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Topics include evaluating definite integrals, analyzing convergence or divergence of several series using valid tests, computing series sums, finding intervals of convergence, and deriving Taylor series about x=0. The exam also includes multivariable geometry tasks for angles, triangle area, and plane equations, plus parametric and vector calculus applications such as tangent lines and an initial-value differential equation set.",{"@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/math-0230-analytic-geometry-and-calculus-ii-sample-final-exam-2/300649/",{"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-0230-analytic-geometry-and-calculus-ii-sample-final-exam-2/300649.png","ImageObject",300,407,{"name":92,"@type":93},"Bintang","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-03","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":56},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What calculus topics are included in the sample final exam?","Question",{"text":112,"@type":113},"The exam includes definite integrals, convergence/divergence of series, Taylor series, and applications such as volume/work problems and geometric relations in 3D.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are the series problems handled in the exam?",{"text":117,"@type":113},"Students must determine convergence or divergence using valid tests; when a series is convergent, the sum is found only when the problem explicitly asks for it.",{"name":119,"@type":110,"acceptedAnswer":120},"Does the exam include differential equations problems?",{"text":121,"@type":113},"Yes. It includes initial-value problems that require showing all work and identifying the type of differential equation before solving.","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},300649,1790183894,{"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":56,"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":144,"read_time":145,"preview_page_count":4},962085564381,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","MATH 0230-Analytic Geometry and Calculus II SAMPLE FINAL EXAM 2  \nTime: 1 hour 50 min  \n􀀙=3  \n1. (10 points) Evaluate the integral I = Z sin3 x cos2 x dx.  \n0  \n2. Evaluate the integral if it converges. If it diverges, show divergence.  \n7  \n(a) (10 points) I = Z ~~p~~71􀀀~~ ~~x dx  \n3  \n1  \n(b) (10 points) I = Z 4~~ ~~x2 dx  \n1  \n3. (10 points) Use the method of cylindrical shells to 􀀌nd the volume generated by rotating the region bounded by the curves y = x2 and y = 4x about the y-axis. No credit will be given if a di􀀋erent method is used.  \n4. (10 points) A water tank has the shape of a sphere of radius 2 m with a spout at the top of length of 1 m, as in the following picture below. The spherical part of the tank is full of water and the spout is empty.  \nSet up but do not evaluate an integral to 􀀌nd the work required to empty the tank by pumping all of the water out of the spout. You may use 􀀚 = 1000 kmg3 for the density of water and g = 9:8~~m~~s2 for the acceleration due to gravity.  \nUse the direction of the x-axis and the origin provided in the picture.  \nx  \n1 m  \n0   \nsphere  \n2 m  \n5. For the given points P (1 ; 􀀀3 ; 1), Q(2 ; 􀀀2 ; 􀀀1), and R (􀀀2 ; 􀀀3 ; 4) .  \n􀀖 􀀖 􀀖  \n(a) (5 points) 􀀌nd the angle 􀀒 between vectors  = PQ and b = PR. Leave the answer inan exact form using cos 􀀀1 .  \n(b) (5 points) 􀀌nd the area of the triangle PQR.  \n(c) (5 points) 􀀌nd a linear equation of the plane that passes through the points P , Q, and R.  \n(You may use results found in previous parts) .  \n6. (10 points) Find the equation of the line tangent to the curve x = sint, y = (t + 1)2 at the point where t = 0 .  \n7. (a) (10 points) Find the area enclosed by one leaf of the graph of r = 2 sin 5􀀒 .  \n(b) (5 points) Set up but do not evaluate an integral to 􀀌nd the length of the boundary of one leaf of the graph.  \n8. For series in parts (a) and (b) below, determine whether the series is convergent or divergent. Clearly show reason by a valid test. If it is convergent, 􀀌nd its sum.  \n1 2n  \n(a) (5 points) X 5n  \nn=1  \n(b) (5 points)  2nn2  \n1 (􀀀1)n  \n(c) (5 points) For the series X n + 2  determine whether it is convergent or divergent.  \nn=1  \nIf the series is convergent, then DO NOT 􀀌nd its sum.  \n9. Determine the sum of the given series.  \n(a) (10 points)  n2~~ ~~+~~ ~~13n~~ ~~+~~ ~~2  \n1 2n  \n(b) (5 points) X 5n n! n=0  \n10. (10 points) Determine the interval of convergence of the series  \n1 ( 1)n xn X ~~ ~~􀀀n~~ ~~4n~~ ~~n=1  \n11. For the functions below, determine their Taylor series about x = 0 . You may apply any general series you know. You should use sigma P notation.  \n 1   \n(a) (5 points)  \n5 + x  \n(b) (5 points) ln(5 + x)  \n(c) (5 points) 􀀀x2 cos(2x)  \n12. (10 points) Find the Taylor series for the function f (x) = sin x about a = 􀀙 . You should use sigma P notation.  \n13. Solve the initial-value problem. Show all the work. Mention a type of the given di􀀋erential equation.  \n(a) (10 points) ~~ ~~y30 = x2 y; y(0) = 5 .  \n(b) (10 points) xy0 􀀀 2y = x3 ; y(1) = 4 .  \n(c) (10 points) y00 + 6y0 + 9y = 0; y(0) = 2; y0 (0) = 􀀀5.","cbCaivWZTjZt79nd","https://ap.wps.com/l/cbCaivWZTjZt79nd","pdf",132566,11,"English","# Section 1: Integrals\n## Evaluate definite integrals and apply convergence checks\n# Section 2: Applications of calculus\n## Volume by cylindrical shells; work to empty a tank\n## Angles, triangle area, and plane/line equations\n# Section 3: Series and Taylor expansions\n## Convergence/divergence, sums, intervals of convergence\n## Taylor series about x=0 and about a\n# Section 4: Differential equations\n## Solve initial-value problems","[{\"question\":\"What calculus topics are included in the sample final exam?\",\"answer\":\"The exam includes definite integrals, convergence/divergence of series, Taylor series, and applications such as volume/work problems and geometric relations in 3D.\"},{\"question\":\"How are the series problems handled in the exam?\",\"answer\":\"Students must determine convergence or divergence using valid tests; when a series is convergent, the sum is found only when the problem explicitly asks for it.\"},{\"question\":\"Does the exam include differential equations problems?\",\"answer\":\"Yes. It includes initial-value problems that require showing all work and identifying the type of differential equation before solving.\"}]","MATH 0230-Analytic Geometry and Calculus II Sample Final Exam 2 | PDF",1789775905,28]