Q: Are monsters good at math?
A: Not unless you Count Dracula.
ESC195: Calculus II builds on the skills you learned from ESC194: Calculus I. You’ll start by studying methods of integration—essentially, ways to compute the “unsolvable” integrals you might have encountered in Fall semester. You’ll also be introduced to sequences and series before diving into the world of multivariable calculus and vector functions. All these concepts are fundamental to science and engineering collectively.
Like ESC194, this is a theoretical course that covers a lot of material at a fast pace and in great depth, so keeping up with the work and further developing your problem-solving skills is key.
Note from your advisors: ESC195 might be the most important course in first- and second-year EngSci, at least from a course progress point of view. ESC195 is a prerequisite for five second-year courses; that makes it an important course to pass, but beyond that, understanding the course material is very important. While you are taking the course, put effort into actually understanding the content, rather than just blindly memorizing. Treat it like you’d treat a second language: the more you practice, the more comfortable you’ll get using it, and the better you’ll be able to do.
Professor
Professor James Davis
The instructor for ESC195 is Professor Davis, whom you’ll recognize from ESC194.
Professor Interview
“All of physics really came down to one equation – F = ma with calculus […] Everything from things like the Bernoulli equation governing fluid flow, to the rocket equation governing how big rockets have to be – it’s all just F = ma with calculus.”
“A student will only learn by doing. You’re not gonna learn by reading the textbook, you’re not gonna learn by attending the lectures alone. You actually have to do the work, and that’s the only way you’ll be able to learn this material”
“Anything that gets in between a student and pencil and paper is detrimental to the learning of mathematics.”
Course Highlights
- You’ll learn integration by parts, trig substitution, and partial fractions. These may sound complicated now, but you’ll get the hang of them.
- Infinite sequences and series – you’ll learn about some of their properties and applications, including how Fourier series can represent any periodic function.
- A lot of multivariable calculus! You can now solve problems in three dimensions. 😄
- Sketching polar graphs (all the complicated-sounding graphs like limaçons, lemniscates, and cardioids) and 3D surfaces (all the even more complicated-sounding graphs like paraboloids and hyperbolas).
Week in the Life of an ESC195 Student
Lectures
ESC195 has three hours of lecture each week. It may not seem like much, but the lectures move very quickly. They cover derivations of course concepts and many worked examples. Students occasionally have trouble taking notes in this course because the professor tends to write quickly on the chalkboard. If you can’t keep up with his notes, we recommend you at least copy down the examples. Knowing how professors solve examples can help you solve similar problems on your own. The course textbook (Stewart’s Calculus, same as in ESC194) supplements your course notes well.
Tutorials
Like ESC194, there are no practicals. There is one hour of tutorial every week. As in ESC194, you’ll be in smaller classes (20-25 students). TAs will work through problems like the assigned homework questions. At the end of each tutorial, you’ll also do a quiz. Each quiz is worth 2-3% of the final grade and are a great way you’re of staying on top of the material. The questions for the quiz stem directly from the problem set for that week, so it incentivizes you to do your homework!
Assessments
As in ESC194, there aren’t any formal assignments for this course beyond the weekly quizzes. However, you’re provided with recommended practice problems every week. DO THESE! They’ll build your calculus skills and help prepare you for assessments.
The assigned problems are all from the Stewart textbook (same textbook as in ESC194). The Stewart textbook already contains the final answers to all odd-numbered questions. You can use it to check your work, discover alternative solution methods, and help yourself if you get stuck.
Our recommendation to you, as in ESC194, is practice, practice, practice. Spending a few extra hours per week on calculus questions will make a huge difference for you!
Midterms & Exams
The midterms and final exam in ESC195 are like those in ESC194. Check out the ESC194 course overview for advice on how to prepare and manage exams. Key takeaway: practice with the Stewart textbook as well as past midterms/exams and be strategic when writing the assessments. Questions aren’t necessarily ordered by difficulty, so if you get stuck, skip ahead and return later to harder questions.
How to Succeed
Quick Tips & Equations
Note: You’re not expected to know the following technical information. You’ll learn it all in the course.
- A Taylor series is an infinite series of polynomial terms that can be used to approximate complicated functions such as exponential, logarithmic, and sinusoidal. As the degree of the polynomial (and number of terms) increases, the Taylor series becomes a better approximation for the function. Make sure you know how to derive a Taylor series (and possibly memorize some common ones) and how to calculate its error.
- Do A LOT of integration problems involving many different methods. Unlike derivatives, some integration problems will need trial and error to solve efficiently. Regular practice will let you solve them faster during exams.
- Practice sketching polar curves. Polar coordinates are essential to solving problems involving circles, cardioids, and limaçons, among others. And the more comfortable you become at solving problems in the polar coordinates, the easier it’ll be for you to work with cylindrical and spherical coordinates, introduced in second-year courses.
- Gain an intuition for gradients, which are like derivates in higher dimensions, and how they work. You should be able to derive these from first principles. Practice plenty of gradient-based problems, including tangent planes and Lagrange multipliers. Although you’ll learn these topics in the last week of class, don’t neglect them, as they are likely to appear on the final exam.
More Details
All the tips from ESC194 will be useful here too, so check them out in the Calculus I overview. However, we still have one piece of advice specific to ESC195.
ESC195 is a very formula-heavy course. Even though you aren’t allowed to bring a formula sheet to the midterms or the final exam, we highly recommend you make one to help with review, which can both help you keep track of what formulas you need to memorize and help you with the actual memorization.
A specific example would be sequence and series techniques. When analyzing series, such as the convergence and/or limit of series, there will be many techniques to remember. To make it easier, we suggest creating a cheat sheet: list all the techniques (formally called “series convergence tests”) and when to use them. Then, as you’re doing practice problems, you can reference this cheat sheet.
Another example: when doing vector function problems, there are many formulas that must be used and memorized, such as the formulas for the unit tangent vector or curvature of a plane. It is a good idea to keep all these formulas written down so that you can reread them and memorize them quicker.
Beyond First Year
- After ESC195, you’ll be able to appreciate not only more advanced scientific fields, but also more advanced math jokes—you know, jokes about cows and bears and all that.
- On a more serious note: the advanced math that you’ll learn in this course will help you understand and work in more specialized fields. Integration techniques and polar coordinates are used extensively in ECE259 (Electromagnetism) next year. Partial derivatives become important in a variety of other fields, including physics, and vector functions are extremely important in computer science.
Note: The course code for Calculus II used to be MAT195. You may still see it referred to as such on some websites (e.g. courses.skule.ca).
