[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-522910-105":59,"doc-detail-522910-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","mean-value-theorem-for-integrals-worksheet-mean-value-mvt-for-integrals","Mean Value Theorem for Integrals - Worksheet - Mean Value & MVT for Integrals","","Worksheet problems for Mean Value Theorem for Integrals in which each task finds the average value of a given function over a specified interval and then determines the value(s) c that satisfy the theorem. The problems use polynomial functions with designated endpoints, requiring computing the integral average and solving the resulting equation for c. Includes multiple numbered items across different intervals and final summaries for average value and the corresponding c values.",{"@graph":69,"@context":122},[70,84,105],{"@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/mean-value-theorem-for-integrals-worksheet-mean-value-mvt-for-integrals/522910/",{"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/mean-value-theorem-for-integrals-worksheet-mean-value-mvt-for-integrals/522910.png","ImageObject",300,407,{"name":92,"@type":93},"วิน","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-11","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does each problem ask you to find first?","Question",{"text":112,"@type":113},"Each problem first asks for the average value of the given function over the specified interval [a, b].","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the second part of the problem after the average value?",{"text":117,"@type":113},"After finding the average value, the problem asks for the value(s) c in the interval that satisfy the Mean Value Theorem for Integrals.",{"name":119,"@type":110,"acceptedAnswer":120},"How are the intervals represented in these worksheets?",{"text":121,"@type":113},"Each function is paired with an interval written in brackets like [−4, 0] or [1, 5], indicating the lower and upper bounds for the averaging integral and the search for c.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},522910,1791536327,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":52,"preview_page_count":4},2336475104736,"https://ap-avatar.wpscdn.com/avatar/22000c4c5e0e5b17e70?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786591360781797222","Kuta Software-Infinite Calculus Name   \nMean Value Theorem for Integrals Date  Period   \nFor each problem, find the average value of the function over the given interval.  \n1) f (x) = −x2 − 2x + 5; [−4, 0]  \n2) f (x) = −x4 + 2x2 + 4; [−2, 1]  \nFor each problem, find the values of c that satisfy the Mean Value Theorem for Integrals.  \n3) f (x) = − ~~ ~~x22~~ ~~ + x + ~~ ~~32~~ ~~ ; [−3, 1]  \n4) f (x) = ~~ ~~4x2~~ ~~ ; [−4, −2]  \nFor each problem, find the average value of the function over the given interval.  \n5) f (x) = −x3 + 7x2 − 11x + 3; [1, 5] 6) f (x) = −x5 + 3x3 ; [0, 1]  \n1  8) f (x) = x 5 − 2x3 + x; [−1, 0]  \n7) f (x) = 4x 2 ; [0, 3]  \n 1  10) f (x) = x 5 − 4x3 + 2x − 1; [−2 2]  \n9) f (x) = ; [2, 3] , x  \n11) f (x) = −x5 + 4x3 − 5x − 3; [−2, 0] 12) f (x) = x 5 − 2x3 − 2; [−1, 1]  \nFor each problem, find the average value of the function over the given interval. Then, find the values of c that satisfy the Mean Value Theorem for Integrals.  \n13) f (x) = −x + 2; [−2, 2] 14) f (x) = −x2 − 8x − 17; [−6, −3]  \n12~~ ~~ 16) f ( x) =  4  ; [−6 −5]  \n15) f (x) = −3(2x − 6) ; [3, 5] (2x + 6)2 ,  \nKuta Software-Infinite Calculus Name   \nMean Value Theorem for Integrals Date  Period   \nFor each problem, find the average value of the function over the given interval.  \n1) f (x) = −x2 − 2x + 5; [−4, 0]  \n131 ≈ 3.667  \n2) f (x) = −x4 + 2x2 + 4; [−2, 1]  \n195 = 3.8  \nFor each problem, find the values of c that satisfy the Mean Value Theorem for Integrals.  \n3) f (x) = − ~~ ~~x22~~ ~~ + x + ~~ ~~32~~ ~~ ; [−3, 1]  \n3~~ ~~−~~ ~~43 ≈ −1.309  \n4) f (x) = ~~ ~~4x2~~ ~~ ; [−4, −2]  \n−2 2 ≈ −2.828  \nFor each problem, find the average value of the function over the given interval.  \n5) f (x) = −x3 + 7x2 − 11x + 3; [1, 5]  \n103 ≈ 3.333  \n 1   \n7) f (x) = 4x 2 ; [0, 3]  \n8~~ ~~3~~ ~~ ≈ 4.619  \n9) f (x) = ~~ ~~1x~~ ~~ ; [2, 3]  \nln 3 − ln 2 ≈ 0.405  \n11) f (x) = −x5 + 4x3 − 5x − 3; [−2, 0]  \n− ~~ ~~23 ≈ −0.667  \n6) f (x) = −x5 + 3x3 ; [0, 1]  \n712 ≈ 0.583  \n8) f (x) = x 5 − 2x3 + x; [−1, 0]  \n− ~~ ~~16 ≈ −0.167  \n10) f (x) = x 5 − 4x3 + 2x − 1; [−2, 2]  \n−1  \n12) f (x) = x 5 − 2x3 − 2; [−1, 1]  \n−2  \nFor each problem, find the average value of the function over the given interval. Then, find the values of c that satisfy the Mean Value Theorem for Integrals.  \n13) f (x) = −x + 2; [−2, 2]  \nAverage value of function: 2  \nValues that satisfy MVT: 0  \n 1   \n15) f (x) = −3(2x − 6) 2 ; [3, 5]  \nAverage value of function: −4  \nValues that satisfy MVT: ~~ ~~359 ≈ 3.889  \n14) f (x) = −x2 − 8x − 17; [−6, −3]  \nAverage value of function: −2  \nValues that satisfy MVT: −5, −3  \n16) f ( x) =  4  ; [−6, −5](2x + 6)2  \nAverage value of function: ~~ ~~16 ≈ 0.167  \nValues that satisfy MVT: −3 − 6 ≈ −5.449  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaidQyHAizCb1H","https://ap.wps.com/l/cbCaidQyHAizCb1H","pdf",73109,"English","# Average value over an interval\n## Computing average value from the integral\n# Applying the Mean Value Theorem for Integrals\n## Solving for c in each interval","[{\"question\":\"What does each problem ask you to find first?\",\"answer\":\"Each problem first asks for the average value of the given function over the specified interval [a, b].\"},{\"question\":\"What is the second part of the problem after the average value?\",\"answer\":\"After finding the average value, the problem asks for the value(s) c in the interval that satisfy the Mean Value Theorem for Integrals.\"},{\"question\":\"How are the intervals represented in these worksheets?\",\"answer\":\"Each function is paired with an interval written in brackets like [−4, 0] or [1, 5], indicating the lower and upper bounds for the averaging integral and the search for c.\"}]","Mean Value Theorem for Integrals - Worksheet - Mean Value & MVT for Integrals | PDF",1791426301]