Honors Calculus II Online Course for Academic Credit
Honors Calculus II [DMAT 264 - Honors STEM Calculus II - 5 credits] is the honors-level second course in the freshman (STEM) calculus sequence investigating integration techniques, sequences, series, Taylor's Theorem, and selected topics from the beginning of multivariable calculus.
Nationally, "Honors" courses usually are centered on a mathematically rigorous development of the concepts of calculus, bringing many advanced topics from upper division courses such as Advanced Calculus and Real Analysis into the freshman honors calculus courses. While this may be a worthwhile approach for students who seek to become mathematics majors, it creates, for many students, an inflated level of course difficulty of questionable benefit to related fields of study.
Our approach to Honors courses is built upon a distinctly different educational philosophy:
- Freshman & Sophomore Calculus is NOT the correct math level to increase rigor
We believe that mathematical rigor is learned after exposure to the calculus, in the upper division, after some time and maturing of mathematical thought is allowed to organically develop. No student ever understood the concept of a derivative because the natural numbers were first axiomatically developed. - Honors means DEEPER, not just HARDER
In mathematics, the potential for making any course harder is a rather simple proposition. Work SMARTER, not HARDER mandates that an honors course should not be more difficult just to say it is. A true honors student wants to go deeper into the topics, not flirt with academic demoralization via some mathematical bootcamp experience. - Technical Writing Curriculum
While axiomatic development has its place in upper division math, core improvement of technical writing skills will benefit all students in all disciplines with immediate effect. If calculus is supposed to be mathematical preparation for science, technology, and engineering related fields, then development of technical writing skills should be as important as computational prowess. - Course Term Paper
Each Honors course student will write a 10-20 page term paper on a topic chosen in collaboration with the course instructor, empowering the student to simultaneously improve technical writing skills and deepen knowledge in the student's chosen academic field via a uniquely creative exercise that will transcend the traditional course boundaries.
In summary, our Honors courses go deeper and broader in the curriculum, offer a notch more challenging course work set, and featuring a technical writing curriculum that truly prepares the student for further academics in the sciences.
Completion of DMAT 264 - Honors STEM Calculus II earns 5 academic credit semester hours with an official academic transcript from Roger Williams University, in Providence, Rhode Island, USA, which is regionally accredited by the New England Commission of Higher Education (NECHE), facilitating transfer of credits nationwide to other colleges and universities.
DMAT 264 - Honors STEM Calculus II is offered in the Fall, Winter, Spring, and Summer semesters with "revolving enrollment", which means you may enroll at any time, and start your course whenever you wish, independent of the traditional academic calendar.
DMAT 264 Honors Calculus II differs from the mainstream DMAT 263 Calculus I course in the following ways:
Honors STEM Calculus II (DMAT 264 - 5 credits) is the second course in the freshman (engineering) calculus sequence, usually referred to as a course on "Integration Theory". With Calculus I concentrating mainly on the derivative and getting to the Fundamental Theorem of Calculus, Calculus II investigates integrals both how compute them algebraically (when you can), numerically (using numerical methods), and then using infinite polynomials (Power Series) to approximate integrals when you cannot compute them algebraically.
Completion of Math 264 - Honors STEM Calculus II earns 5 academic credit semester hours with an official academic transcript from Roger Williams University, in Providence, Rhode Island, USA, which is regionally accredited by the New England Commission of Higher Education (NECHE), facilitating transfer of credits nationwide to other colleges and universities.
HONORS Calculus II Course Information Video
Integration by Substitution
Double Folium Curve
Honors Calculus II Course Introduction
The second semester of 1st year Calculus is the preparatory course for all of the 2nd year courses (Multivariable Calculus, Differential Equations, Linear Algebra). Often referred to as "the first course on integral calculus", Calculus II focuses on a more practical, computational command of the integral, which is only introduced as a concept in Calculus I.
Calculus II is comprised of two distinct parts:
- Techniques of Integration
When you (or a computer) can algebraically integrate a function, how is that accomplished? Essentially, the Rules of Differentiation are "inverted" to the integral, providing the main strategies for attacking the algebraic integral - when possible. - When Algebraic Integration Just Can't Be Done
The vast majority of functions cannot be algebraically integrated - there just is no algebraic antiderivative for such functions. The development of "Plan B" for attacking these types of algebraic integrals comes in the form of expanding the way we describe functions, not just with the elementary class of functions including such friends as sin(x), ex, x1/2, etc. but with a more generalized description based upon infinite polynomials. This raises all kinds of questions that have to be studied, but once accomplished, we are able to conquer these algebraic integrals.
