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... Robert Morris University Multivariable Calculus & Analytic Geometry (a.k.a. Calc 3) MATH3090-A Spring 2022 Summary: This course is the third and final course in the calculus sequence at RMU. The aim of the course is to examine calculus concepts in three dimensions. Since some of the problems involve a three-dimensional picture or a lengthy computation, we will introduce and use mathematical software and graphing calculators. An important aspect of mathematical proficiency is visual reasoning and this course will emphasize the connection between the graphical interpretation of concepts with the symbolic, numerical, and real-world viewpoints. Course Learning Objectives: Throughout this course we will work together to be able to carry out the following skills using graphical, numeric, symbolic, and verbal representations: 1. Utilize graphing calculators and computer packages in the solution and representation of problems. 2. Communicate, recognize, and graph points, vectors, multi-variable functions, parametric curves, and vector fields in R^2 and R^3 using cartesian, cylindrical, and spherical coordinate systems. 3. Compute and apply the vector operations: dot product, cross product, scalar multiplication, vector addition, and vector subtraction to identify and represent lines and planes in R^3 4. Compute and interpret the derivative and integral of a parametric curve in R^3 5. Explain the relationships between, illustrate, and compute velocity, acceleration, TNB-frames of a position function given by a parametric curve in R^3 6. Identify and compute geometric properties of parametric curves such as tangent lines, curvature, and arc length 7. Apply vectors parametric curves, partial derivatives, directional derivatives, gradients, multiple integrals, line integrals, and vector fields to solve problems in other fields such as engineering, probability, computer science, and physics. 8. Compute and interpret partial derivatives, gradients, directional derivatives, multiple integrals, and line integrals, recognizing assumptions that need to be met concerning the domain and continuity. 9. Apply changes in coordinate systems to solve problems (e.g. cylindrical, spherical, and Jacobians) 10. Formulate and solve optimization problems using partial derivatives and Lagrange Multipliers. 11. Identify and compute geometric properties and applications of multivariable functions including tangent planes, areas, volumes of solids, surface areas, center of mass, average value, and probability. This syllabus was created by Monica VanDieren (Robert Morris University), with contributions by faculty of the mathematics department, in the summer of 2021. Support for the project was generously provided by the Pennsylvania Grants for Open and Affordable Learning (PA GOAL) program, Grant #27. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License and may not be reproduced for commercial purposes. Required Textbook: Active Calculus - Multivariable+Vectors. The online version is free. If you click on this link, you will see that there are options available to purchase a print copy of the majority of the book at a small cost (the last few sections that we will cover are online only at this time). Required Access to Online Assignments: WeBWorK: Some, but not all, of the assignments will be on the free software WeBWorK which can be accessed at WeBWorK. Flipgrid: Some assignments will involve filming a presentation of a mathematics question. These video presentations will be conducted through flipgrid, which is also free. Required Materials/Technology: 1. Students should have a working camera and microphone for full participation in synchronous sessions and for recording flipgrid assignments. 2. Binder, notebook and/or folder to organize and store class notes, handouts and homework. Some students chose to have a digital notebook while others use paper. 3. Students are required to have the ability to take a photo and/or scan their written work to upload to blackboard and email to the instructor. This helps students communicate questions and work to the instructor and classmates. It also is how assessments will be submitted to the instructor. Supplementary Suggested Materials (these are not required) 1. Calculus Early Transcendental Functions, any edition. (you do not need to purchase the webassign code). This is the textbook that is often used in Calculus I and Calculus II at RMU. 2. Students are recommended to have access to a printer to print out class notes and exam questions, since it is more efficient to write mathematical symbols by hand than to type it. 3. To communicate with other students and the instructor online, some students prefer to use a tablet with a stylus rather than a mouse to draw mathematical symbols and graphs. This is really a personal preference. On-line 3-d Graphing Resource: To help visualize multivariable concepts and to build an intuition for them, we will be using the on-line graphing applet found at https://c3d.libretexts.org/CalcPlot3D/index.html. Feel free to use this applet when you are working on your homework or studying for this class. This syllabus was created by Monica VanDieren (Robert Morris University), with contributions by faculty of the mathematics department, in the summer of 2021. Support for the project was generously provided by the Pennsylvania Grants for Open and Affordable Learning (PA GOAL) program, Grant #27. