Math 19B (Online) Calculus for Science Engineering and Mathematics University of California Santa Cruz

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1 Math 19B (Online) Calculus for Science Engineering and Mathematics University of California Santa Cruz Instructors and Course Creators Tony Tromba, PhD, Distinguished Professor, Mathematics Department, UC Santa Cruz Frank Bäuerle, PhD, Lecturer, Mathematics Department, UC Santa Cruz Katrina Fullman, Instructional Designer, UC Online Education, UCOP Course Description (from the UCSC course catalog) The definite integral and the fundamental theorem of calculus. Areas, volumes. Integration by parts, trigonometric substitution, and partial fractions methods. Improper integrals. Sequences, series, absolute convergence and convergence tests. Power series, Taylor and Maclaurin series. Pre-Requisites and Entrance Requirements Course 19A or AP Calculus AB exam score of 4 or 5, or BC exam score of 3 or higher, or IB Mathematics Higher Level exam score of 5 of higher. Course Learning Objectives 1. To understand the concept of the area under a graph. 2. To understand how areas can be calculated using the concept of the antiderivative of a function. 3. To understand the definite and indefinite integral concept. 4. To learn how to find the anti derivatives of elementary algebraic and trigonometric functions. 5. To understand how to apply the integral to finding volumes using Cavalieri's Principle. 6. To understand how to find volumes using the method of cylindrical shells. 7. To understand the application of the integral concept to the concepts of work and energy in physics.

2 8. To apply the integral to determine lengths of arcs and surface area. 9. To understand Taylor polynomials and the Taylor remainder formula. 10. To understand infinite series, power series and Taylor series. Learning Activities The course uses a learning management system (LMS) to organize content and act as a portal to various other platforms including an interactive e-textbook, a homework platform with immediate feedback, and a discussion forum. There are ten weekly modules in the course, but these can be reconfigured to accommodate a shorter summer session term. Within each module, there are 1-3 lessons and each of these lessons outline four learning activities. To be successful in this course, students must actively engage in all four of the following activities: 1. Read the textbook 2. Watch the video lectures 3. Do the homework 4. Interact with instructors, TA's and fellow students 1. Readings Each lesson indicates the required reading and links to the interactive e-textbook based on Jon Rogawski's textbook, Calculus, Early Transcendentals, 2nd ed. In addition, as students read each section in the e-book, they encounter graded progress-check questions as well as supplemental exercises. Progress-check questions provide immediate feedback and are set up to allow three attempts. 2. Videos Each lesson also links to short video lectures that explain major course concepts and techniques and provide examples. Students are encouraged to alternate between viewing the videos and reading the corresponding sections in the e-book. Students can pause or rewind lecture videos if they don t understand content covered. They can also come back for review at any time. In the first module, students are encouraged to watch supplemental lectures that provide context and historical background to the mathematics covered in the course. Page 2

3 While this content is not covered on assessments, it is an integral part of the course, providing students with a deeper understanding of the concepts covered in the regular lectures. 3. Homework Each lesson indicates the required homework assignment links to the online homework system. This system provides immediate feedback on answers submitted and allows unlimited attempts. In preparation for exams, students are provided with optional ungraded practice materials. 3. Interaction The course includes an on-line discussion forum centered on questions relating to the video lectures, homework, reading, and course logistics. In addition, the teaching staff holds regular on-line and in-person office hours, as well as an all day drop-in section. Finally the course includes student-run, location-based study groups with corresponding online group interaction spaces in the LMS. Details on how to participate are outlined on the lesson pages and other dedicated pages in the course learning management system. Course Modules. Get Started Prepare for Online Learning (Essential course information on getting connected to course tools and getting help with logistical, content and technical issues) Syllabus (Table view of due dates and syllabus with course policies) Office Hours (Schedule and information on participating in online and in-person office hours) Exam Information (Dates and expectations) Academic Integrity (Campus academic integrity rules and policies) MSI Session Poll (Tutoring session times) Course Contract (Pledge to abide by academic integrity policy) To access the course materials, students must complete the course contract by the end of the first week of instruction Module 1 Understand the notion of area under a graph and its informal definition Page 3

