[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-522864-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-522864-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","mean-value-theorem-worksheets","Mean Value Theorem - Worksheets","","Mean Value Theorem worksheet problems from Infinite Calculus, asking students to find values of c that satisfy the Mean Value Theorem for given functions on specified intervals. It includes tasks to decide whether the theorem can be applied by checking continuity and differentiability on the interval or interior. One critical thinking problem uses the theorem to prove an inequality involving sin a and sin b for real numbers a and b with a ≠ b, guiding learners through the MVT framework and bounding with properties of cos.",{"@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/mean-value-theorem-worksheets/522864/",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/mean-value-theorem-worksheets/522864.png","ImageObject",300,407,{"name":42,"@type":43},"Angel","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-12","2026-10-08",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",6,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What does the worksheet ask you to find for the first set of problems?","Question",{"text":63,"@type":64},"For each problem, you find the values of c that satisfy the Mean Value Theorem on the given interval.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How do you decide whether the Mean Value Theorem can be applied?",{"text":68,"@type":64},"Check whether the function is continuous on the closed interval and differentiable on the open interval; if not, the worksheet indicates why the theorem cannot be applied.",{"name":70,"@type":61,"acceptedAnswer":71},"How does the critical thinking problem use the Mean Value Theorem?",{"text":72,"@type":64},"It sets f(x)=sin x on the interval [a,b], applies the MVT to obtain a relationship involving cos c, and then uses cos c ≤ 1 to derive sin a − sin b ≤ a − b for a ≠ b.","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},522864,1791552309,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,104,109,114,119,122,126],{"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":55,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},"Technology",50,"technology",{"id":105,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},7,"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},687207412472,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Kuta Software-Infinite Calculus Name   \nMean Value Theorem Date  Period   \nFor each problem, find the values of c that satisfy the Mean Value Theorem.  \n1) y = −x2 + 8x − 17; [3, 6]  \n2) y = x 3 − 9x2 + 24x − 18; [2, 4]  \n3) y = − ~~ ~~x22~~ ~~ + x − ~~ ~~12~~ ~~ ; [−2, 1] 4) y = x22~~ ~~ − 2x − 1; [−1, 1]  \n5) y = x 3 + 3x2 − 2; [−2, 0] 6) y = −x3 + 4x2 − 3; [0, 4]  \nx 2 − 9   x 2   \n7) y = ; [1, 4] 8) y = ; [−4, 1]  \n3x 2x − 4  \n 1   1   \n9) y = −(−2x + 6)2 ; [−2, 3] 10) y = −(−5x + 25)2 ; [3, 5]  \nFor each problem, determine if the Mean Value Theorem can be applied. If it can, find all values of c that satisfy the theorem. If it cannot, explain why not.  \n11) y = − ~~ ~~4xx~~ ~~8~~ ~~ ; [−3, −1] 12) y = −x24~~ ~~9~~ ~~ ; [1, 3]  \n2   2  \n13) y = −(6x + 24) 3 ; [−4, −1] 14) y = (x − 3) 3 ; [1, 4]  \nCritical thinking question:  \n15) Use the Mean Value Theorem to prove that  sin a − sin b  ≤  a − b  for all real values of a and b where a ≠ b.  \nKuta Software-Infinite Calculus Name   \nMean Value Theorem Date  Period   \nFor each problem, find the values of c that satisfy the Mean Value Theorem.  \n1) y = −x2 + 8x − 17; [3, 6]  \n{~~ ~~92~~ ~~ }  \n2) y = x 3 − 9x2 + 24x − 18; [2, 4]  \n{~~ ~~9~~ ~~+3~~ ~~3~~ ~~ , ~~ ~~9~~ ~~−3~~ ~~3~~ ~~ }  \n3) y = − ~~ ~~x22~~ ~~ + x − ~~ ~~12~~ ~~ ; [−2, 1] 4) y = x22~~ ~~ − 2x − 1; [−1, 1]  \n{− ~~ ~~12~~ ~~ }  \n{0}  \n5) y = x 3 + 3x2 − 2; [−2, 0]  \n{~~ ~~−3~~ ~~~~ ~~3~~ ~~ , −3~~ ~~−3~~ ~~3~~ ~~ }  \n7) y = x~~ ~~23−x~~ ~~9~~ ~~ ; [1, 4]  \n{2}  \n6) y = −x3 + 4x2 − 3; [0, 4]  \n{~~ ~~83~~ ~~ }  \n8) y = 2xx~~ ~~4~~ ~~ ; [−4, 1]  \n{2 − 6 }  \n1   \n9) y = −(−2x + 6)2 ; [−2, 3]  \n{~~ ~~74~~ ~~ }  \n 1   \n10) y = −(−5x + 25)2 ; [3, 5]  \n{~~ ~~92~~ ~~ }  \nFor each problem, determine if the Mean Value Theorem can be applied. If it can, find all values of c that satisfy the theorem. If it cannot, explain why not.  \n11) y = − ~~ ~~4xx~~ ~~8~~ ~~ ; [−3, −1] 12) y = −x24~~ ~~9~~ ~~ ; [1, 3]  \nThe function is not continuous on [−3, −1] { 3 }  \n2   2  \n13) y = −(6x + 24) 3 ; [−4, −1] 14) y = (x − 3) 3 ; [1, 4]  \n{− ~~ ~~289~~ ~~ } The function is not differentiable on (1, 4)  \nCritical thinking question:  \n15) Use the Mean Value Theorem to prove that  sin a − sin b  ≤  a − b  for all real values of a and b where a ≠ b.  \nLet f (x) = sin x. Use the interval [a,b] . By the MVT, we know that there is at least one c  \nsin b − sin a sin b − sin a  \nsuch that  b − a  = cos c. We know cos c ≤ 1 for all c. Therefore,  b − a  ≤ 1, ~~ ~~~~ ~~sin~~ ~~aasinb~~ ~~b~~ ~~ ≤ 1, and  sin a − sin b  ≤  a − b  .  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaiprFFJkQuOOh","https://ap.wps.com/l/cbCaiprFFJkQuOOh","pdf",42557,"English","# Mean Value Theorem Problems\n## Find c values\n## Determine applicability\n## Critical thinking: sin a - sin b inequality","[{\"question\":\"What does the worksheet ask you to find for the first set of problems?\",\"answer\":\"For each problem, you find the values of c that satisfy the Mean Value Theorem on the given interval.\"},{\"question\":\"How do you decide whether the Mean Value Theorem can be applied?\",\"answer\":\"Check whether the function is continuous on the closed interval and differentiable on the open interval; if not, the worksheet indicates why the theorem cannot be applied.\"},{\"question\":\"How does the critical thinking problem use the Mean Value Theorem?\",\"answer\":\"It sets f(x)=sin x on the interval [a,b], applies the MVT to obtain a relationship involving cos c, and then uses cos c ≤ 1 to derive sin a − sin b ≤ a − b for a ≠ b.\"}]","Mean Value Theorem - Worksheets | PDF",1791426249]