ESC195: Calculus II


Q: Are monsters good at math?
A: Not unless you Count Dracula.


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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


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. 

  • Multivariable calculus! You can now solve problems in three dimensions. 😄

  • Sketching polar graphs, i.e., all the complicated-sounding graphs like limaçons, lemniscates, and…cardiods!  

Week in the Life of an ESC195 Student

Lectures

ESC195 has three hours worth of lecture each week. While three hours may not seem like that much, 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.

Tutorials

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, like in Calc I. Each quiz is only worth 2% but they serve as 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 every week.

Practicals

There are no practicals for ESC195. 

Assignments

As in ESC194, there are no formal assignments for this course. However, you are provided with recommended practice problems every week. DO THESE!! It 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 alternate 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!

Exams

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 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. 


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. 

  • Know the Taylor expansions for basic functions (exp, cos, sin, etc).

  • Do lots of integration problems and practice using different techniques. 
XKCD webcomic. A satirical representation of integration: flow chart showing the messy steps and dead-ends of integration.

Integration sometimes feels like this…but you’ll figure it out eventually! [Edited – Source]

  • Practice sketching polar curves: 
A compilation of generic polar curves, along with their equations.

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  • Multivariate chain rule: 
Pictorial representation of the multivariate chain rule, using a tree diagram.

[Source]

  • Understand what a gradient is and how to use it.
Graph of an arbitrary 3D function f(x,y), with the partial derivative and gradient vectors drawn at an arbitrary point.

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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.

Make a cheat sheet for sequences and series

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.  


What Will You Take Out of It?

  • 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.
Meme: Image of a polar bear with the words

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  • 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. 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).