Math 19A (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, Continuing Lecturer, Mathematics Department, UC Santa Cruz Katrina Fullman, Instructional Designer, UC Online Education, UCOP Course Description (from the UCSC course catalog) The limit of a function, calculating limits, continuity, tangents, velocities, and other instantaneous rates of change. Derivatives, the chain rule, implicit differentiation, higher derivatives. Exponential functions, inverse functions, and their derivatives. The mean value theorem, monotonic functions, concavity, and points of inflection. Applied maximum and minimum problems. Pre-­‐Requisites and Entrance Requirements Passing grade in Math 3 (Precalculus) or Applied Mathematics and Statistics 3, or mathematics placement examination (MPE) score of 400 or higher; or AP Calculus AB exam score of 3 or higher. Course Learning Objectives 1. Understand the concept of an instantaneous rate of change and the derivative of a function 2. Learn how to calculate derivatives explicitly and implicitly and to master how derivatives affect the behavior of a function 3. Master the application of the derivative notion to optimization problems 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 Math 19A (Online)
University of California Santa Cruz – Course Outline
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 activities: 1. Read the textbook 2. Watch the video lectures 3. Do the homework 4. Interact with others 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. 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. 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 and 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 Page 2
Math 19A (Online)
University of California Santa Cruz – Course Outline
hold 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 LMS. 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) • Calculus Diagnostic Test (Optional test to determine readiness for Calculus) • 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 Learning Objectives: • Understand the foundations for the study of the material that lies ahead • Understand why mathematics was and continues to be so crucial in the development of our modern society and why calculus is an essential part of a science curriculum. • Understand the notion of limit and its informal definition • Demonstrate an introductory understanding of the instantaneous velocity of a moving object Topics: • Introductory Lectures: The Birth of Calculus • Lesson 2.1: Limits, Rates of Change, Tangent Lines • Lesson 2.2: Limits: A Numerical and Graphical Approach Module 2 Learning Objectives: Page 3
Math 19A (Online)
University of California Santa Cruz – Course Outline
Understand basic properties of limits Demonstrate basic techniques to calculate limits effectively. Gain a basic understanding of what it means for a function to be continuous at a point Topics: • Lesson 2.3 Basic Limit Laws • Lesson 2.4: Limits and Continuity • Lesson 2.5: Evaluating Limits Algebraically Module 3 Learning Objectives: • Understand and apply the squeeze theorem to calculate trigonometric limits • Understand and calculate limits at infinity • Understand and apply the Intermediate Value Theorem Topics: • Lesson 2.6: Trigonometric Limits • Lesson 2.7: Limits at Infinity • Lesson 2.8: The Intermediate Value Theorem Module 4 Learning Objectives: • Understand how to precisely define instantaneous velocity, and more generally, instantaneous rates of change. • Understand a limit definition of the central notion of the course: the derivative. • Gain a basic understanding of the notion of the derivative as a function. • Apply the first set of rules on how to effectively calculate derivatives • Apply alternate notations for the derivative • Understand the definition of the number "e" and gain a basic understanding of its significance. Topics: • Chapter 3 Introduction • Lesson 3.1: Definition of the Derivative • Lesson 3.2: The Derivative as a Function • MATH19A Midterm1Review Module 5 Learning Objectives: • Understand the product and quotient rules. • Understand derivatives of derivatives called higher derivatives. •
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Math 19A (Online)
University of California Santa Cruz – Course Outline
• Calculate the derivatives of the trigonometric functions. Topics: • Lesson 3.3: Product and Quotient Rules • Lesson 3.5: Higher Derivatives • Lesson 3.6: Trigonometric Functions Module 6 Learning Objectives • Gain a basic understanding of the chain rule for calculating the derivative of a composition of differentiable functions. • Apply the chain rule to determine the derivatives of inverse functions of differentiable functions as well as the derivatives of general exponential and logarithmic functions. • Understand and apply a comprehensive way to calculate the derivative of all elementary functions (such as combinations and compositions of polynomial, rational, trigonometric, inverse trigonometric, logarithmic and exponential functions). Topics: • Lesson 3.7: The Chain Rule • Lesson 3.8: Derivatives of Inverse Functions • Lesson 3.9: Derivatives of General Exponential and Logarithmic Functions Module 7 Learning Objectives • Understand how to the chain rule is used to calculate the derivatives of implicitly defined functions. • Apply the chain rule to the computation of "Related Rates,” our first real application of the derivative. Topics: • Lesson 3.10: Implicit Differentiation • Lesson 3.11: Related Rates • MATH19A Midterm 2 Review Module 8 Learning Objectives • Gain a basic understanding of the notion of maximum and minimum of a function. • Gain a basic understanding of "Mean Value Theorem” Page 5
