Teaching Template Post

4 minute read

This is a template syllabus for future use in a Calculus 1 course. This entire page is fictional and was written so I can be lazy in the future.

Basic Info:

Lecture Times: Tue/Thu 11:40-12:55
Location: TBA
Instructor: Alex Xu
Email: axu [at] math.columbia.edu ; please include the course number in the subject header
Office Hours: Wed 11:40-12:55 or by appointment at (Columbia has not assigned first year grads offices yet)
Teaching Assistants: TBA

Course description

In this class we will cover

  • Limits of sequences and series, and continuity of functions
  • Derivatives of functions and their implications
  • Integration of functions and the fundamental theorem of calculus

Ultimately, the goal is to gain an intuition for these abstract mathematical concepts and be able to apply them in contexts outside of a math class.


The standard reference is Calculus: Early Transcendentals, 8th edition, by John Stewart. This will be used as a reference for examples and exercises.

An interesting supplementary textbook will be The Calculus: A Genetic Approach, by Otto Toeplitz. A PDF is freely available online here. This book will be used to help you learn the concepts more concretely


Weekly homework sets will be posted online (for a total of 11) on weeks that do not have midterms. The lowest 3 grades will be dropped, for a total of 8 assignments that count towards your grade. But you should still do all of the homework for your own sake. Math classes generally build each lecture on top of the previous one so it will only hurt you if you decide to abuse this policy and skip homework.

Homework is due in class by the end of the first class the following week (Mon/Tues). Alternatively you can also drop them off before class in the drop box on the 4th floor of the Math building. Late homeworks will not be accepted without a note from a doctor or dean documenting a medical or family emergency. Grades will be posted on CourseWorks. You are encouraged to discuss the homework with other students but you must write your solutions individually, in your own words.

In addition, the material to be covered will be posted online in advance and I highly recommend that you at least skim the textbook so you can get more out of lecture.


There will be 2 midterms and a final exam. Since later concepts build upon previous concepts, these exams will necessarily be cumulative. Practice problems will be released the week before. If you require special accommodations due to religious or disability reasons, please try to reach out at least 2 weeks beforehand to we can find an arrangement that works.


Approximate topics covered each class. The material for the week will be approximately taken from Stewart; extra reading from other sources is optional but helpful for intuition and historical contexts.

Class # (Date) Topics Notes
1 Review of special functions Stewart Ch 1
2 Inverse and piecewise functions  
3 Sequences and their limits Stewart Ch 2.1-2.2, Toeplitz Ch 6-10
4 Continuity and limits of functions  
5 Limit laws, squeeze theorem Stewart Ch 2.3-2.8
6 The derivative, motivation and definition  
7 The derivative of special functions Stewart Ch 3.1-3.3,3.6
8 Product and quotient rules  
9 The chain rule and implicit differentiation Stewart Ch 3.4-3.5
10 Related rates and linear approximation Stewart Ch 3.9-3.10
11 Midterm 1 review  
12 Midterm 1  
13 Extreme value theorem, graph sketching Stewart Ch 4.1-4.3, Toeplitz Ch 18-21
14 Mean value theorem  
15 Indeterminate forms and L’Hospital’s rule Stewart Ch 4.4-4.6
16 Second derivatives and convexity  
17 Optimization problems Stewart Ch 4.7
18 Midterm 2 review  
19 Midterm 2  
20 Antiderivatives Stewart Ch 4.9
21 Area and distance Stewart Ch 5.1-5.2, Toeplitz Ch 11-16
22 Riemann integrals and computations  
23 Fundamental theorem of calculus Stewart Ch 5.3, Toeplitz Ch 23
24 Substitution rule and integration by parts Stewart Ch 5.5, 7.1
25 Indefinite integrals Stewart Ch 5.4
26 Improper integrals Stweart Ch. 7.2
27 Review of first half of the course  
28 Review of second half of the course  


  • Hw 1: (Due by Lecture 3)
  • HW 2: (Due by Lecture 5)
  • HW 3: (Due by Lecture 7)
  • HW 4: (Due by Lecture 9)
  • HW 5: (Due by Lecture 11)
  • Hw 6: (Due by Lecture 15)
  • HW 7: (Due by Lecture 17)
  • HW 8: (Due by Lecture 21)
  • HW 9: (Due by Lecture 23)
  • HW 10: (Due by Lecture 25)
  • HW 11: (Due by Lecture 27)


Category Weight
Homework 40% (5% per homework)
Midterm Exams 40% (20% per midterm)
Final Exam 20%

If your final exam score is higher than the lowest of your midterm exam scores, then the corresponding midterm exam score will be replaced with your final exam score. Grading for this course will be as follows

Grade Score
A 90%-100%
B 80%- 89%
C 70%- 79%
D 60%- 69%
F 0% - 59%


Here are some resources that you might find helpful