Calculus 2 - The Longest Course
Calculus II is the longest course in the Calculus sequence.
Many students feel that Calculus II is the most difficult course in the Calculus sequence as well.
Our Calculus II course has the following components:
- Integral Calculus Refresher
Calculus II starts with an intensive 40 assignment refresher of introductory integral calculus, starting from the definition of the integral via signed area, ranging to beginning algebraic integration, the Fundamental Theorem of Calculus, and initial applications of the integral. As many students start in our Calculus II course, having studied Calculus I elsewhere, it is important for all students to have this common foundation moving forward into the Calculus II course. - Techniques of Integration
One of the goals of Calculus II is to become an expert in algebraic integration: finding antiderivatives. Computer algebra tools can find antiderivatives automagically, so an exploration of the techniques of antiderivatives must contain an meaningful mixture of integration concepts, manual skills, and usage of computer algebra software. Traditional Calculus II courses explore these techniques purely from the paper/pencil standpoint, which has merits and drawbacks in this modern age. We strive for a balance between classical and modern computational mathematics in a unique way. For example, a more advanced integration technique known as Integration via Differentiation is presented, which is absent from all traditional textbooks, since it is computationally difficult with only manual tools - it is the leverage of computer algebra tools that makes this technique come alive. - Double Integrals and Gauss-Green Theorem
Calculus II starts the dimensional generalization of integration theory, into double (or 2D) integrals, which can be used to measure volume and other applications. Double integrals initially come in two varieties: over 2D regions that are essentially rectangular, and over 2D regions that are not rectangular, but whose boundary curve can be formulated. In the latter case, the Gauss-Green Theorem is utilized; it is common for this theorem to be explored only in Multivariable Calculus, but we do this theorem early as an introduction to the higher dimensional Fundamental Theorem of Calculus. - When Algebraic Integration Cannot Be Completed
The second half of Calculus II explores the problem: What to do when you cannot find an antiderivative to a function, like f(x) = x * tan(x) ? The limitation is not a matter of cleverness - one can prove that this function has no algebraic integral in terms of elementary functions (exponential, log, trig, roots, finite polynomial, etc.) This leads to the concept of a function polynomial expansion - a way to approximate a function by polynomials (bridging to equality when considering infinite polynomials). Moving to this new way of describing functions beyond elementary functions allows integration to be accomplished. As a result of polynomial expansions, infinite sequences and series are studied - a flipped order as found in all traditional calculus textbooks. - Introduction to Vector Calculus
To meet transferability requirements, Calculus II has a few introductory lessons on 3D geometry: lines, planes, vectors, dot products, cross products.
Why is the Calculus II so long? Historically and nationally, it has always been so. Calculus II has the reputation of being the barrier course to 2nd year Calculus - colloquially, the "weed-em out course". One reason for so many topics is for articulation/transferability to the majority of colleges/universities in the U.S.
DMAT 264 - Learning Outcomes
- To understand and compute algebraic integrals using a variety of symbolic techniques, including substitution, integration by parts, integration via differentiation, iteration methods
- To understand and compute with the Fundamental Theorem of Calculus
- To understand and compute solutions to applications problems involving integrals
- To compute measurements of volumes of geometric objects using integrals (slides, surfaces of rotation
- To understand and compute with Green's Theorem
- To understand and compute parametric and Polar function integrals
- To understand and compute double integrals
- To understand and compute splines and polynomial approximations
- To understand and compute with Taylor's Theorem
- To understand and compute with L'Hopital's Rule and using expansions to compute limits
- To understand and compute sequences and series
- To understand and compute convergence or divergence of sequences and series using various tests (Ratio, Integral, p-Test)
- To understand and compute 3D vector analysis, dot product, planes, and cross products
- To understand and compute partial derivatives and tangent planes to a surface
- Honors Topics:
- * To understand the concept of integration in finite terms and its connection to integration techniques
- * To understand and compute basic solutions of differential equations in relation to integration techniques.
- * To understand and compute polynomial approximations to solutions of differential equations.
- * To understand and compute near-finite-term integrals and their expression in power series.
- * To understand and compute integrals using complex integration techniques
- * To understand and compute numerical integration techniques of Newton, Midpoint, and Runge-Kutta, and higher RK approximations.
