[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-466351-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-466351-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":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","sample-calculus-exam-1","Sample Calculus Exam #1","","Sample Calculus Exam #1 includes instructions on calculator use and grading rules for short-answer versus work-requiring questions. It covers second derivatives, simplification of derivatives, tangent line equations and linear approximation, particle motion on the x-axis using position-to-speed and direction questions, derivatives involving inverse trigonometric forms and points of nonexistence, and related rates/differential equation transformations. The later problems emphasize uniqueness using sketches, Newton’s Method iteration formulas, derivative analysis of root functions, and optimization of trapezoid area, followed by function behavior, limits, sketching with asymptotes, and modeling physical processes with differential equations in tire pressure and conical water tanks.",{"@graph":14,"@context":72},[15,34,55],{"@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/sample-calculus-exam-1/466351/",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/sample-calculus-exam-1/466351.png","ImageObject",300,407,{"name":42,"@type":43},"4398046744996","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-29","2026-09-30",true,{"@type":52,"interactionType":53,"userInteractionCount":30},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"When are calculators allowed, and are they necessary?","Question",{"text":62,"@type":63},"Calculators allowed in the Principles of Mathematics 12 provincial examination may be used, but they will not be necessary.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"Do I need to show work for all questions?",{"text":67,"@type":63},"For questions 1-8 only the final result is marked for short-answer questions. For questions 9-14, all work must be shown to receive full credit.",{"name":69,"@type":60,"acceptedAnswer":70},"What topics are included in the exam?",{"text":71,"@type":63},"The exam includes derivatives and second derivatives, tangent lines and linear approximation, motion and speed direction, inverse trig derivatives and existence of derivatives, Newton’s Method, uniqueness via sketches, optimization, function increase/decrease and concavity, limits, and differential-equation word problems about tire pressure and a conical water tank.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},466351,1793135421,{"code":4,"msg":81,"data":82},"success",[83,87,91,94,99,104,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":92,"slug":93},70,"exam",{"id":95,"doc_module":4,"doc_module_name":25,"category_name":96,"show_sort_weight":97,"slug":98},5,"Comic",60,"comic",{"id":100,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},6,"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":95,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"user_id":132,"nickname":42,"user_avatar":11,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":133,"file_id":134,"file_url":135,"file_type":136,"file_size":137,"view_count":30,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":30,"language":138,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":139,"faqs":140,"seo_title":141,"seo_description":12,"update_tm":142,"read_time":110},4398046744996,"Sample Calculus Exam \\#1  \nInstructions Any calculator that is allowed in the Principles of Mathematics 12 provincial examination may be used, but will not be necessary. Answers that are \\calculator ready,\" like 3 + ln 7 or ep 2, are fully acceptable.  \nQuestions 1{8 are \\short answer\" questions, and only the 􀀌nal result will be marked; in particular, you need not show your work. In questions 9{14, you need to show your work in detail to receive full credit.  \n1. [8] a) Let f (x) = tan x. Find f 00 (x), the second derivative of f(x) .  \nb) Let f (x) = x arctan x 􀀀 (1=2) ln(1 + x2 ) . Find f 0 (x) and simplify.  \n2. [4] Find g0 (3) if g(x) = x2h (x) where h(3) = 􀀀2 and h0 (3) = 5 .  \n3. [4] Find the slope of the tangent line to the curve y + xln y 􀀀 2x = 0 at the point (1=2 ; 1) .  \n4. [6] Let f (x) =  2x   \nx2 +3 .  \na) Write an equation of the tangent line to the curve y = f (x) at x = 1 .  \nb) Use linear approximation to give an approximate value for f (1:2) .  \n5. [6] A particle moves along the x-axis so that its position at time t is given by x = t3 􀀀 4t2 +1.  \na) At t = 2, what is the particle's speed?  \nb) At t = 2, in what direction is the particle moving?  \nc) At t = 2, is the particle's speed increasing or decreasing?  \n6. [5] Let g (x) = arcsin(cos x) .  \na) Calculate and simplify the derivative g0 (x) .  \nb) At what points does g0 (x) fail to exist?  \n7. [5] Suppose that f 0 (2) = 3 . Find lxi2 fp(x~~x~~)~~ ~~􀀀~~ ~~f~~ ~~(2~~p2~~) .  \n8. [6] Suppose that the function y satis􀀌es the di􀀋erential equation dydt = 􀀀5y, and  \nlet z = y 2 . Then z satis􀀌es a di􀀋erential equation of the form dzdt = f (z) . Find f (z) .  \nFor questions 9{14, please show all your work.  \n9. [8] a) By using a sketch, or otherwise, explain why the equation e 􀀀x = x has exactly one solution.  \nb) In using Newton's Method to 􀀌nd the solution of the equation e 􀀀x = x, the current estimate is xn . Find an expression (in terms of xn ) for the next estimate xn+1 .  \n10. [8] Let f (x) = x1=x for x > 0.  \na) Find f 0 (x) .  \nb) At what value of x does the curve y = f (x) have a horizontal tangent line?  \nc) Using the results of parts a) and b) (not a calculator), determine which is larger, 3 1=3 or 􀀙 1=􀀙 . Explain.  \n11. [10] Three of the sides of a trapezoid have length 1 . What should be the length of the fourth side if the area of the trapezoid is to be as large as possible?  \n1  \n12. [10] A function f (x) de􀀌ned on the whole real line satis􀀌es the following conditions  \nf (0) = 0; f (2) = 2 ; lim f (x) = 0 ;  \nx!+1  \nf 0 (x) = k(2x 􀀀 x2 )e 􀀀x for some positive constant k.  \na) Determine the intervals on which f is increasing and decreasing and the location of any local maximum and minimum values of f.  \nb) Determine the intervals on which f is concave upward or downward, and the x-coordinates of any in􀀍ection points of f.  \nc) Determine lim f (x) .  \nx!􀀀1  \nd) Sketch the graph of y = f (x), showing any asymptotes and the information determined in parts a){c) .  \n13. [10] The air pressure in an automobile's spare tire was initially 3000 millibar. Unfortunately, the tire had a slow leak. After 10 days the pressure in the tire  \nhad declined to 2800 millibar. If P (t) is the air pressure in the tire at time t, then P (t) satis􀀌es the di􀀋erential equation  \ndPdt = 􀀀k(P (t) 􀀀 A) ;  \nwhere k is a constant and A is the atmospheric pressure. For simplicity, take atmospheric pressure to be 1000 millibar. When will the pressure in the tire be 2500 millibar?  \n14. [10] A water tank has the shape of a vertex-down right circular cone. The depth of the tank is 9 meters, and the top of the tank has radius 6 meters. Water 􀀍ows into the tank from a hose at a constant rate of 14􀀙 cubic metres per hour, and leaks out of a hole at the bottom of the tank at a rate of kh cubic metres per hour when the depth of water in the tank is h metres. Here k is a constant. When the water is 3 metres deep in the tank, its surface is rising atthe instantaneous rate of 2 metres per h","cbCaifkAUlXCBGKv","https://ap.wps.com/l/cbCaifkAUlXCBGKv","pdf",103091,"English","# Instructions\n## Questions 1-8\n## Questions 9-14","[{\"question\":\"When are calculators allowed, and are they necessary?\",\"answer\":\"Calculators allowed in the Principles of Mathematics 12 provincial examination may be used, but they will not be necessary.\"},{\"question\":\"Do I need to show work for all questions?\",\"answer\":\"For questions 1-8 only the final result is marked for short-answer questions. For questions 9-14, all work must be shown to receive full credit.\"},{\"question\":\"What topics are included in the exam?\",\"answer\":\"The exam includes derivatives and second derivatives, tangent lines and linear approximation, motion and speed direction, inverse trig derivatives and existence of derivatives, Newton’s Method, uniqueness via sketches, optimization, function increase/decrease and concavity, limits, and differential-equation word problems about tire pressure and a conical water tank.\"}]","Sample Calculus Exam #1 | PDF",1790771108]