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

> A study guide for Columbia MATH UN1101 Calculus I: functions, limits, derivatives and an introduction to integrals, with practice for every topic.

Source: https://practicecalculus.com/courses/columbia-math-un1101
Updated: 2026-11-03

## At a glance

- **Course code:** MATH UN1101
- **Points:** 3
- **Offered:** Fall and spring
- **Prerequisites:** A working understanding of precalculus (see the department's guidance for first-year students)
- **Covers:** Functions, limits, derivatives and an introduction to integrals
- **Next course:** MATH UN1102 Calculus II
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH UN1101 is Columbia's Calculus I, a 3-point course offered in both fall and spring.
- It covers functions, limits, derivatives and an introduction to integrals, and assumes a solid grasp of precalculus.
- It is the first course in every one of the department's calculus sequences.
- The next course is MATH UN1102 Calculus II.

**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

| UN1101 topic | Practice | Read |
| --- | --- | --- |
| Functions review | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Exponential functions](/functions/exponential-functions), [logarithmic functions](/functions/logarithmic-functions) |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit), [limit laws](/rules/limit-laws) |
| Evaluating limits | [Factor and cancel](/learn/calculus-1/limits/factoring-zero-over-zero) | [How to evaluate limits](/blog/how-to-evaluate-limits), [squeeze theorem](/theorems/squeeze-theorem) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity), [Intermediate Value Theorem](/theorems/intermediate-value-theorem) |
| The derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiation rules | [The quotient rule](/learn/calculus-1/derivative-rules/quotient-rule) | [Product and quotient rules](/blog/product-and-quotient-rules), [chain rule](/blog/chain-rule) |
| Applications of derivatives | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates](/blog/related-rates), [optimization problems](/blog/optimization-problems) |
| Introduction to integrals | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [Derivatives of integrals](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |

## Columbia's calculus sequences

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

| Path | Courses after UN1101 |
| --- | --- |
| Standard | [UN1102](/courses/columbia-math-un1102), [UN1201](/courses/columbia-math-un1201), [UN1202](/courses/columbia-math-un1202), then UN2010 Linear Algebra |
| Accelerated | UN1102, UN1205 Accelerated Multivariable Calculus, then UN2010 |
| Honors | UN1102, then UN1207 and UN1208 Honors Mathematics A and B |

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 $u = 3x$:

$$\lim_{x \to 0} \frac{\sin 3x}{x} = \lim_{x \to 0} 3 \cdot \frac{\sin 3x}{3x} = 3$$

**2. A quotient rule derivative.**

$$\frac{d}{dx}\left(\frac{x}{x^2 + 1}\right) = \frac{(x^2 + 1) - x(2x)}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}$$

**3. The Fundamental Theorem, part 1.**

$$\frac{d}{dx} \int_0^x \cos(t^2)\,dt = \cos(x^2)$$

## How to do well in MATH UN1101

- **Make precalculus fluent before week two.** The bulletin assumes it. The [functions warm-up](/learn/calculus-1/functions-warmup/function-notation) and [key trig values](/learn/calculus-1/functions-warmup/key-trig-values) lessons are a fast check.
- **Learn limits by technique.** Direct substitution, factoring, rationalizing and the [squeeze theorem](/theorems/squeeze-theorem) cover almost every exam limit.
- **Drill the derivative rules** with the [power rule](/rules/power-rule), [quotient rule](/rules/quotient-rule) and [chain rule](/rules/chain-rule) until they are automatic, keeping the [formula sheet](/formulas) handy.
- **Know the common slips.** Read [common calculus mistakes](/blog/common-calculus-mistakes) and test yourself with [practice exams](/practice) 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.

## Frequently asked questions

### What is Columbia MATH UN1101?

MATH UN1101 is Calculus I at Columbia. The bulletin lists its topics as functions, limits, derivatives and an introduction to integrals. It is worth 3 points.

### What do I need to know before UN1101?

The bulletin says an understanding of precalculus is assumed and points students to the department's guidance for first-year students. In practice that means algebra, functions, exponentials, logarithms and trigonometry.

### What comes after UN1101?

MATH UN1102 Calculus II, which covers integration methods, applications of the integral, Taylor's theorem and infinite series. UN1101 also satisfies the stated prerequisite for MATH UN1201 Calculus III, although the department strongly urges taking Calculus II first.

### Is Calculus I at Columbia hard?

It is a standard first course in calculus, but the pace is quick and it assumes fluent precalculus. Students who rebuild their algebra and trig skills early and practice a little every day usually find it manageable.

### Can I skip UN1101 with AP credit?

The bulletin says students with strong AP scores may start at Calculus II or III, and describes how AP credit is granted after a later course is passed. Check the department's placement guidance for your score.