- * To understand and explore higher integral functions, such as those defined by elliptical and hyperbolic integrals
- * To explore initial topics in Analytical Number Theory
- *To understand higher special functions defined by either series or integral formulations
- *To understand and compute more advanced polynomial and rational polynomial approximation techniques (Chebyshev, et al)
- * To develop mathematical technical writing skills, culminating in a term paper on an approved topic
DMAT 264 - Syllabus of Topics
Calculus II Example Student Profiles
Case 1: Calculus I Done, Now Calculus II
For many students, planning for the Calculus II course is simply the natural progression from completing the Calculus I course, as part of their particular educational plan of study.Often students have the sights set on one of the second year courses:
- Multivariable Calculus
Usual course for majors in Economics, Physics, Engineering, Science - Differential Equations
An essential course for physics and engineering students - Linear Algebra
An essential course for physics, engineering, and computer science students. - Calculus-Based Probability
Required course for advanced economics, finance, bio-statistics, and other math-heavy graduate degrees.
| Typical Completion Timelines for MAT 2620 - Calculus II | ||||
| Hours Dedicated | Calculus I Grade | Dedication | Completion Time | Advisory |
|---|---|---|---|---|
| 5-10 hours/week | C,D | 1-2 hours/day | 16 weeks | Reasonable |
| 7-12 hours/week | C,D | 2-3 hours/day | 14 weeks | Reasonable |
| 10-15 hours/week | C+,B,B+ | 3-4 hours/day | 10 weeks | Reasonable |
| 15-20 hours/week | B | 5-6 hours/day | 8 weeks | Stretched |
| 20-25 hours/week | A | 5-7 hours/day | 6 weeks | Stretched |
| 25-35 hours/week | A+ | 6-8 hours/day | 4 weeks | World's Record |
Case 2: Undergraduate Science Major
Rafael is a undergraduate chemistry major, and part of his degree program dictates that he finish Calculus I and II. Rafael did pretty well in Calculus I, but wants to try something different for Calculus II, and to take the course via Distance Calculus.How fast can Rafael complete the Calculus II course?
The answer to this question depends mostly on when Rafael plans to work on the course.
The most successful time for Rafael would be during winter, spring, or summer breaks from his regular course load. By modifying his schedule to work on the Calculus II course during these "off times", Rafael will be able to concentrate and focus just on the Calculus II course, and his tendency towards success will increase dramatically.
If Rafael must complete the Calculus II course during the regular semester, as part of the other 3-5 courses he is taking, trying to fit Calculus II in there will be challenging. Here are some timelines for such a plan:
| Calculus II Course Plan With Full Course Load Otherwise | ||||
| Hours Dedicated | Calculus I Grade | Dedication | Completion Time | Advisory |
|---|---|---|---|---|
| 9 hours/week | C | 3 hours/day, 3 days/week | 18 weeks | Reasonable |
| 12 hours/week | C | 3 hours/day, 4 days/week | 14 weeks | Reasonable |
| 12 hours/week | B | 3 hours/day, 4 days/week | 12 weeks | Reasonable |
| 15 hours/week | B | 3 hours/day, 5 days/week | 10 weeks | Stretched |
| 15 hours/week | A | 3 hours/day, 5 days/week | 8 weeks | Stretched |
| 18 hours/week | A+ | 3 hours/day, 6 days/week | 6 weeks | Ambitious |
Case 3: Returning To Graduate School: Rushed
Kathy just got accepted into Graduate School in Economics towards a Ph.D. She has a conditional acceptance, and must finish Calculus II in a very short time. Kathy also works full time.What must Kathy do to finish Calculus II?
The "short time" needs to be on the order of 4-6 weeks for Calculus II. Anything less is quite impossible. Calculus II is just too long and difficult to complete in any less time. Kathy will also need to plan to be on the C grade path - which will lower her syllabus requirement by 30%. As the graduate school only needs demonstration that the Calculus II course was completed with a grade of C or above, and this grade will not transfer to Kathy's GPA, a C grade is planned for. (If Kathy finishes her C syllabus earlier than expected, she may go back and complete the additional assignments to move back to the B or A grade paths).
Kathy will need to plan for large blocks of time to dedicate towards the course. If shooting for 4 weeks completion, then at least 8-10 hours per day. If shooting for 6 weeks completion, then 6-8 hours per day, every day. It will be a very challenging 4-6 weeks.
Kathy should also be ready that the course takes a little longer than the planned 6 weeks. Going a week or two over to 8 weeks is quite common. Usually graduate schools are quite understanding of the situation of "I'm almost done", and in many cases, the graduate school will give a short extension to aid the student with harsh deadlines.
Kathy must understand that even though there are some tough deadlines with this plan, that does not mean that "corners are cut" in the Calculus II course because of these extenuating circumstances. There is only 1 way through the course, and that is the right way.
Case 4: Returning To Graduate School: Methodical
Halley is applying to Graduate School in Biostatistics towards a Ph.D., and will start in 1 year. Halley knows she must complete Calculus II prior to starting that course work (and even before submitting the graduate school application).What will the Calculus II course look like for Halley?
