[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-394201-105":59,"doc-detail-394201-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","midterm-3-worksheet-section-a-solutions","Midterm 3 Worksheet Section A - Solutions","","Peer Exam Review Session materials for Math 231 - Calculus II, providing solutions to selected midterm worksheet problems. The content guides students through determining convergence of power series, computing Maclaurin series expansions, extracting first non-zero terms, evaluating limits using series, and expanding esin x via derivative-based Maclaurin coefficients. Additional tasks include using Taylor series for derivatives centered at a, polar region area calculations, and matching parametric equations to graphs.",{"@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/midterm-3-worksheet-section-a-solutions/394201/",{"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/midterm-3-worksheet-section-a-solutions/394201.png","ImageObject",300,407,{"name":92,"@type":93},"Ben ","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-07","2026-09-26",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",17,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do you find the radius and interval of convergence for a given power series?","Question",{"text":113,"@type":114},"Compute convergence criteria using the ratio/root test to determine where the series converges. Then express the interval by converting the inequality in x around the series center into x bounds.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What is the process for finding a Maclaurin series expansion (including first non-zero terms)?",{"text":118,"@type":114},"Use known series for basic functions, then apply algebraic operations (substitution, multiplication) and collect like powers. For the “first two non-zero terms,” identify the smallest powers with nonzero coefficients.",{"name":120,"@type":111,"acceptedAnswer":121},"How is the polar region area calculated from equations like r = 3 + 2sin(θ) and r = 2?",{"text":122,"@type":114},"Find intersection angles by solving the polar equations for θ. Then integrate A = (1/2)∫(r_outer^2 − r_inner^2)dθ over the relevant θ interval.","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},394201,1790528315,{"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":44,"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,"preview_page_count":4},2336478951081,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","Center for Academic Resources in Engineering (CARE)  \nPeer Exam Review Session  \nMath 231 − Calculus II  \nMidterm 3 Worksheet Solutions  \nThe problems in this review are designed to help prepare you for your upcoming exam. Questions pertain to material covered in the course and are intended to reflect the topics likely to appear in the exam. Keep in mind that this worksheet was created by CARE tutors, and while it is thorough, it is not comprehensive. In addition to exam review sessions, CARE also hosts regularly scheduled tutoring hours.  \nTutors are available to answer questions, review problems, and help you feel prepared for your exam during these times:  \nSession 1: Apr 8, 7-9PM (Hriday, Tommy)  \nSession 2: Apr 26, 2-4PM (Sushrut, Amy)  \nCan’t make it to a session? Here’s our schedule by course:  \n[https://care.grainger.illinois.edu/tutoring/schedule-by-subject](https://care.grainger.illinois.edu/tutoring/schedule-by-subject)  \nSolutions will be available on our website after the last review session that we host.  \nStep-by-step login for exam review session:  \n1. Log into Queue @ Illinois: [https://queue.illinois.edu/q/queue/844](https://queue.illinois.edu/q/queue/844)  \n[2](2). Click “New Question”  \n3. Add your NetID and Name  \n4. Press “Add to Queue”  \nPlease be sure to follow the above steps to add yourself to the Queue.  \nGood luck with your exam!  \n1. You are given the power series  \n∞ (x − 7)n  \nX 3n (2n + 1)  \nn=1  \n.  \n(a) Find the radius of convergence.  \n(b) Find the interval of convergence.  \n(a)  \n (x − 7)n   (x − 7)n+1   \nan = 3n (2n + 1), an+1 = 3n+1(2n + 3)  \nln 􀀌 an+an1 􀀌 = ln 􀀌 3n(7n)13) 3n((2x~~ ~~1)n 􀀌  \n= ln 􀀌 2n2n~~ ~~~~ ~~13 x~~ ~~−3~~ ~~7 􀀌  \n􀀌 x − 7 􀀌  \n= 􀀌 ~~ ~~ 􀀌 \u003C 1  \n􀀌 3 􀀌  \n=⇒ |x − 7| \u003C 3 =⇒ R = 3  \n(b)  \n−3 \u003C x − 7 \u003C 3  \n4 \u003C x \u003C 10  \n∞ (4 − 7)n ∞ (−1)n 3n If x = 4 ,  3n (2n + 1) =  3n (2n + 1)  \n∞ (−1)n =X 2n+ 1  \nn=1  \nThis series at x = 4 converges by the alternating series test.  \n∞ (10 7)n ∞ 1 If x = 10 , X 3n~~ ~~(2−n+~~ ~~1) =X 2n + 1  \nn=1 n=1  \nThis series at x = 10 diverges by the limit comparison test with ~~1~~n . Therefore, the interval of convergence is [4 , 10) .  \n2. Find the Maclaurin Series for:  \n(a)  \n x f (x) =  \n1 + 4x2  \n(b)  \nf (x) = x3 cos 􀀀x2 􀀁  \n(a) Notice power series:  \nx 􀀒 ~~ ~~1~~ ~~+14x2 􀀓 → x ∞X(−1)n (4x2 )nn=0  \n=  \n∞  \nX (−1)n 4n (x2n+1) n=0  \n(b)  \n∞ x2ncos(x) =⇒ (−1)n  (2n)! cos 􀀀x2 􀀁 =⇒ (−1)n ~~ ~~(x42nn)! x3 cos 􀀀x2 􀀁 =⇒ ∞X(−1)nx(4n+2n)3!   \nn=0  \n3. Find the first two non-zero terms of the Maclaurin Series for (a) This function f (x)  \nf (x) = sinx(x2~~ ~~)  \n(b) This function g(x)  \ng (x) = x cos 􀀀x2 􀀁  \n(c) Use parts (a) and (b) to write down the first two non-zero terms in the Maclaurin Series for the product f (x)g (x) .  \n(d) Use your work in parts (a) and (b) to evaluate:  \nf (x) − g(x)  \nlim  \nn→0 x5  \n(a)  \nsin 􀀀x2 􀀁 = x2 − (x23!)3 + ··· = x2 − x36!=⇒ f (x) ≈ x2~~ ~~−x~~ x3~~6~~!  ~~ = x − x35!   \n(b)  \ng (x) = x(cos 􀀀x2 􀀁)= x[1 − (x22!)2 + ... ]  \n= x[1 − x24! + ... ]  \nx5 ≈ x − ~~  ~~2!  \n(c)  \nf (x)g (x) ≈ x2 − x26! − x36! + x3!102! ≈ x2 − 2  \n3 x6  \n(d)  \n= ln x~~ ~~−~~ x3~~5~~! ~~−x5~~ ~~x~~ ~~+~~ x2~~5~~!  ~~ = ln x5~~ ~~(~~ 12~~x−5~~ 16~~) =  \n\n| 1 |\n| --- |\n| 3 |\n\n4. The Maclaurin series expansion of esin x is:  \nx2   x4  \n(a) 1 + x − + − · · · 2 12 x2 x4  \n(b) 1 − x + ~~ ~~ − ~~ ~~ + ··· 2 8 x2 x4  \n(c) 1 + x + ~~ ~~ − ~~ ~~ + ··· 2 8 x2 x4  \n(d) 1 + x + ~~ ~~ − ~~ ~~ + ··· 2 12  \nLet f (x) = esin x.  \nf′(x) = esin x cos x = f(x)cos x  \nf′′(x) = f′(x)cos x − f(x)sin x  \nf′′′(x) = f′′(x)cos x − f′(x)sin x − (f(x)cos x + f′(x)sin x)  \n= f′′(x)cos x − 2f′(x)sin x − f′(x)  \nf 4 (x) = f′′′(x)cos x − f′′(x)sin x − 2(f′(x)cos x + f′′(x)sin x) − f′′(x)  \n= f′′′(x)cos x − 3f′′(x)sin x − 2f′(x)cos x − f′′(x)  \nCalculating higher order derivatives at x = 0,  \nf(0) = 1  \nf′(0) = 1 × 1 = 1  \nf′′(0) = 1 − 0 = 1  \nf′′′(0) = 1 − 0 − 1 = 0  \nf 4 (0) = 0 − 0 − 2 − 1 = −3  \nf 5 (0) = f4 (0)cos0 − 3f′′(0)cos0 − 2f′′(0)cos0 − f′′′(0)= −3 − 3 − 2 − 0 = −8  \n","cbCaipVtg5hNcql1","https://ap.wps.com/l/cbCaipVtg5hNcql1","pdf",460211,"English","# Solutions\n## Power series convergence\n## Maclaurin series and expansions\n## Series-based limits\n## Higher-order derivatives for Maclaurin\n## Taylor series centered at a\n## Polar area calculation\n## Parametric equations and graph matching","[{\"question\":\"How do you find the radius and interval of convergence for a given power series?\",\"answer\":\"Compute convergence criteria using the ratio/root test to determine where the series converges. Then express the interval by converting the inequality in x around the series center into x bounds.\"},{\"question\":\"What is the process for finding a Maclaurin series expansion (including first non-zero terms)?\",\"answer\":\"Use known series for basic functions, then apply algebraic operations (substitution, multiplication) and collect like powers. For the “first two non-zero terms,” identify the smallest powers with nonzero coefficients.\"},{\"question\":\"How is the polar region area calculated from equations like r = 3 + 2sin(θ) and r = 2?\",\"answer\":\"Find intersection angles by solving the polar equations for θ. Then integrate A = (1/2)∫(r_outer^2 − r_inner^2)dθ over the relevant θ interval.\"}]","Midterm 3 Worksheet Section A - Solutions | PDF",1790400007,23]