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License and may not be reproduced for commercial purposes. Participation and Being Prepared Professors Expectations: Students are expected to read all of the assigned readings and watch all the assigned videos posted each week. Students will participate in the weekly activities and complete all assignments. Students are expected to participate actively in group projects and synchronous sessions. Students are expected to turn in assignments by the due date. If circumstances are such that a student is unable to complete the assignment on time, they should reach out to the instructor. Assessments and Teaching Methods: In mathematics, true understanding includes: standard problem-solving ability; presentation of solutions; and comprehension of fundamental concepts including graphical interpretations. A correct answer with incomplete, inconsistent, or incorrect work will not be assigned full credit. Just being able to solve problems is expected; however, it does not constitute excellent work. Demonstrating mathematical and personal maturity, sharing insight in problem solutions, helping your peers, and exceeding minimum requirements comprise excellent work. Therefore your understanding of the learning objectives will be assessed in several ways: 1. Practice Homework: I am concerned that you master the course material and develop your problem-solving skills. Since material in this class builds on previous material covered in the class, it is imperative to complete the practice problems in a timely manner, and preferably prior to viewing the subsequent lessons. Homework is your opportunity to test out and develop your mathematical proficiency. Expect to make mistakes, and have several false-starts at a problem. This is a natural part of the learning process. You are provided multiple attempts to solve each practice problem. 2. Challenge Problems: Since mathematics is more than following the steps in an algorithm, in addition to the skills that you will develop with the practice homework, you will be assigned challenge problems in each unit. These problems often require several steps, may involve applying multiple concepts, often involve communicating mathematics, and/or may involve multiple representations of a mathematical concept. Full credit for these problems may involve explaining the problem solving process and solution of the problems in flipgrid videos. Most of these submissions will be private, but select assignments may be presentations viewable by the entire class (similar to a class presentation in a face to face class). 3. Labs: We will have a mathematical version of a science lab. During these sessions you will be exploring, and sometimes experimenting with, mathematical concepts. These labs are a mandatory component of the class. Most of the labs are scheduled for the first half of the semester, with no labs in the last few weeks. 4. Video and Reading Journal: You will be assigned questions associated with most video and reading assignments. These will be used to assess your understanding of the material and provide you an opportunity to reflect on your learning. These assignments will be checked weekly and are available in WeBWorK. 5. Participation a. Synchronous Sessions: These mandatory sessions will be highly interactive; students contribute to the classroom discussion and problem solving activities. Students are expected to log in on time, and unmute their microphone when responding to questions. For most of the time, students will be working together on a virtual whiteboard. This syllabus was created by Monica VanDieren (Robert Morris University), with contributions by faculty of the mathematics department, in the summer of 2021. Support for the project was generously provided by the Pennsylvania Grants for Open and Affordable Learning (PA GOAL) program, Grant #27. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License and may not be reproduced for commercial purposes. 6. 7. 8. 