Halley is not in a rush, yet she wants to finish the course methodically, in a timely manner, as quickly as possible. As Halley is a Ph.D. bound student, she is highly motivation and task-completion oriented. Some timelines for Halley for Calculus II might look like this:
| Graduate-Bound Student Timelines for MAT 2620 - Calculus II | ||||
| Hours Dedicated | Calculus I Grade | Dedication | Completion Time | Advisory |
|---|---|---|---|---|
| 5-7 hours/week | C | 1-2 hours/day | 18 weeks | Relaxed |
| 8-12 hours/week | B | 2-3 hours/day | 10 weeks | Reasonable |
| 10-14 hours/week | B+/A | 3-4 hours/day | 8 weeks | Reasonable |
| 14-20 hours/week | A | 4-5 hours/day | 6 weeks | Ambitious |
Case 5: Lack of Success in Classroom Calculus II Course
Al took Calculus I and did OK in the course, earning a C grade. Then Al went onto Calculus II, hoping to improve his grades, but found Calculus II to be considerably more difficult than Calculus I, and ended up earning a D grade (or not passing).Al loathes the thought of returning to the classroom lecture for Calculus II. The lectures were not that much fun going through them the first time, and trying to sit through them a second time will be more than just painful.
Al wants to change gears, and try something new - taking Calculus II via Distance Calculus.
Typically, students like Al will take the course over the summer months, and be able to focus just on the Calculus II course, without being overwhelmed by multiple courses all vying for Al's attention.
Al should plan to take a step back, a deep breath, and engage the course as if starting on a marathon race. Just because Al "had Calculus II already", this poor foundational knowledge will not help Al succeed in this second (and quite different attempt).
Al would be approaching Calculus II from a completely different viewpoint than his previous attempt, which certainly goes a long way to minimizing the "review" nature of repeating a course. When going through the new Calculus II course, all will see things he saw before (e.g. integrals), but completely different types of questions that he probably saw in the classroom/textbook course. An open mind, and genuine academic curiousity to investigate and embrace the new style of work will be required for success.
Al should plan for a minimum of 12 weeks to engage the course, even during the summer. It will take time for the course concepts and materials to "sink in" and become comfortable.
We have had many students like Al succeed in Calculus II where their classroom experience with Calculus II was a failure. However, we have also had many students like Al who discover that the reason they didn't succeed in their first attempt at Calculus II was an indicator that their math skills are not strong, their interest in math is weaker, and their chosen course study plan may need to change upon failure in two different course paradigms.
Case 6: AP Calculus BC Equivalency
Anwar is a hot-shot high school student. He has completed AP Calculus AB (equivalent to Calculus I), and instead of taking the AP Calculus BC exam, he wants to jump ahead and take the Calculus II course from Distance Calculus, earn collegiate credits directly, and transfer them to his planned university (that he currently has applications out to).Anwar will most likely be able to finish the Calculus II course very quickly with a dedicated effort that is quite natural for him. 6-8 weeks is the common time frame.
However, Anwar may discover that Calculus II via Distance Calculus is an adult course, and quite unlike the high school course paradigm that he has been so successful in previous. Some students like Anwar can find Calculus II more difficult than they anticipated, and they have to dedicate more time and effort than they had planned to budget for.
While Anwar can earn collegiate credits for Calculus II in a similar way that AP Calculus BC would earn, Anwar would not benefit from the inflated GPA multiplier that AP courses help with high school GPAs.
On the plus side, quite often schools (mostly graduate schools) will not accept AP Calculus course credit for satisfaction of program prerequisites. Some schools are starting to insist that higher courses - like Calculus II - be completed a real collegiate courses, and not "waived via AP Exam". We have seen more than a few students have to return to Distance Calculus to re-take Calculus II again as a real collegiate course, as they graduate program would not accept the AP Calculus BC exam completed in high school as proof of Calculus II completion.
Distance Calculus Referenced Colleges & Universities (29 Years - 393+ Institutions)
Distance Calculus students have transferred course credits to these colleges and universities:
CALCULUS II: ACADEMICS
80% Computer Algebra, 20% Pencil/Paper, 0% Multiple Choice
Through the usage of a computer algebra system like LiveMathâ„¢ - you will never miss a minus sign again!
Although the driving of a computer algebra system requires some up-front time to learn and master, once completed (rather quickly for most students), the time saved from having to be a "minus sign accountant" adds to the productivity of your study time. If you have ever spent hours looking for that "little numerical error", you know what we mean.
