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The test covers multivariable calculus and vector calculus topics including planes and lines, Lagrange multipliers for constrained distance, determinants and linear systems, parametrized curves and tangent/curvature computations, tangent planes and linearization-based approximations, critical points and second-derivative classification, coordinate transformations for double integrals, flux and surface integrals, and center-of-mass calculations using cylindrical or spherical coordinates, plus gradient and potential field work.",{"@graph":14,"@context":73},[15,34,56],{"@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/1802-final-exam-problem-set/496344/",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/1802-final-exam-problem-set/496344.png","ImageObject",300,407,{"name":42,"@type":43},"ahduag","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-22","2026-10-02",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",7,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What are the exam rules and point allocation?","Question",{"text":63,"@type":64},"The exam states that no books, notes, or calculators are allowed. It contains 15 problems worth a total of 250 points.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What core topics are included in the problems?",{"text":68,"@type":64},"The problems cover planes and lines in 3D, matrix determinants and solution sets, parametric curve tangent computations, tangent planes and linearization, critical point analysis, Lagrange multipliers, chain rule in spherical coordinates, double integrals with region and variable changes, flux calculations, and center-of-mass computations.",{"name":70,"@type":61,"acceptedAnswer":71},"Which methods are requested for constrained optimization and distances?",{"text":72,"@type":64},"Problem 7 specifies using the Lagrange multiplier method to set up equations for the point on a plane closest to an external point, without solving the system.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},496344,1791826890,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},"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":96,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":96,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":143,"read_time":144,"preview_page_count":4},3985747870414,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","18.02 Final Exam  \nNo books, notes or calculators.  \n15 problems, 250 points.  \nUseful formula: cos2 (θ) = ~~1~~2 (1 + cos(2θ))  \nProblem 1. (20 points)  \na) (15 pts.) Find the equation in the form Ax + By + Cz = D of the plane P which contains the line L given by x = 1 − t, y = 1 + 2t, z = 2 − 3t and the point (- 1, 1, 2) .  \nb) (5 pts.) Let Q be the plane 2x + y + z = 4. Find the component of a unit normal vector for Q projected on a unit direction vector for the line L of part(a) .  \nProblem 2. (15 points)  \nLet L denote the line which passes through (0,0,1) and is parallel to the line in the xy-plane given by y = 2x.  \na) (5 pts.) Sketch L and give its equation in vector-parametric form.  \nb) (5 pts.) Let P be the plane which passes through (0,0,1) and is perpendicular to the line L of part(a) . Sketch in P (above) and give its equation in point-normal form.  \nc) (5 pts.) Suppose that the point P  (0 , 0 , 1) lies on L. Write down the method or formula you would use to find the point P ∗ which is: (i) on L; (ii) the same distance away from the point (0,0,1) as P ; and is (iii) on the other side of P from P.  \nProblem 3. (20 points)  \nGiven the 3 × 3 matrix: Aa =  \n⎡⎣  \n1  \n−2  \n−1  \n0 1  \n1  \n3  \n−1  \na  \n⎤  \n⎦ :  \na) (5 pts.) Let a = 2: show that | A2 | = 0 b) (7 pts.) Find the line of solutions to A2  \n⎡⎣  \nx y  \nz  \n⎤⎦  \n=  \n⎡⎣  \n0 0  \n0  \n⎤⎦  \nc) (8 pts.) Suppose now that a = 1, and that A 1 −1 =  \n⎡⎣  \n∗  \n−3  \n∗  \n∗  \np  \n∗  \n∗  \n5  \n∗  \n⎤  \n⎦ .  \nFind p.  \nProblem 4. (10 points)  \nLet r (t) = 􀁨cos(et) , sin(et) , et 􀁩 .  \n r􀀰 (t)   \na) (5 pts.) Compute and simplify the unit tangent vector T (t) = | .  \nr􀀰 (t) |  \nb) (5 pts.) Compute T􀀰 (t)  \nProblem 5. (20 points)  \nConsider the function F (x, y , z) = z 􀁰x2 + y + 2 yz :  \na) (10 pts.) The point P0 : (1 , 3 , 2) lies on the surface F (x, y , z) = 7. Find the equation of the tangent plane to the surface F = 7 at P0 .  \nb) (5 pts.) If starting at P0 a small change were to be made in only one of the variables, which one would produce the largest change (in absolute value) in F ? If the change is this variable was of size 0. 1, approximately how large would the change in F be ?  \nc) (5 pts.) What distance from P0 in the direction ±􀁨−2 , 2 , −1􀁩 will produce an approximate change in F of size 0.1 units, according to the (already computed) linearization of F?  \nProblem 6. (15 points)  \nLet f (x, y) = x + 4y + 2xy .  \na) (10 pts.) Find the critical point(s) of f(x, y)  \nb) (5 pts.) Use the second-derivative test to test the critical point(s) found in part(a) .  \nProblem 7. (10 points)  \nLet P be the plane with equation Ax + By + Cz = D and P0 = (x0 , y0 , z0 ) be a point which is not on P.  \nUse the Lagrange multiplier method to set up the equations satisfied by the point (x, y , z) on P which is closest to P0 . (Do not solve.)  \nProblem 8. (15 points)    \na) (10 pts.) Let F(x, y , z) be a smooth function of three variables for which 􀁲F (1 , −1 , √ 2) =􀁨1 , 2 , −2􀁩 .  \nUse the Chain Rule to evaluate ~~ ~~Fφ~~ ~~ at (ρ, φ, θ) = (2 , ~~π~~4 , − ~~π~~4 ) .  \n(Use x = ρ sin φ cos θ , y = ρ sin φ sin θ , z = ρ cos φ.)  \nb) (5 pts.) Suppose f(x, y) is a smooth, non-constant function. Is it possible that, for all points (x, y), the gradient of f at the point (x, y) is equal to the vector 􀁨−y , x􀁩 ? Justify (briefly) .  \nProblem 9. (10 points)  \n􀁚􀁚R f dA = 􀁚02 􀁚x22√ 2x f (x, y) dy dx .  \na) (5 pts.) Sketch the region R.  \nb) (5 pts.) Rewrite the double integral as an iterated integral with the order interchanged.  \nProblem 10. (15 points)  \nSet up the integral 􀁚􀁚R f (x, y) dA where R is the region bounded by the four curves x2 y = 4, x2 y = 9, yx = 1, and yx = 2 as a double integral in the variables u = x2 y and v = yx . (Your answer should be completely ready to integrate, once the function f is given.)  \n1  1  1  2  \nNote: the inverse transformation is given by x = u 3 v − 3 , y = u 3 v 3 .  \nProblem 11. (15 points)  \nF (x, y) = x (i + j), and let C be the closed curve in the xy-pl","cbCaiuJzruJwzvyJ","https://ap.wps.com/l/cbCaiuJzruJwzvyJ","pdf",179175,"English","# Instructions\n# Problem 1 - Planes and projections\n# Problem 2 - Lines and perpendicular planes\n# Problem 3 - Matrices and solutions\n# Problem 4 - Parametric curves and tangents\n# Problem 5 - Tangent plane and linearization\n# Problem 6 - Critical points and classification\n# Problem 7 - Lagrange multipliers (closest point)\n# Problem 8 - Chain rule and gradients\n# Problem 9 - Double integrals and region changes\n# Problem 10 - Change of variables in double integrals\n# Problem 11 - Flux integrals over closed curves\n# Problem 12 - Center of mass of a solid cone\n# Problem 13 - Gradient fields and line integrals\n# Problem 14 - Surface integrals over cylinder patch","[{\"question\":\"What are the exam rules and point allocation?\",\"answer\":\"The exam states that no books, notes, or calculators are allowed. It contains 15 problems worth a total of 250 points.\"},{\"question\":\"What core topics are included in the problems?\",\"answer\":\"The problems cover planes and lines in 3D, matrix determinants and solution sets, parametric curve tangent computations, tangent planes and linearization, critical point analysis, Lagrange multipliers, chain rule in spherical coordinates, double integrals with region and variable changes, flux calculations, and center-of-mass computations.\"},{\"question\":\"Which methods are requested for constrained optimization and distances?\",\"answer\":\"Problem 7 specifies using the Lagrange multiplier method to set up equations for the point on a plane closest to an external point, without solving the system.\"}]","18.02 Final Exam - Problem Set | PDF",1790931019,13]