[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-464426-105":59,"doc-detail-464426-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","ap-precalculus-free-response-questions-section-ii-part-a-question-1-2","AP® Precalculus Free-Response Questions - Section II, Part A - Question 1-2","","AP Calculus/Precalculus-style free-response exam questions for AP® Precalculus, Section II with Parts A and B, presented in a timed format. The document includes calculator-required and calculator-prohibited problems on composing functions, solving equations from graph constraints, exponential/logarithmic modeling with average rate of change and error analysis, and trigonometric modeling of a rolling tire with sinusoidal parametrization. Instructions specify graph interpretation, limit notation, inverse-function reasoning, and required work for each subpart.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/ap-precalculus-free-response-questions-section-ii-part-a-question-1-2/464426/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/ap-precalculus-free-response-questions-section-ii-part-a-question-1-2/464426.png","ImageObject",300,407,{"name":92,"@type":93},"Adam","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-23","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"Which questions require a graphing calculator in Section II, Part A?","Question",{"text":113,"@type":114},"Section II, Part A states that a graphing calculator is required for the questions in that part.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How is the video game daily sales function modeled in Question 2?",{"text":118,"@type":114},"Sales are modeled by G(t) = a + b ln(t + 1), where G(t) is measured in thousands of units and t is days since the initial sales day.",{"name":120,"@type":111,"acceptedAnswer":121},"What sinusoidal modeling information is used for the rolling tire problem in Question 3?",{"text":122,"@type":114},"Question 3 models the height of point W using a sinusoidal function h(t) = a sin(b(t + c)) + d, based on periodic maxima and the given contact times and maximum height.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},464426,1790816235,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":52,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},1374404737137,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","2024  \nAP® Precalculus Free-Response Questions  \n© 2024 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](collegeboard.org).  \n[AP Central is the official online home for the AP Program: apcentral.collegeboard.org](AP Central is the official online home for the AP Program: apcentral.collegeboard.org).  \nPRECALCULUS  \nSECTION II, Part A  \nTime—30 minutes  \n2 Questions  \nA GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS.  \nGO ON TO THE NEXT PAGE.  \n1. The figure shows the graph of the function f on its domain of −3.5 £ x £ 3.5. The points (−3, 1) , (0, 1), and (3, 1) are on the graph of f . The function g is given by g(x) = 2.916 · (0 .7) x.  \n(A) (i) The function h is defined by h(x) = (g 􀀂 f )(x) = g( f (x)). Find the value of h(3) as a decimal approximation, or indicate that it is not defined.  \n(ii) Find all values of x for which f (x) = 1, or indicate that there are no such values.  \n(B) (i) Find all values of x , as decimal approximations, for which g(x) = 2, or indicate that there are no such values.  \n(ii) Determine the end behavior of g as x increases without bound. Express your answer using the mathematical notation of a limit.  \n(C) (i) Determine if f has an inverse function.  \n(ii) Give a reason for your answer based on the definition of a function and the graph of y = f (x) .  \nWrite your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.  \nGO ON TO THE NEXT PAGE.  \n2. On the initial day of sales (t = 0) for a new video game, there were 40 thousand units of the game sold that day. Ninety-one days later (t = 91), there were 76 thousand units of the game sold that day.  \nThe number of units of the video game sold on a given day can be modeled by the function G given  \nby G (t) = a + b ln(t + 1), where G (t) is the number of units sold, in thousands, on day t since the initial day of sales.  \n(A) (i) Use the given data to write two equations that can be used to find the values for constants a and b in the expression for G (t) .  \n(ii) Find the values for a and b as decimal approximations.  \n(B) (i) Use the given data to find the average rate of change of the number of units of the video game sold, in thousands per day, from t = 0 to t = 91 days. Express your answer as a decimal approximation. Show the computations that lead to your answer.  \n(ii) Use the average rate of change found in (i) to estimate the number of units of the video game sold, in thousands, on day t = 50 . Show the work that leads to your answer.  \n(iii) Let At represent the estimate of the number of units of the video game sold, in thousands, using the average rate of change found in (i). For A50 , found in (ii), it can be shown that A50 \u003C G(50) . Explain why, in general, At \u003C G (t) for all t , where 0 \u003C t \u003C 91.  \n(C) The makers of the video game reported that daily sales of the video game decreased each day after t = 91. Explain why the error in the model G increases after t = 91.  \nWrite your responses to this question only on the designated pages in the separate Free Response booklet. Write your solution to each part in the space provided for that part.  \nGO ON TO THE NEXT PAGE.  \nEND OF PART A  \nPRECALCULUS  \nSECTION II, Part B  \nTime—30 minutes  \n2 Questions  \nNO CALCULATOR IS ALLOWED FOR THESE QUESTIONS.  \n3. The tire of a car has a radius of 9 inches, and a person rolls the tire forward at a constant rate on level ground, as shown in the figure. Point W on the edge of the tire touches the ground at time t = 12 second. The tire completes a full rotation, and the next time W touches the ground is at time t = 52 seconds. The maximum height of W above the ground is 18 inches. As the tire rolls, the height of W above the ground periodically increases and decreases.  \nThe sinusoidal function h models the h","cbCaifJ2DQfdHkRO","https://ap.wps.com/l/cbCaifJ2DQfdHkRO","pdf",360348,"English","# Section II, Part A\n## Question 1\n## Question 2\n# Section II, Part B\n## Question 3\n## Question 4","[{\"question\":\"Which questions require a graphing calculator in Section II, Part A?\",\"answer\":\"Section II, Part A states that a graphing calculator is required for the questions in that part.\"},{\"question\":\"How is the video game daily sales function modeled in Question 2?\",\"answer\":\"Sales are modeled by G(t) = a + b ln(t + 1), where G(t) is measured in thousands of units and t is days since the initial sales day.\"},{\"question\":\"What sinusoidal modeling information is used for the rolling tire problem in Question 3?\",\"answer\":\"Question 3 models the height of point W using a sinusoidal function h(t) = a sin(b(t + c)) + d, based on periodic maxima and the given contact times and maximum height.\"}]","AP® Precalculus Free-Response Questions - Section II, Part A - Question 1-2 | PDF",1790768182,25]