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 course is a theoretical course that covers a lot of material at a fast pace and great depth, so keeping up with the work and further developing your problem-solving skills is key.
Professor
The instructor for ESC195 is Professor Davis, whom you will recognize from ESC194.
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 will 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
Classes
ESC195 has three hours of lecture each week. It may not seem like that much but the lectures move very quickly. They cover derivations of course concepts and many worked examples. Sometimes students find it hard to take 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 all 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.
While there are no practicals, ESC195 has an hour of tutorial every week. As in ESC194, you will be in smaller classes (25-30 students). TAs will work through problems similar to the assigned homework questions. At the end of each tutorial, you will also do a quiz. Each quiz is worth 2% and they are a great way to check that you are staying on top of the material. The questions for the quiz stem directly from the homework set for that week, so it incentivizes you to do the homework each week.
Assessments
As in ESC194, there are no formal assignments for this course. However, you are provided with recommended practice problems every week. DO THESE! They will build your calculus skills and help prepare you for assessments. Indeed, some questions on quizzes, midterms, and exams are similar to those assigned.
The assigned problems are all from the Stewart textbook (same textbook as in ESC194). A great thing about the textbook is that you can buy it in a package with a student solution manual. This contains worked solutions to all odd-numbered problems. You can use it to check your work, discover alternative solution methods, and help yourself if you get stuck.
Note: Although it can be useful, the student solution manual is not required for this course. The Stewart textbook already contains the final answers to all odd-numbered questions.
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!
Assessments in ESC195 are very similar to 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. These tend to go fast for students, so start with the questions you know how to solve before moving to the hard ones. ESC195 exams have fewer questions than those in ESC194, but the questions can be significantly harder.
How to Succeed
Quick Tips & Equations
Note: You are not expected to know the following technical information. You will learn it all in the course.
- Taylor series is an infinite series of polynomial terms that can be used to approximate complicated functions involving exp, cos, sin, and ln. 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. Some integration problems will need trial and error to solve efficiently, unlike derivatives. Regular practice will let you solve them faster during exams.
- Practice sketching polar curves. Polar coordinates are essential to solving problems involving circles, cardioids, limaçons, etc. And the more comfortable you are in solving problems in the polar coordinates, the easier it will be for you to work with cylindrical and spherical coordinates, introduced in 2nd year courses.
- Gain an intuition for gradients, which are like derivates in higher dimensions, and how they work. You should be able to derive them from first principles and from an algorithm, and practice many problems with gradients such as tangent planes and Lagrange multipliers; you will learn about these methods in the last week of class, but do not ignore them as they will come up 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.
Working with sequences and series is an important tool for engineers, so make sure you practice and understand these topics. When analyzing 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 are doing practice problems, you can reference this cheat sheet.
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 will 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 physics fields 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).