9. b. Virtual Study Groups: You will be assigned to a study group on flipgrid to ask and respond to questions. Each week you should contribute at least one posting to your group. You are welcome (and encouraged) to form your own study groups with classmates outside of flipgrid. If you would like a separate space within flipgrid for your individually formed group, just email your instructor. Group Projects: Instead of quizzes and chapter tests, there will be group projects in this class. All of the group projects will have an individual component which must satisfactorily be completed before the student is assigned to a group. Failure to complete the individual assignment by the deadline may result in a 0 for the group project. Final Exam: The final exam will be cumulative and will involve a timed and untimed component. The timed component will take place virtually on Thursday during the Final Exam week from 8:30am-10:30 am. Extra Credit: There is no extra credit in this course. Late Work: Due to the stresses of living in a pandemic, Prof. V. understands that sometimes homework deadlines may be difficult to meet for a variety of reasons. Therefore, if you would like to request a homework extension on an assignment, please fill out this form prior to the due date. Evaluation: Graded Activities Practice Problems Challenge Problems & Labs Video and Preview Problems Participation Group Projects Comprehensive Final Percent of Course Grade 15% 25% 15% 10% 20% 15% Grades: The following scale will be used as a starting point for determining course grades. Total Letter grade Percentage 93-100 A 90-92 A87-89 B+ 83-86 B 80-82 B77-79 C+ 70-76 C 60-69 D 0-59 F Communication Expectations: E-mail is the primary form of communication between class sessions. It is expected that you will frequently check your Robert Morris University e-mail account, since the instructor will be using this account to contact you as needed. Please let the instructor know if this is problematic for you. This syllabus was created by Monica VanDieren (Robert Morris University), with contributions by faculty of the mathematics department, in the summer of 2021. Support for the project was generously provided by the Pennsylvania Grants for Open and Affordable Learning (PA GOAL) program, Grant #27. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License and may not be reproduced for commercial purposes. Professionalism: Students are expected to maintain appropriate behavior in the class meetings and other activities that reflect our program and university. Appropriate attire, decorum, and attitudes will be assumed. Do not, for example, schedule a routine doctor appointment during our class time, nor purchase a plane ticket departing prior to our final examination. Do not take phone calls or text-message during class. If you are experiencing difficulties accessing course material or attending required meeting, please contact the instructor. Covid-19 Attendance Policy Covid-19 Attendance Policy As a result of the Covid-19 pandemic, all students are encouraged to remain home or in their residence hall room when experiencing any signs of illness. Students who test positive for the virus or who must be quarantined after exposure to the virus will be excused from class attendance. Instructors will be notified by the Dean of Students Office if a student is in quarantine or has contracted the virus. A student who is absent due to observed symptoms of Covid-19, is in quarantine due to suspected exposure, or who has a confirmed case of Covid-19, is entitled to makeup work missed if the student fulfills the instructor notification requirements of the policy. Students are not to be penalized for any missed assignments, projects, examinations, tests, etc. or to have their daily grades automatically reduced when covered by this policy. While the faculty member must allow the student to make up or complete any assignments, etc., that were missed due to officially sanctioned obligations, faculty members are under no obligation to tutor or otherwise provide missed instruction. Faculty will determine when make-up exams are scheduled and when missed assignments are due. Students must notify the Dean of Students Office at 412-397-6483 to be excused from class attendance and for this policy to be in effect. Instructors will be notified by the Dean of Students Office. This syllabus was created by Monica VanDieren (Robert Morris University), with contributions by faculty of the mathematics department, in the summer of 2021. Support for the project was generously provided by the Pennsylvania Grants for Open and Affordable Learning (PA GOAL) program, Grant #27. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License and may not be reproduced for commercial purposes.  ...

... MATH 108 Lab 13 More On Optimization The following problems are programmed in WeBWorK with embedded GeoGebra applets. Problems 1 and 5 are open response questions and graded by the instructor. Problems 2-4 are randomized for each student and auto-graded by WeBWorK. 