Columbia MATH UN1101 Calculus I: Topics, Study Guide and Practice

Columbia MATH UN1101 is Calculus I at Columbia University, the first course in all of the mathematics department's calculus sequences. It is where students who have not yet had calculus begin, whatever their major. This guide maps each UN1101 topic to a lesson you can practice with instant feedback.

What Columbia MATH UN1101 covers

The Columbia bulletin keeps its description short: functions, limits, derivatives, and an introduction to integrals, with an understanding of precalculus assumed.

That matches a standard Calculus 1 course. The "introduction to integrals" part usually means antiderivatives, Riemann sums and the Fundamental Theorem, with integration methods and applications saved for UN1102. On PracticeCalculus, that is the Calculus 1 path through the integrals unit.

Because the official description is brief, expect the derivative portion to carry most of the semester: the limit definition, the product, quotient and chain rules, derivatives of trigonometric, exponential and logarithmic functions, and applications such as related rates, curve sketching and optimization. If you are unsure whether a topic is on your exam, check your section's syllabus, since instructors set the detailed schedule.

UN1101 topic map

Columbia's calculus sequences

The bulletin lists three paths through calculus and linear algebra, and UN1101 starts all of them:

Credit is allowed for only one of these sequences, so it is worth deciding early which one fits your plans. The bulletin also notes that students with strong AP scores may start at Calculus II or III, and that a 5 on AP Calculus BC allows starting directly in Calculus III. If you are placed into UN1101, the course assumes no previous calculus, so the first weeks are the best time to build habits: short daily practice, careful algebra and checking each answer.

Example problems at the UN1101 level

1. A trig limit. Using \lim_{u \to 0} \frac{\sin u}{u} = 1 with u = 3x:

2. A quotient rule derivative.

3. The Fundamental Theorem, part 1.

How to do well in MATH UN1101

• Make precalculus fluent before week two. The bulletin assumes it. The functions warm-up and key trig values lessons are a fast check.
• Learn limits by technique. Direct substitution, factoring, rationalizing and the squeeze theorem cover almost every exam limit.
• Drill the derivative rules with the power rule, quotient rule and chain rule until they are automatic, keeping the formula sheet handy.
• Know the common slips. Read common calculus mistakes and test yourself with practice exams before each midterm.

Tip: a limit that gives \frac{0}{0} is not finished, it is a signal to simplify. Factor, cancel and substitute again.