Math 19A (Online)
University of California Santa Cruz – Course Outline
• Understand how the derivative helps us understand the shape of graphs Topics: • Chapter 4 Introduction • Lesson 4.2: Extreme Values • Lesson 4.3: The Mean Value Theorem and Monotonicity • Lesson 4.4: The Shape of a Graph Module 9 Learning Objectives: • Gain a further understanding of how the derivative helps us understand the shape of graphs • Gain an understanding of L’Hôpital’s Rule and use it to compute limits Topics: • Lesson 4.5: L’Hôpital’s Rule • Lesson 4.6: Graph Sketching and Asymptotes Module 10 Learning Objectives • Apply the derivative to basic optimization problems • Understand how to model a physical phenomenon by a function y=f(x) and apply our methods to optimize f(x) independent of what f(x) actually represents • (Optional) Gain an overview of "Newton's method,” a technique that uses the derivative to address the problem of solving equations numerically, a topic which is very important in engineering applications Topics: • Lesson 4.7: Applied Optimization • Lesson 4.8: Newton's Method Final Exam Review • Exam Information • Math 19A Practice Finals • Math 19A Practice Problems for Final Course Resources • Course Overview • Course Introduction • Instructor Introductions • All Lecture Videos • Homework (CalcPortal) Page 6
Math 19A (Online)
University of California Santa Cruz – Course Outline
Support Options • College-­‐Based Study Groups • Tutoring Options • MSI (Modified Supplemental Instruction) in Santa Cruz • Disability Accommodations • Study Tips • Office Hours Instructional Strategy Instructional Element Canvas CalcPortal Intended Learning Experience • Primary course portal that students use to view videos and find general information and announcements on the course. Faculty update course information as necessary. • Canvas is 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. • Homework platform that students use to complete all homework assignments, quizzes, and exams if offered online. • Faculty 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 ported over to Canvas for export to appropriate Student Information Systems • Engagement can be evaluated through course analytics that provide data on 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 Page 7
Math 19A (Online)
University of California Santa Cruz – Course Outline
Instructional Element Launchpad Piazza Brightcove/Video Lectures Office Hours/Adobe Connect Intended Learning Experience areas that need further study or exploration. • E-­‐textbook platform that provides interactive reading assignments for each lesson. • Course designers customized the e-­‐book by editing passages and/or deleting extraneous material, embedding additional interactive features, creating progress check questions, etc. • Engagement/performance can be assessed through 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 • Discussion forum that allows students to interact with classmates and faculty • Teaching staff will facilitate and provide regular presence by answering questions and endorsing helpful student posts. Teaching staff will also create tags and folders to organize posts, and provide seed questions to stimulate discussion. • Engagement can be assessed 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. • Online video hosting platform. • Engagement will 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. • Offered both online and in-­‐person several times a week by Faculty and TAs. One-­‐on-­‐one appointments available. • Online Office Hours conducted using Adobe Connect, a web-­‐
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Math 19A (Online)
University of California Santa Cruz – Course Outline
Instructional Element Drop-­‐in Sections Tutoring College-­‐Based Study Groups Intended Learning Experience based 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. 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. • Offered once a 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. • 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). 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. • Groups are encouraged to meet in person or online at least once a week. • Assigned Canvas Group space to exchange notes, create wiki, make announcements, meet online. • Groups are 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. Page 9
Math 19A (Online)
University of California Santa Cruz – Course Outline
Assessment Activities Component Description Percentage Homework Online Homework problems for each lesson accessed through CalcPortal 15% •
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Quizzes Periodic online quizzes in CalcPortal •
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Reading •
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Final 30% 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) •
5% 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) •
10% 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). 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) •
2 Proctored Midterms 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. 40% Summative assessment used to evaluate student learning. Total: 100.00% Page 10