1. (Applet for Lab 13, Problem 1, can be found at https://www.geogebra.org/m/fxj9y8vu) Use the applet to complete the following questions. You will use one checkbox at a time to complete each of parts (a) to (d). Before you begin, go to the bottom of this problem to review the relevant terminology and mathematical content related to the Extreme Value Theorem (EVT). Answer using complete sentences. (a) Select Checkbox 1 only. The applet now displays the graph of f (x). Identify the interval on which f (x) is defined. Next, identify the absolute extreme output values of f (x) (if they exist) and the input value(s) at which each absolute extreme value occur. If no such value(s) exist, explain why the conditions required in the EVT are not satisfied for f (x). (b) Select Checkbox 2 only. The applet now displays the graph of g(x). Identify the interval on which g(x) is defined. Next, identify the absolute extreme output values of g(x) (if they exist) and the input value(s) at which each absolute extreme value occur. If no such value(s) exist, explain why the conditions required in the EVT are not satisfied for g(x). (c) Select Checkbox 3 only. The applet now displays the graph of j(x). Identify the interval on which j(x) is defined. Next, identify the absolute extreme output values of j(x) (if they exist) and the input value(s) at which each absolute extreme value occur. If no such value(s) exist, explain why the conditions required in the EVT are not satisfied for j(x). (d) Select Checkbox 4 only. The applet now displays the graph of k(x). Identify the interval on which k(x) is defined. Next, identify the absolute extreme output values of k(x) (if they exist) and the input value(s) at which each absolute extreme value occur. If no such value(s) exist, explain why the conditions required in the EVT are not satisfied for k(x). General Information: Terminology: 1  A closed, bounded interval is an interval that contains its endpoints and both endpoints are finite values. For example, intervals of the form [a, b], where a and b are both real numbers with a < b are closed and bounded. We can think of a function as being continuous on [a, b] if it is defined on all of [a, b] and its graph has no holes or jumps. Recall: Extreme Value Theorem (EVT): A continuous function on a closed, bounded interval will have an absolute maximum value and an absolute minimum value on that interval. Fact: If f is continuous on [a, b] and f (c) is an absolute extreme value, then c = a, c = b, or c is a critical number between a and b 2. (Applet for Lab 13, Problem 2, can be found at https://www.geogebra.org/m/cbufmady) Consultant Company X is an organization that contracts with colleges and universities around the US to help them develop open educational resources for specialty courses. Company Xs annual revenue in hundreds of thousands of dollars is given by the model R(t), where t is the number of years since 2002. The applet shows the graph of Company Xs marginal revenue function, M R(t). Observe that there is an input box for t that will give you the coordinates of point A on M R(t) and the corresponding point the graph of R(t). Use the applet to answer the questions below. Note: If you are asked about an extreme value that you determine does not exist, type none in all the associated and relevant boxes. (a) According to the graph, R(t) and M R(t) each have a domain of Thus Company Xs revenue model is valid from the the year year . . to the (b) In 2010, Company X posted a profit of 112.61 hundred thousand dollars. Company Xs total revenue in 2010 was hundred thousand dollars, meaning that their total costs in that same year were hundred thousand dollars. 2 (c) The critical number(s) for R(t) are . These critical values correpsond and . In each of these years, the marginal to the years revenue was hundred thousand dollars. (d) (Answer chronologically) Company Xs revenue was increasing between the years and and then was increasing again between the years and . Company Xs revenue was decreasing from the year to the year and then was decreasing again between the years and . (e) Company Xs revenue was at a relative maximum in the year hundred thousand dollars. the companys total revenue was when (f) Company Xs revenue was at a relative minimum twice. The first relative minwhen the companys total revenue was imum occurred in the year hundred thousand dollars. The second relative minimum occurred in the year when the companys total revenue was hundred thousand dollars. (g) In the year , Company Xs revenue was at an absolute minimum of hundred thousand dollars. The annual revenue for Company X was at an absolute maximum value of hundred thousand dollars in the . year 3. (Applet for Lab 13, Problem 3, can be found at https://www.geogebra.org/m/upkuwykv) ADTF Distributors manufactures and supplies the 12,000 BTU Commando 8 air conditional to retailers across the country. Their average costs in hundreds of dollars per full truck load shipped (ftl) is given by the model C(a) for a ftls manufactured and 0 shipped. The applet above shows the graph of C (a). Observe that there is an input 0 box for a that will give you the coordinates of point A on C (a) and the corresponding point the graph of C(a). Use the applet to answer the questions below. Note: If you are asked about an extreme value that you determine does not exist, type none in all the associated and relevant boxes. 0 (a) According to the applet, C(a) and C (a) each have a domain of . The output units for C(a) are , while the output units for the derivative 0 C (a) are . 