[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-394033-105":59,"doc-detail-394033-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","math-5c-spring-2010-practice-exam-3","Math 5C Spring 2010 Practice Exam 3","","Practice exam 3 for Math 5C (Spring 2010), consisting of multiple-choice and free-response questions. The exam covers topics including Fourier series coefficients, harmonic functions and properties in higher dimensions, power series radius of convergence, Taylor series interval of convergence, and Riemann–Lebesgue lemma implications for Fourier behavior. Free-response problems require evaluation, use of generating functions for Catalan numbers, summation of series, and Parseval’s identity to compute trigonometric/integral-related expressions, plus determining a harmonic function’s value at the origin from boundary data.",{"@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/math-5c-spring-2010-practice-exam-3/394033/",{"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/math-5c-spring-2010-practice-exam-3/394033.png","ImageObject",300,407,{"name":92,"@type":93},"Asher","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-28","2026-09-26",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How are points allocated on this exam?","Question",{"text":113,"@type":114},"The exam has 125 points total, with multiple-choice questions worth 5 points each and free-response questions worth 20 points each. A blank free-response problem is awarded 5 points.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Are calculators or notes allowed?",{"text":118,"@type":114},"No notes, books, or electronic devices are allowed.",{"name":120,"@type":111,"acceptedAnswer":121},"What key topics are tested in the multiple-choice section?",{"text":122,"@type":114},"Topics include Fourier series coefficient identification, radius of convergence for modified power series, properties of harmonic functions with boundary values, harmonic function definitions, series expansions near zero, and interval of convergence for a Taylor series.","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},394033,1790528336,{"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":34,"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},687197207639,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Math 5C Spring 2010 Practice Exam 3  \nName   \nPerm No.   \nDirections:  \n\n| M. Choice |  |\n| --- | --- |\n| F. Resp. 1 |  |\n| F. Resp. 2 |  |\n| F. Resp. 3 |  |\n| F. Resp. 4 |  |\n| Total |  |\n\n1. There are 125 points on this exam; 100 points = 100% .  \n2. Each multiple choice problem is 5 points.  \n3. Each multiple choice problem has exactly one best answer.  \n4. No multiple choice problem requires heavy computation.  \n5. Each free response problem is 20 points.  \n6. Free response questions require justi􀀌cation; no work, no credit.  \n7. A blank free-response problem is awarded 5 points.  \n8. No notes, books, or electronic devices are allowed.  \nMultiple Choice  \n1. Using the standard inner product for functions on [􀀀􀀙; 􀀙], what is k sin xk?(a) 0  \n(b) sin2 x  \n(c) 1  \n(d) Unde􀀌ned  \n(e) None of the above  \n2. The function f (x) = x2 , 􀀀􀀙 \u003C x \u003C 􀀙 is expanded into a Fourier series. What is the coe􀀎cient of sin(3x)?  \n(a) 0  \n(b) 3  \n(c) 1=3  \n(d) 9  \n(e) 1=9  \n3. The power series P an xn has a 􀀌nite radius of convergence R > 0. What is the radius of convergence of Pan (x=2)n?  \n(a) R  \n(b) 2R  \n(c) R=2  \n(d) 1  \n(e) Cannot be determined without more information  \n4. Suppose that f is harmonic on R3 and that f = 6 for all points on the unit sphere centered at the origin. What is the largest possible value of f (1=2 ; 0 ; 0)?  \n(a) 0  \n(b) 6  \n(c) 84􀀙  \n(d) It could be arbitrarily large  \n(e) None of the above  \n5. By de􀀌nition, a harmonic function f is one which satis􀀌es which of the following?  \n(a) rf = 0  \n(b) 􀀁f = 0  \n(c) r 􀀂 rf = 0  \n(d) 􀀁 2 f = 0  \n(e) df = 0  \n6. Which of these is a series expansion of 􀀀 ln(1 􀀀 x) for x near zero?  \n(a) x + x2 =2 + x3 =3 + 􀀁 􀀁 􀀁  \n(b) x 􀀀 x2 =2 + x3 =3 􀀀 􀀁 􀀁 􀀁  \n(c) 􀀀x + x2 =2 􀀀 x3 =3 + 􀀁 􀀁 􀀁  \n(d) 􀀀x 􀀀 x2 =2 􀀀 x3 =3 􀀀 􀀁 􀀁 􀀁  \n(e) None of the above  \n7. When 1=(1 􀀀 x) is expanded in a Taylor series at zero, what is its interval of convergence?  \n(a) (􀀀1 ; 1)  \n(b) (􀀀1 ; 1]  \n(c) [􀀀1 ; 1)  \n(d) [􀀀1 ; 1](e) R  \n8. The Riemann-Lebesgue lemma states that, if Z􀀀􀀙􀀙 j f(x)j dx \u003C 1 , then ln Z􀀀􀀙􀀙 f (x) cos nxdx = ln Z􀀀􀀙􀀙 f (x) sinnx dx = 0 :  \nWhat does this tell about the Fourier series of f?  \n(a) Nothing  \n(b) Sums and integrals can be interchanged  \n(c) The series converges  \n(d) Parseval's identity  \n(e) The coe􀀎cients tend to zero  \n9. The 2􀀙-periodic function f (x) = x3 , 􀀀􀀙 \u003C x \u003C 􀀙, is expanded in a Fourier series. What is the value of the series at x = 􀀙?  \n(a) 0  \n(b) 􀀙 3  \n(c) 􀀀􀀙3  \n(d) The series diverges  \n(e) None of the above  \nFree Response  \n1. Evaluate  \nln n2 \" 􀀒1 + 1n􀀓 n+1=2 􀀀 e\\#  \n2. The generating function for the Catalan numbers is  \n n~~ ~~~~ ~~1 􀀒 2nn􀀓 xn = 1~~ ~~􀀀~~ ~~p~~ ~~12x~~ ~~􀀀~~ ~~4x  \nUse this to sum the series  \n1  \nX  \nn=0  \n􀀒 2nn􀀓 2􀀀4n  \n3. Given  \n􀀙 􀀀 j xj = 􀀙2 + 4􀀙 􀀒 cos x + cos93x + cos255x + cos479x + 􀀁 􀀁 􀀁 􀀓 ; 􀀀􀀙 \u003C x \u003C 􀀙; use Parseval's identity to evaluate  \n 1   1   1   \n1 + + + + 􀀁 􀀁 􀀁  \n34 54 74  \n4. Let f be a harmonic function in R3 and B be the unit ball centered at the origin. Assume that (in spherical coordinates) f = sin 􀀞 on @B . Find f (0 ; 0 ; 0) .","cbCaimTVAexnOEsZ","https://ap.wps.com/l/cbCaimTVAexnOEsZ","pdf",66587,"English","# Multiple Choice\n## Free Response","[{\"question\":\"How are points allocated on this exam?\",\"answer\":\"The exam has 125 points total, with multiple-choice questions worth 5 points each and free-response questions worth 20 points each. A blank free-response problem is awarded 5 points.\"},{\"question\":\"Are calculators or notes allowed?\",\"answer\":\"No notes, books, or electronic devices are allowed.\"},{\"question\":\"What key topics are tested in the multiple-choice section?\",\"answer\":\"Topics include Fourier series coefficient identification, radius of convergence for modified power series, properties of harmonic functions with boundary values, harmonic function definitions, series expansions near zero, and interval of convergence for a Taylor series.\"}]","Math 5C Spring 2010 Practice Exam 3 | PDF",1790397827,18]