[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-464672-105":59,"doc-detail-464672-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","16bfinalexampractice1sol","16BFinalExam(Practice1)sol","","Practice exam set covering multivariable optimization with constraints, double integrals over regions, and first-order differential equations including solving with initial conditions. Includes differential equation analysis via graphical solution of y'=cos(y), and convergence/divergence assessment using the integral test plus infinite series tests. Also requests Taylor series expansions with at least five non-zero terms, and probability computations for a continuous density on [0, π/2], including the CDF and variance. Final sections address standard normal and normal probability calculations such as Pr(X≤2).",{"@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/16bfinalexampractice1sol/464672/",{"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/16bfinalexampractice1sol/464672.png","ImageObject",300,407,{"name":92,"@type":93},"Fans","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":8},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you solve the constrained optimization problem for y^2 with the given constraint?","Question",{"text":112,"@type":113},"Set up the necessary equation using the constraint and solve for stationary points, then evaluate the candidate points to confirm the maximum exists.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What does the integral test require to determine convergence or divergence?",{"text":117,"@type":113},"You must verify the integral test hypotheses for the given positive term function and then compare the convergence of the corresponding improper integral.",{"name":119,"@type":110,"acceptedAnswer":120},"How are the cumulative distribution function and variance for X computed from f(x)=sin(x) on [0, π/2]?",{"text":121,"@type":113},"Compute F(x) by integrating f(t) from 0 to x, then use the definition of variance with E[X] and E[X^2] obtained from integrals over the same interval.","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},464672,1790768504,{"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":8,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"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":129,"read_time":144},5909892330395,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","| This exam consists of 10 questions.Answer the questions in the  \u003Cbr>spaces provided.   |  |  |\n| --- | --- | --- |\n|  |  |  |\n\n1.(25 points)Find the points(x,y)which maximise the function y²z,subject to the con-straint y²+4acy=1200.You may assume,withoutjustification,that a maximum exists.Solution:  \n3y²一12oo=0==16  →入  ：-5ft i wi2d--2) agpois=A= 5y= 2=>2=40oy=20-2x=,)=2+入十山-120  \ny=-20→x=  \n(16,20)aud(-10,-2)2o².10 >(-z)²(-10 =)(16,20)   Move  \nP  \n2.(25 points)Let R be the region enclosed by y=1/x²,y=1/x and z =2 in the firstquadrant.Determine the following double integral  \nSolution:  \n(/→x1)  \n一儿u(一l2d  \n3.(25 points)Find the general solution to the following differential equation:  \nx²y'=ycos(1/xc)+y  \nHence,or otherwise,find solutions with the following initial conditions:y(1/π)=0 andy(1/π)=-1.  \nSolution:  \nsoltTouy'=yxcm十x-2→y=0的 s  \nx~cmc×=)d= -cmcy) du --SmCu  \n→uly si(一   \n个  \n±e⁴auy na-2eo cmAety ={±ee)  \n七Lu owAa大  soltou  \n4.(25 points)A company expects to have a constant income rate for the next ten years.They will invest it in a saving account with 50%annual interest rate.If they want tohave $10000000 in the account after ten years,what must their income rate be?You donot need to simplify your answer.  \nSolution:  \naunudlk= congt autwtevert rate  \n00、0】eVoluu oevkd七 = 160Accumwatd16  \n=>  \nC个  \nkes(-2+2)=100000  \n yoa  \n5.(25 points)Consider the differential equation y'=cos(y).Graph the solutions with thefollowing initial conditions:y(0)=-π/4,y(0)=π/2 and y(0)=3π/4.Be sure to labeleach solution carefully.  \n6.(a)(15 points)Using the integral test,determine whether  \nis convergent or divergent.Be sure to check all the hypotheses of the integral test.  \nSolution:  \ncouvageCa l to21≠'(v)= 二3\u003C0m CI,∞机 k≥  \n(b)(10 points)Using this,or otherwise,determine if the infinite series  \nis convergent or divergent.  \nSolution:  \n≤Sin²(W≤to  \no  \n7.(25 points)Determine the Taylor series of the following functions.Write at least thefirst five non-zero terms of the sum to illustrate the general pattern.  \n(a)  \nf(x)=e  \nSolution:  \n(b)  \nf(x)=In(1-2x²)  \nSolution:  \n4u(1-2z2)=1u(1+(-2²))  \n8.(25 points)Let X be a continuous random variable on [0,π/2]with probability densityfunction f(x)=sin(x).Calculate the cumulative distribution function of X.CalculateVar(X).  \nF(x   iu(2)x=-(+  \nF(o)二0=)  -cm(0+C0C10  \nF()=1-c(  \n≠'()=1G =一ceg Lem  \n氧x)=x²g(z)=HuL²≠'(2=2z G(2)=-cm (z)个《(x)  \n)个ur  \n九u)  \nA'()=1G(a)= Sinlx)  \n《2}\\sin(x)dx=-x^{2}\\c00(x)+2x\\sin(x)+2o\\c π/z  \n9.(25 points)Let Z be the standard normal random variable.Assume that  \nIf X is a normal random variable with variance 4 and such that,calculate  \nPr(X≤2).  \nSolution:  \n Vaw(X)==92=  \nP√(X≥1)=方= Mediam(x)= I=μ=  \nSAaudawA Nnmd.(X2)  \nA三)=P(已≤1=0  \n→  Pr(×≤2)=00·乙","cbCaiq7AhC6taFKv","https://ap.wps.com/l/cbCaiq7AhC6taFKv","pdf",4500598,11,"English","# Questions\n## Differential equations and solutions\n## Integral tests and series convergence\n## Taylor series expansions\n## Probability and normal distributions","[{\"question\":\"How do you solve the constrained optimization problem for y^2 with the given constraint?\",\"answer\":\"Set up the necessary equation using the constraint and solve for stationary points, then evaluate the candidate points to confirm the maximum exists.\"},{\"question\":\"What does the integral test require to determine convergence or divergence?\",\"answer\":\"You must verify the integral test hypotheses for the given positive term function and then compare the convergence of the corresponding improper integral.\"},{\"question\":\"How are the cumulative distribution function and variance for X computed from f(x)=sin(x) on [0, π/2]?\",\"answer\":\"Compute F(x) by integrating f(t) from 0 to x, then use the definition of variance with E[X] and E[X^2] obtained from integrals over the same interval.\"}]","16BFinalExam(Practice1)sol | PDF",28]