Command of a computer algebra software system is a modern-day necessity of mathematical academics. It is important, however, to retain a meaningful command of paper/pen/pencil manual computations as well. Our blend of curriculum strives for an 80%/20% split between computer algebra usage and manual computation and written skills. With each module in our curriculum, a concluding Literacy Sheet assignment ensures that each student has written mathematical competency in the subject area.
The proctored final exam is a written exam away from the computer. It is these Literacy Sheet assignments, and the continuing bridge from modern computer algebra software back to classical, manual mathematics that prepares the student for this written final exam.
We do not have any multiple-choice work. We are a real collegiate-level course program - not a "canned" set of multiple-choice question sheets which are common from large publishers and degree-mill schools.
A modern-day study of the topics in Calculus II demands the usage of a computer. Although the learning of integration techniques is meant to support a manual approach to the calculation, the goal is to learn the concepts of the techniques, and to drive the computer algebra tool to computations using those techniques. Writing out long computations on the solution of
is rather pointless considering the technology available today; but understanding the process and strategy towards computing this antiderivative, and the inherent skills acquired that will generalize to other solution skills, is essential, and the usage of computer algebra technology allows for this study to replace time wasted on multi-page hand calculations.
Calculus II - Example Curriculum
Videotext - A Modern Replacement of the Textbook
What is a videotext? It is like a textbook, except instead of being based upon printed information, this "text" is based upon video presentations as the core method of explaining the course topics. Instead of a huge, thick 1000-page Calculus textbook to lug around in your backpack, all of this new "videotext" can be loaded into your iPods or iPhones (and soon, the iPad!).Example Videos are in MP4/H.264 format, which play in most modern browsers without additional software. When additional software is required, a backup Flash player will play the video. As a backup to Flash, you may also use iTunes and/or VLC.
Although we are anywhere from a few miles to a few thousand miles apart, watching these screencast videos is like sitting next to the course instructor, watching his computer, learning the topics of Calculus at the same time as learning how to drive the computer algebra and graphing software LiveMath™. These LiveMath™ screencast videos make up the majority of the video presentations in the videotext.
Calculus II - Screencast Video Questions
One extremely powerful aspect of the Distance Calculus course technologies is the usage of screencast video (and audio) recordings made by the students and the instructors, exchanged just as easily as emails back and forth.
If a picture is worth a thousand words, then a screencast movie is worth a million words - and saves boatloads of time and effort.
Instead of trying to type out a math question about a particular topic or homework question, the ease of "turning on the screen recorder" and talking and showing your question - in the span of a few minutes - can save hours of time trying to convert your question into a typed (and coherent) narrative question.
Example Student/Instructor Question/Answer Movies
When a student asks a question in a homework notebook, sometimes the best way to explain the answer is via a screen movie. We utilize easy-to-use software to create screen movies with audio so that you can communicate with your instructor "as if they were sitting at the computer with you". Students don't ask questions like this everytime they have a question, but it is a good way to ask questions after text chats aren't working.If a picture is worth a thousand words, and a movie is worth a thousand pictures, then a movie with audio is worth 1000*1000*1000 = 1,000,000,000 (billion) words!
Calculus II - Example Student Work and Grading
Distance Calculus - Student Reviews
Dist Calc saved a ton of hassle and time having to take calc at a local community college, which realistically would have delayed my entrance to BU LEAP by at least a year.
Would strongly recommend it.
Pros:
-- Exceptional Instruction and Support: Dr. Curtis was incredibly responsive and helpful whenever I had questions. The TA was also very supportive, and thanks to their guidance, I was proud to earn a 100% in the course—even without having taken pre-calculus beforehand.
-- Innovative Software Platform: The custom software used in the course made a huge difference for me. I found it intuitive and engaging, and it helped reinforce the concepts in a way traditional textbooks never did.
-- Thorough, Rigorous Curriculum: The structure of the course really pushed me to stay organized and plan ahead. I felt like I was being challenged in all the right ways.
Potential Considerations for Others:
-- Requires Strong Time Management: If you haven’t taken pre-calc, like me, you’ll need to be extra proactive. The course can move quickly if you need, and pacing yourself is essential.
-- Software Learning Curve: While I personally loved the software, students who aren’t comfortable adapting to new digital tools might need a bit of extra time upfront to get used to it.
-- Helpful to Have Supplementary Resources: One improvement might be to offer a short list of "starter resources" (videos, concept overviews, etc.) for students who need a broader intro to calculus before diving in.
Overall, I highly recommend this course to motivated students, especially those comfortable with self-paced learning and open to using new tools. Dr. Curtis is a fantastic instructor, and the course setup really works.
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