# NYU MATH-UA 121 Calculus I: Topics, Study Guide and Practice

> A study guide for NYU MATH-UA 121 Calculus I: derivatives, antiderivatives, the definite integral and areas and volumes, with practice for every topic.

Source: https://practicecalculus.com/courses/nyu-math-ua-121
Updated: 2026-11-03

## At a glance

- **Course code:** MATH-UA 121
- **Credits:** 4
- **Offered:** Fall, spring and summer
- **Prerequisites:** A grade of C or better in MATH-UA 9, or a qualifying SAT, ACT, AP or placement score
- **Covers:** Derivatives, antiderivatives and integrals of functions of one variable, graphing and optimization, the definite integral and the Fundamental Theorem, areas and volumes
- **Next course:** MATH-UA 122 Calculus II
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH-UA 121 is the first course in the NYU College of Arts and Science calculus sequence, worth 4 credits and offered in fall, spring and summer.
- It covers a standard Calculus 1: derivatives and their applications, antiderivatives, the definite integral and areas and volumes.
- Students place in with a C or better in MATH-UA 9 or a qualifying test or placement score.
- The next course is MATH-UA 122 Calculus II.

**NYU MATH-UA 121** is Calculus I in NYU's College of Arts and Science, the starting point for students heading into math, science, economics and pre-health tracks that require calculus. It is a full one-semester introduction to single-variable calculus. This guide lines up each MATH-UA 121 topic with a lesson you can practice right away, with instant feedback on every answer.

## What NYU MATH-UA 121 covers

The NYU bulletin describes MATH-UA 121 as a course on derivatives, antiderivatives and integrals of functions of one variable. Applications include graphing and optimization, and the course builds to the definite integral, the Fundamental Theorem of Calculus, and using integrals to find areas and volumes.

That is a textbook **Calculus 1**: limits and continuity first, then derivatives and their rules, then applications, then integration. It matches the full Calculus 1 path on PracticeCalculus almost unit for unit.

## MATH-UA 121 topic map

| MATH-UA 121 topic | Practice | Read |
| --- | --- | --- |
| Limits and continuity | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits), [continuity](/blog/continuity) |
| 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 | [Product rule](/learn/calculus-1/derivative-rules/product-rule) | [Power rule](/rules/power-rule), [chain rule](/rules/chain-rule) |
| Exponential, log and trig derivatives | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Derivatives of trig functions](/blog/derivatives-of-trig-functions) |
| Graphing with derivatives | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Curve sketching](/blog/curve-sketching-with-derivatives) |
| Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) |
| Antiderivatives | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives](/blog/antiderivatives) |
| The definite integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Areas and volumes | [Volumes with disks](/learn/calculus-1/applications-of-integrals/volumes-disks) | [Area between curves](/blog/area-between-curves), [volumes of revolution](/blog/volumes-of-revolution) |

## Where MATH-UA 121 sits in the NYU sequence

| Course | Focus | Prerequisite (from the bulletin) |
| --- | --- | --- |
| MATH-UA 121 Calculus I | Derivatives, integrals, areas and volumes | MATH-UA 9 with a C, or a qualifying score |
| [MATH-UA 122 Calculus II](/courses/nyu-math-ua-122) | Integration techniques, polar and parametric equations, series | MATH-UA 121 with a C, or AP AB/BC 4, or placement |
| [MATH-UA 123 Calculus III](/courses/nyu-math-ua-123) | Multivariable and vector calculus | MATH-UA 122 with a C, or AP BC 5, or placement |

NYU also runs a separate Mathematics for Economics sequence (MATH-UA 131, 132, 133). The bulletin notes that students cannot take both that sequence and the standard Calculus I to III sequence, so check which one your major requires before you register.

## Example problems at the MATH-UA 121 level

**1. A zero over zero limit.** Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.

Factor and cancel: $\frac{(x-3)(x+3)}{x-3} = x + 3$, so the limit is $3 + 3 = 6$.

**2. A product rule derivative.** Differentiate $f(x) = x^3 e^x$.

$$f'(x) = 3x^2 e^x + x^3 e^x = x^2 e^x (3 + x)$$

**3. A definite integral with the Fundamental Theorem.**

$$\int_0^2 (3x^2 + 1)\,dx = \left[x^3 + x\right]_0^2 = (8 + 2) - 0 = 10$$

## How to do well in MATH-UA 121

- **Start with the warm-up.** If MATH-UA 9 was a while ago, the [functions warm-up](/learn/calculus-1/functions-warmup/function-notation) rebuilds exponent, log and trig skills before limits begin.
- **Make the derivative rules automatic.** Most of the semester leans on the [product rule](/rules/product-rule), [quotient rule](/rules/quotient-rule) and [chain rule](/rules/chain-rule). Drill them until you do not have to think.
- **Watch for the classic slips.** Our list of [common calculus mistakes](/blog/common-calculus-mistakes) covers dropped chain rule factors, sign errors and forgotten constants.
- **Keep the formula sheet nearby** while you practice. The [calculus formula sheet](/formulas) has every rule on one page.
- **Rehearse before each exam** with timed [practice exams](/practice), and follow our guide on [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam).

> **Tip:** optimization problems are the most common place students lose points. Write the quantity you are maximizing as a function of one variable first, then differentiate.

## Frequently asked questions

### What is NYU MATH-UA 121?

MATH-UA 121 is Calculus I at NYU. It covers derivatives, antiderivatives and integrals of functions of one variable, with applications such as graphing and optimization, the Fundamental Theorem of Calculus, and areas and volumes.

### What do I need before taking MATH-UA 121?

The NYU bulletin lists a C or better in MATH-UA 9 (the precalculus course) or a qualifying SAT, ACT, AP or placement score. Solid algebra and comfort with functions, exponents and trig values matter most.

### What comes after MATH-UA 121?

MATH-UA 122 Calculus II, which moves on to integration techniques, polar and parametric equations and infinite series. After that comes MATH-UA 123 Calculus III, which is multivariable calculus.

### Is MATH-UA 121 hard?

It is a full Calculus 1 course in one semester, so the pace is steady and new rules arrive every week. Most students who struggle are tripped up by algebra rather than calculus, so short daily practice and fixing algebra slips early make a real difference.

### How can I prepare for MATH-UA 121?

Review function notation, exponent and log rules and the key trig values, then start on limits. The free functions warm-up and limits units on PracticeCalculus cover exactly that ground.