4 Understand the formal definition of rea under a graph and apply it to calculating areas of regions in specific examples Understand the definition of the definite integral and its interpretation as a difference of areas Lesson 5.1: Approximating and Computing Area Lesson 5.2: The Definite Integral Module 2 Understand the proof of the Fundamental Theorem Understand the basic idea of calculating definite integral using the Fundamental Theorem Understand the use of the substitution method in calculating definite integrals Lesson 5.3 The Fundamental Theorem of Calculus, Part I Lesson 5.4: The Fundamental Theorem of Calculus, Part II Lesson 5.6: Substitution Method Module 3 Understand how to calculate areas between curves Understand Cavalieri's principle in using integrals to calculate volumes Learning to use Cavalieri's principle to compute the volumes of solids of revolution Lesson 6.1: Area Between Two Curves Lesson 6.2: Setting Up Integrals: Volume Lesson 6.3: Volumes of Solids of Revolution Module 4 Understand how to calculate volumes of solids of revolution using the method of cylindrical shells Understand the applications of calculus to a first understanding of the physics concepts of work and energy Lesson 6.4: The Method of Cylindrical Shells Lesson 6.5: Work and Energy Page 4

5 MATH19B Midterm1Review Module 5 Understand the method of integration by parts to calculate integrals Understand how to calculate integrals involving trigonometric expressions Understand the method of trigonometric substitution to evaluate integrals involving algebraic integrands Lesson 7.1: Integration by Parts Lesson 7.2: Trigonometric Integrals Lesson 7.3: Trigonometric Substitution Module 6 Learning Objectives Understand the method of partial fraction decomposition in evaluating integrals involving rational functions. Understand the concept of an improper integral. Understand how to apply the integral to compute the length of a curve and the surface area of solids of revolution. Lesson 7.5: The Method of Partial Fractions Lesson 7.6: Improper Integrals Lesson 8.1: Arc Length and Surface Area Module 7 Learning Objectives Understand how to approximate differentiable functions by polynomials. Understand the concept of sequences of numbers Lesson 8.4: Taylor Polynomials Lesson 10.1: Sequences MATH19B Midterm 2 Review Module 8 Learning Objectives Gaining an understanding of infinite sums of numbers Understanding when infinite sums of positive numbers exist Page 5

6 Lesson 10.2: Summing an Infinite Series Lesson 10.3: Convergence of Series with Positive Terms Module 9 Gain an understanding of the different types of convergence of a series and their relationship Understand and learn to apply tests of convergence to infinite series such as the ratio and root test Lesson 10.4: Absolute and Conditional Convergence Lesson 10.5: The Ratio and Root Tests Module 10 Learning Objectives Understand the concept of a power series and its radius of convergence Understand the special case of the Taylor series arising as a power series of a function that is differentiable infinitely often Gaining practice in calculating Taylor series for standard elementary function Lesson 10.6: Power Series Lesson 10.7: Taylor Series Final Exam Review Exam Information Math 19B Practice Finals Math 19B Practice Problems for Final Course Resources Course Overview Course Introduction Instructor Introductions All Lecture Videos Homework (CalcPortal) Support Options College-Based Study Groups Tutoring Options MSI (Modified Supplemental Instruction) in Santa Cruz Page 6

7 Disability Accommodations Study Tips Office Hours Instructional Strategy Instructional Element Intended Learning Experience Canvas Primary course portal that students will use to view videos and find general information and announcements on course. Faculty will update course information as necessary. Will be used by students for every study session students access the ebook, discussion forum, and homework portal through Canvas Engagement can be evaluated through course analytics that provide data on how frequently students login, number of pages visited, etc. Pedagogy: Canvas is a central hub through which students can access all course information and communicate with teaching staff and peers. It provides organization and structure to the course. CalcPortal Homework platform that students will use to complete all homework assignments, quizzes, and exams if offered online. Faculty will engage regularly with CalcPortal to create exam questions, grade exams, and calculate overall grade. The grade book will be maintained in this platform, but final grades for all students will be transferred over to Canvas for export to appropriate Student Information Systems Engagement can be evaluated by how frequently students login, time spent on homework questions, completion of assignments, etc. Pedagogy: CalcPortal allows students to test their understanding of course content frequently. It provides immediate feedback on homework questions and hence reinforces knowledge and flags areas that need further study or exploration. Launchpad E-textbook platform that provides regular interactive reading assignments for each lesson. Course designers have created a customized e-book closely Page 7