3 (b) ADTF Distributors average cost when manufacturing and shipping 7 ftls is hundered dollars per ftl. Equivalently, the average cost is dollars per ftl. This means that when 7 full truck loads of air conditioners are shipped, ADTF Distributors has a total cost of dollars. . This answer corresponds to (c) The critical number(s) for C(a) are shipments where the rate of change of average cost is hundred dollars per ftl per ftl. (d) ADTF Distributors average cost was increasing from ftl. ADTF Distributors average cost was decreasing from ftl. ftl to ftl to (e) ADTF Distributors had a relative maximum average cost of hundred ftl. ADTF Distributors had a dollars per ftl when they shipped relative minimum average cost when ftl were manufactured shipped. hundred dollars per ftl. The relative minimum average cost was (f) When ADTF Distributors ships ftl, they have an an absolute minimum average cost of hundred dollars per ftl. By way of comparison, their absolute maximum average cost is hundred dollars per ftl when ftl. a= 4. (Applet for Lab 13, Problem 4, can be found at https://www.geogebra.org/m/nghaus9q) The Cheese Emporiums annual profit is given by the model P (m) in tens of thousands of dollars, where m represents the number of varieties of cheese in tens that they stock in a given year. For example, the value m = 1 indicates that the Emporium stocked 10 varieties of cheese. The applet above shows the graph of P 0 (m). Observe that there is an input box for m that will give you the coordinates of point E on P 0 (m) as well as the corresponding points on P (m) and P 0 (m). The points B and D are the intercepts for P 0 (m) (showing the m values where the derivative is 0). Use the applet to answer the questions below. Note: If you are asked about an extreme value that you determine does not exist, type NONE in all the associated and relevant boxes. (a) According to the graph, the function P (m) has a domain of 4 . Thus, the Emporiums profit model is valid when they stock between varieties of cheese. rieties and va- (b) Suppose that when the Emporium stocks 120 varieties of cheese, their annual revenue is 153 ten thousand dollars. The profit model implies that the annual profit when the Emporium stocks 120 varieties of cheese is ten thousand dollars. Thus, their annual cost to stock 120 varieties of cheese must be tne thousand dollars. (c) The critical number(s) for P (m) are . For each of these critical numbers, the Emporiums annual profit was changing at a rate of dollars per each 10 additional cheese varieties stocked. (d) The Emporiums profit increases from m = and then increases again from m = poriums profit is decreasing from m = to m = to m = to m = . The Em. (e) Using the second derivative test, the Emporium had a relative minimum profit ten thousand dollars when varieties of cheese were of stocked since P 00 ( ) is . Again, using the second derivative test, the Emporium had a relative maximum ten thousand dollars when varieties of cheese profit of were stocked since P 00 ( ) is . varieties of cheese are stocked in a year, the Emporium has an (f) When an absolute maximum profit of ten thousand dollars. By way of comten thousand parison, the Emporiums absolute minimum profit is dollars when m = . 5. (Applet for Lab 13, Problem 5, can be found at https://www.geogebra.org/m/azv35dkv) Company Z opened its doors for business in 1979. The model C(t), gives Company Zs annual cost in hundreds of dollars, where t is the number of years since 1979. The applet above shows the graph of Company Zs marginal cost function, M C(t). Observe that there is an input box for t that will give you the coordinates of point A on M C(t) and the corresponding point the graph of C(t). Use the applet to answer the questions below. Answer the following using COMPLETE SENTENCES. 5 (a) Between what years was Company Zs costs increasing. Explain how the graph of M C(t) can be used to determine this. (b) Between what years was Company Zs costs decreasing. Explain how the graph of M C(t) can be used to determine this. (c) In what year did Company Z have a relative maximum cost? What was the relative maximum cost? Explain how the graph of M C(t) can be used to determine this. (d) In what year did Company Z have a relative minimum cost? What was the relative minimum cost? Explain how the graph of M C(t) can be used to determine this. (e) In what year did Company Z have an absolute minimum cost? What was the absolute minimum cost? In what year did Company Z have an absolute maximum cost? What was the absolute maximum cost? This assignment was created by Alfred Dahma, Timothy Flowers, and Valerie Long (Indiana University of PA) in Spring 2022. Support for the project was generously provided by the Pennsylvania Grants for Open and Affordable Learning (PA GOAL) program, Grant #21. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License and may not be reproduced for commercial purposes. To view a copy of this license visit http://creativecommons.org/licenses/by-nc-sa/4.0/ 6  ...