_count":4},687207412472,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","2026  \nAP® Calculus AB Free-Response Questions  \n© 2026 College Board. College Board, Advanced Placement, AP, AP Central, and the acorn logo are registered [trademarks of College Board. Visit College Board on the web: collegeboard.org](trademarks of College Board. Visit College Board on the web: collegeboard.org).  \nAP Central is the official online home for [the AP Program: apcentral.collegeboard.org](the AP Program: apcentral.collegeboard.org).  \nCALCULUS AB  \nSECTION II PART A  \nTIME – 30 MINUTES  \nDirections:  \nSection II, Part A has 2 free-response questions and lasts 30 minutes.  \nA graphing calculator is required for the questions on this part ofthe exam. You may use a handheld graphing calculator or the calculator available in this application. Make sure your calculator is in radian mode.  \nYou may use the available paper for scratch work and planning, but only work written in the freeresponse booklet will be scored. Any work done on scratch paper will not be scored. In the freeresponse booklet, write your solution to each part of each question in the space provided for that part. For questions that have sub-parts, be sure to label those clearly in your solution. Use a pencil or a pen with black or dark blue ink.  \nYou are permitted to use your calculator to solve an equation, find the derivative of a function ata point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your question, namely the equation, function, or integral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produce your results. Show all of your work, even though a question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit.  \nYour work must be expressed in standard mathematical notation rather than calculator syntax. For example, y15 x2 dx may not be written as fnInt^ X2, X,1, 5h.  \nUnless otherwise specified, answers (numeric or algebraic) need not be simplified. If you use decimal approximations in calculations, your work will be scored on accuracy. Unless otherwise specified, your final answers should be accurate to three places after the decimal point.  \nUnless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f^x h is a real number.  \nYou can go back and forth between questions in this part until time expires. The clock will turn red when 5 minutes remain—the proctor will not give you any time updates or warnings.  \nNote: This exam was originally administered digitally. It is presented here in a format optimized for teacher and student use in the classroom.  \n1. Male birds of a certain species arrive at a nesting area over a thirty-day period. The rate at which the male birds arrive at the nesting area at time t days is modeled by a differentiable function M, where M^t h is measured in number of birds per day. Selected values of M^t h are shown in the table.  \n\n| t (days) | 0 | 5 | 10 | 15 | 20 | 25 | 30 |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| M^t h (birds per day) | 2 | 7 | 16 | 6 | 5 | 2 | 0 |\n\n(Note: Your calculator should be in radian mode.)  \nPart A  \nApproximate M l ^7 .5h using the average rate of change of M over the interval 5 \\# t \\# 10. Show the work that leads to your answer, and indicate units of measure.  \nPart B  \ni. Use a midpoint Riemann sum with the three subintervals 60, 10@, 610, 20@, and 620, 30@ to approximate y030 M^t hdt . Show the work that leads to your answer.  \nii. Interpret the meaning of y030 M^t hdt in the context of the problem.  \nPart C ","cbCaitqWcgOhnSSS","https://ap.wps.com/l/cbCaitqWcgOhnSSS","pdf",311142,"English","# Section II Part A\n## Directions\n## Question 1\n### Part A\n### Part B\n### Part C\n### Part D\n## Question 2\n### Part A\n### Part B\n### Part C\n### Part D","[{\"question\":\"How long is Section II Part A, and how many free-response questions does it contain?\",\"answer\":\"Section II, Part A lasts 30 minutes and includes 2 free-response questions.\"},{\"question\":\"What calculator settings are required for this exam section?\",\"answer\":\"A graphing calculator is required, and it must be in radian mode.\"},{\"question\":\"What counts for grading when using scratch paper or the free-response booklet?\",\"answer\":\"Only work written in the free-response booklet is scored; work on scratch paper is not graded.\"}]","AP® Calculus AB - Section II Part A - Free-Response Questions | PDF",1789894285,25]