8 Instructional Element Intended Learning Experience synchronized with the lectures, embedding additional interactive features, creating progress check questions, etc. Engagement/performance can be assessed through course analytics that provide data on how frequently students login, scores on progress check questions, pages visited, etc. Pedagogy: customization provides students with a text that is tied closely to other course elements. Interactive elements engage students and lead to greater comprehension of reading. Portability allows students to view textbook from mobile devices Piazza Discussion forum that allows students to interact with classmates and faculty Faculty/TAs will facilitate and provide regular presence by answering questions and endorsing helpful student posts. Teaching staff will also create tags and categories to organize posts and provide seed questions to stimulate discussion. Engagement will be accessed through analytics that provide data on how frequently a student logins and how frequently they post. Pedagogy: discussion boards lead to less isolation in online courses, provide students with opportunities to collaborate and form community, and allow for much quicker response times to student questions. By responding to classmates questions, students have an opportunity to demonstrate knowledge and explain complicated concepts in their own words, thus leading to greater understanding and comprehension of the subject matter. Brightcove Online video hosting platform. Engagement can be assessed through analytics showing number and duration of video views. Pedagogy: online lectures and other course videos create sense of instructor presence, allow for the presentation of complex material, allow students to access course material whenever they wish, and allow students to review (rewind) lectures when they don t Office Hours/Adobe Connect understand a concept presented. Offered both online and in-person several times per week by Faculty and TA. One-on-one appointments available. Online Office Hours conducted using Adobe Connect, a webbased conferencing software that allows for VOIP, text-based chat, whiteboard that allows both faculty and students to write equations, screen sharing, and video and powerpoint presentation. Page 8

9 Instructional Element Intended Learning Experience Pedagogy: Office Hours give both remote and local students an opportunity to connect with teaching staff and peers and get individualized instruction on content with which they are struggling. Drop-in Sections Several sections offered per week extended meeting period where students can drop in to work face-to-face with other students and TAs. TAs may provide mini-lectures on concepts covered in weekly modules. Extra meetings scheduled during exam periods. Pedagogy: provides an additional opportunity for face-to-face interaction and small group instruction. Tutoring Face-to-Face Tutoring available at UCSC through MSI (Modified Supplemental Instruction, a campus program), Math Department, and ACE (a divisional program for students of underrepresented groups) College-Based Study Groups Pedagogy: provides students with additional support if they are struggling with comprehension of the material. Can be used by both lower and higher performing students. Informal student-led study group set up for each residential college and/or remote campus. One to three student volunteer facilitators appointed. Encouraged to meet in person or online at least once per week. Assigned Canvas Group space to exchange notes, create wiki, make announcements, meet online. Encouraged to do homework together, watch lectures, work on supplemental problem worksheets, etc. Pedagogy: provides a student-centered learning activity with opportunities for both face-to-face and online interaction, provides an opportunity for students to share knowledge and form community. Assessment Activities Component Description Percentage Homework Online Homework problems for each lesson accessed through 15% CalcPortal Page 9

10 Component Description Percentage Quizzes Reading 2 Proctored Midterms Final Low stakes questions with unlimited attempts and immediate feedback in CP. Students can check their understanding of the content in each lesson and are incentivized to complete homework through points earned toward final grade. Periodic online quizzes in CalcPortal The online quizzes are distinguished from on-line homework by being limited in time and by the absence of hints and feedback. Students have only one attempt on each question. There will be partial credit (where appropriate or feasible). TAs and instructors will check answers and may assign partial credit after the computer score has been calculated. A final score on a quiz or other on-line test may be higher than what students receive immediately after submitting the test to CalcPortal. Reading progress check questions in Launchpad (e-textbook platform) Low stakes progress check questions with 3 attempts on each question. Located in each lesson. Encourages students to complete the readings and gives them an opportunity to check their understanding. Online and in-person options (online exams proctored by Proctor U) Formative assessments that give students an opportunity to evaluate progress and make improvements where necessary. Provides students with examples of how questions are formatted on final. Comprehensive final exam. Online and in-person options (online exams proctored by Proctor U) Summative assessment used to evaluate student learning. Total: % 10% 5% 30% 40% Page 10

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