[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-522878-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-522878-en":130},{"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","rolles-theorem-worksheet-infinite-calculus","Rolle's Theorem - Worksheet - Infinite Calculus","","Worksheet problems for applying Rolle’s Theorem. Students evaluate several functions and intervals, determining whether the theorem’s conditions are met, then solving for all values of c where f(c) = 0 and the derivative matches the required condition. Tasks include polynomial, trigonometric, and transcendental functions, with specified domains such as closed intervals in x or ranges involving pi. Some items require explaining failure due to discontinuity within the interval.",{"@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/rolles-theorem-worksheet-infinite-calculus/522878/",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/rolles-theorem-worksheet-infinite-calculus/522878.png","ImageObject",300,407,{"name":42,"@type":43},"Sarah ","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-11","2026-10-08",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",7,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How do you find the value(s) of c for a problem using Rolle's Theorem?","Question",{"text":63,"@type":64},"Check the function on the given closed interval for the theorem’s requirements, then determine all c values in the interval where the theorem guarantees the stated condition holds. If the theorem applies, the worksheet asks for all such c values.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"When can Rolle's Theorem not be applied on these worksheets?",{"text":68,"@type":64},"It cannot be applied when the function is not continuous on the entire given interval or when the endpoint requirements are not satisfied. The worksheet explicitly notes cases where the function is not continuous on intervals like [-2, 2] and [-π, π].",{"name":70,"@type":61,"acceptedAnswer":71},"What types of functions appear in this Rolle's Theorem worksheet?",{"text":72,"@type":64},"The problems include polynomial functions (with specified intervals), and trigonometric functions such as sin(2x), -2sin(2x), 2tan(x), and -2cos(2x), each paired with an interval like [-π, π] or other x ranges.","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},522878,1791505987,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,109,114,119,122,126],{"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":55,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":33,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":143,"read_time":123,"preview_page_count":4},962085320529,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Kuta Software-Infinite Calculus Name   \nRolle's Theorem Date  Period   \nFor each problem, find the values of c that satisfy Rolle's Theorem.  \n1) y = x 2 + 4x + 5; [−3, −1]  \n2) y = x 3 − 2x2 − x − 1; [−1, 2]  \n3) y = − x 3 + 2x2 + x − 6; [−1, 2] 4) y = x 3 − 4x2 − x + 7; [−1, 4]  \n5) y = − x 3 + 2x2 + x − 1; [−1, 2] 6) y = x 3 − x 2 − 4x + 3; [−2, 2]  \n− x 2 − 2x + 15   x 2 − 2x − 15   \n7) y = ; [−5, 3] 8) y = ; [−3, 5]  \n− x + 4 − x + 6  \n9) y = −~~ ~~x~~ ~~2~~ ~~+x~~ ~~x4+~~ ~~15~~ ~~; [−3, 5] 10) y = x~~ ~~2−+x~~ ~~−3~~ ~~6~~ ~~ ; [−3, 2]  \n11) y = −2sin (2x) ; [−π , π] 12) y = sin (2x) ; [−π , π]  \nFor each problem, determine if Rolle's Theorem can be applied. If it can, find all values of c that satisfy the theorem. If it cannot, explain why not.  \n13) y = x~~ ~~2~~ ~~−x~~ ~~−4~~ ~~12~~ ~~ ; [−3, 4] 14) y = −~~ ~~x~~ ~~2−−x~~ ~~x3+~~ ~~8~~ ~~; [−4, 2]  \n15) y = −~~ ~~x2x367~~ ~~ ; [−6, 6] 16) y = −~~ ~~x24~~ ~~4~~ ~~ ; [−2, 2]  \n17) y = 2tan (x) ; [−π , π] 18) y = −2cos (2x) ; [−π , π]  \nKuta Software-Infinite Calculus Name   \nRolle's Theorem Date  Period   \nFor each problem, find the values of c that satisfy Rolle's Theorem.  \n1) y = x 2 + 4x + 5; [−3, −1]  \n{−2}  \n2) y = x 3 − 2x2 − x − 1; [−1, 2]  \n{~~ ~~2~~ ~~+3~~ ~~7~~ ~~ , 2~~ ~~−3~~ ~~7~~ ~~ }  \n3) y = − x 3 + 2x2 + x − 6; [−1, 2] 4) y = x 3 − 4x2 − x + 7; [−1, 4]  \n{~~ ~~2~~ ~~−3~~ ~~7~~ ~~ , 2~~ ~~+3~~ ~~7~~ ~~ } {~~ ~~4~~ ~~+~~ ~~3~~ ~~19~~ ~~ , 4~~ ~~−~~ ~~3~~ ~~19~~ ~~ }  \n5) y = − x 3 + 2x2 + x − 1; [−1, 2] 6) y = x 3 − x 2 − 4x + 3; [−2, 2]  \n2~~ ~~−3~~ ~~7~~ ~~ , 2~~ ~~+3~~ ~~7~~ ~~ } {~~ ~~1~~ ~~+~~ ~~3~~ ~~13~~ ~~ , ~~ ~~1~~ ~~−~~ ~~3~~ ~~13~~ ~~ }  \n{  \n7) y = −~~ ~~x~~ ~~2~~ ~~~~ ~~15~~ ~~; [−5, 3] 8) y = x~~ ~~2~~ ~~−2x−6~~ ~~15~~ ~~ ; [−3, 5]  \n{1} {3}  \n9) y = −~~ ~~x~~ ~~2~~ ~~+x~~ ~~x4+~~ ~~15~~ ~~; [−3, 5] 10) y = x~~ ~~2−+x~~ ~~−3~~ ~~6~~ ~~ ; [−3, 2]  \n{−1} {3 − 6 }  \n11) y = −2sin (2x) ; [−π , π] 12) y = sin (2x) ; [−π , π]  \n{− ~~ ~~3π4~~ ~~ , − ~~ ~~π4~~ ~~ , π4~~ ~~ , ~~ ~~3π4~~ ~~ } {− ~~ ~~3π4~~ ~~ , − ~~ ~~π4~~ ~~ , π4~~ ~~ , ~~ ~~3π4~~ ~~ }  \nFor each problem, determine if Rolle's Theorem can be applied. If it can, find all values of c that satisfy the theorem. If it cannot, explain why not.  \n13) y = x~~ ~~2~~ ~~−x~~ ~~−4~~ ~~12~~ ~~ ; [−3, 4] 14) y = −~~ ~~x~~ ~~2−−x~~ ~~x3+~~ ~~8~~ ~~; [−4, 2]  \n{−4 + 2 2 } {3 − 7 }  \n15) y = −~~ ~~x2x367~~ ~~ ; [−6, 6] 16) y = −~~ ~~x24~~ ~~4~~ ~~ ; [−2, 2]  \n{−7 + 13 } The function is not continuous on [−2, 2]  \n17) y = 2tan (x) ; [−π , π] 18) y = −2cos (2x) ; [−π , π]  \nThe function is not continuous on [−π , π] {− ~~ ~~π2~~ ~~ , 0, π2~~ ~~ }  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaiunwKEvMRDZI","https://ap.wps.com/l/cbCaiunwKEvMRDZI","pdf",41691,"English","# Find c values using Rolle's Theorem\n## Problems 1-12\n# Determine whether Rolle's Theorem can be applied\n## Problems 13-18","[{\"question\":\"How do you find the value(s) of c for a problem using Rolle's Theorem?\",\"answer\":\"Check the function on the given closed interval for the theorem’s requirements, then determine all c values in the interval where the theorem guarantees the stated condition holds. If the theorem applies, the worksheet asks for all such c values.\"},{\"question\":\"When can Rolle's Theorem not be applied on these worksheets?\",\"answer\":\"It cannot be applied when the function is not continuous on the entire given interval or when the endpoint requirements are not satisfied. The worksheet explicitly notes cases where the function is not continuous on intervals like [-2, 2] and [-π, π].\"},{\"question\":\"What types of functions appear in this Rolle's Theorem worksheet?\",\"answer\":\"The problems include polynomial functions (with specified intervals), and trigonometric functions such as sin(2x), -2sin(2x), 2tan(x), and -2cos(2x), each paired with an interval like [-π, π] or other x ranges.\"}]","Rolle's Theorem - Worksheet - Infinite Calculus | PDF",1791426263]