# UCLA Math 31A Differential and Integral Calculus: Study Guide

> A study guide for UCLA Math 31A: limits, derivative rules, related rates, optimization, L'Hôpital's rule and antiderivatives, each mapped to free practice.

Source: https://practicecalculus.com/courses/ucla-math-31a
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 31A
- **Units:** 4 (three hours of lecture and one of discussion per week)
- **Requisite:** Mathematics Diagnostic Test, or Math 1 with a grade of C- or better
- **Recommended preparation:** At least three and a half years of high school math, including coordinate geometry and trigonometry
- **Covers:** Limits and continuity, the derivative and its rules, exponential, log and inverse trig derivatives, related rates, linear approximation, the Mean Value Theorem, curve shape, optimization, L'Hôpital's rule and antiderivatives
- **Next course:** MATH 31B (Integration and Infinite Series) or MATH 32A (Calculus of Several Variables)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- Math 31A is UCLA's first calculus course: differential calculus and its applications, plus an introduction to integration.
- It is a 4-unit course that requires the Mathematics Diagnostic Test or Math 1 with a C- or better.
- In the department outline used from Summer 2026, it ends with antiderivatives and initial value problems, and definite integrals open Math 31B.
- Passing 31A with a C- or better unlocks both Math 31B and Math 32A.

**UCLA Math 31A** is the first course in UCLA's calculus sequence. In one ten-week quarter it covers limits, the derivative and its applications, and ends with an introduction to integration. This guide maps every Math 31A topic to free practice so you can learn each idea by solving problems with instant feedback.

## What UCLA Math 31A covers

The UCLA General Catalog summarizes Math 31A as differential calculus and applications, with an introduction to integration. The Mathematics Department's course outline, which uses an early transcendentals text from Summer 2026 on, fills that in with four units. Limits come first: limit laws, continuity and the Intermediate Value Theorem, indeterminate forms, trig limits and limits at infinity. Differentiation follows, from the definition through the product, quotient and chain rules, implicit differentiation, inverse functions and inverse trig, and exponential and logarithmic derivatives. Applications include related rates, linear approximation, extreme values, the Mean Value Theorem, concavity and the shape of a graph, optimization and L'Hôpital's rule. The course closes with indefinite integrals and initial value problems.

That covers a standard **Calculus 1** course up to, but not including, the definite integral, which the current outline places at the start of [Math 31B](/courses/ucla-math-31b).

## Math 31A topic map

| Math 31A topic | Practice | Read |
| --- | --- | --- |
| Limits and limit laws | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits), [limit laws](/rules/limit-laws) |
| Continuity and the IVT | [Continuity](/learn/calculus-1/limits/continuity) | [Intermediate Value Theorem](/theorems/intermediate-value-theorem) |
| Indeterminate forms and trig limits | [Factor and cancel](/learn/calculus-1/limits/factoring-zero-over-zero) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Limits at infinity | [Limits at infinity](/learn/calculus-1/limits/limits-at-infinity) | [Limits at infinity guide](/blog/limits-at-infinity) |
| Definition of the derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Product, quotient and chain rules | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Product and quotient rules](/blog/product-and-quotient-rules), [chain rule](/blog/chain-rule) |
| Implicit differentiation and inverse trig | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Arctangent](/functions/arctangent), [inverse function rule](/rules/inverse-function-rule) |
| Exponential and log derivatives | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Logarithmic differentiation](/rules/logarithmic-differentiation) |
| Related rates | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates guide](/blog/related-rates) |
| Linear approximation and the MVT | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation), [Mean Value Theorem](/theorems/mean-value-theorem) |
| Extreme values and curve shape | [Absolute extrema](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems) |
| L'Hôpital's rule | [L'Hôpital's rule](/learn/calculus-1/applications-of-derivatives/lhopitals-rule) | [L'Hôpital's rule guide](/blog/lhopitals-rule) |
| Indefinite integrals and initial value problems | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives](/blog/antiderivatives) |

## The UCLA calculus sequence

| Course | Units | Requisite | Content |
| --- | --- | --- | --- |
| Math 31A | 4 | Diagnostic test or Math 1 | Differential calculus, intro to integration |
| Math 31B | 4 | 31A | Integration techniques, sequences and series |
| Math 32A | 4 | 31A | Differential calculus of several variables |
| Math 32B | 4 | 31B and 32A | Multiple, line and surface integrals |
| Math 3A | 4 | Diagnostic test or Math 1 | Calculus for life sciences students |

After 31A, continue with the [Math 31B guide](/courses/ucla-math-31b) or the [Math 32A guide](/courses/ucla-math-32a), and later [Math 32B](/courses/ucla-math-32b).

## Example problems at the Math 31A level

**1. A limit at infinity.**

$$\lim_{x\to\infty}\frac{5x^3 - x}{2x^3 + 7} = \frac{5}{2}$$

**2. An inverse trig derivative with the chain rule.**

$$\frac{d}{dx}\arctan(3x) = \frac{3}{1 + 9x^2}$$

**3. An initial value problem.** Solve $f'(x) = 6x^2 - 4$ with $f(1) = 3$. Antidifferentiate: $f(x) = 2x^3 - 4x + C$. Then $f(1) = 2 - 4 + C = 3$ gives $C = 5$, so $f(x) = 2x^3 - 4x + 5$.

## How to do well in UCLA Math 31A

- **Practice after every lecture.** A ten-week quarter leaves little room to catch up. Short daily sessions with our [limits practice](/learn/calculus-1/limits/limits-practice) and [mixed rules practice](/learn/calculus-1/derivative-rules/derivative-rules-practice) keep you current.
- **Earn at least a C-.** Math 31B and 32A both require 31A with a C- or better, so the grade matters for the rest of the sequence.
- **Know the inverse trig and log derivatives cold,** since the current outline moves them into 31A. The [calculus formula sheet](/formulas) lists them all.
- **Prepare for midterms** with timed [practice exams](/practice) and the [common calculus mistakes](/blog/common-calculus-mistakes) list.

## Frequently asked questions

### What is UCLA Math 31A?

Math 31A, Differential and Integral Calculus, is UCLA's first quarter of calculus. It covers limits and continuity, derivatives and their rules, applications such as related rates and optimization, and an introduction to integration through antiderivatives.

### What comes after Math 31A?

Math 31B, Integration and Infinite Series, and Math 32A, Calculus of Several Variables. Both require Math 31A with a grade of C- or better.

### How do I get into Math 31A?

The catalog requires the Mathematics Diagnostic Test or Math 1 with a C- or better, and recommends at least three and a half years of high school math including coordinate geometry and trigonometry.

### Is Math 31A hard?

It moves quickly because UCLA uses ten-week quarters. Students with solid algebra and trig who practice after every lecture usually find the material manageable.

### Math 31A vs Math 3A: which should I take?

Math 3A, Calculus for Life Sciences Students, focuses on modeling and optimization in biology, and it is not open for credit to students with credit in another calculus sequence. Check which sequence your major requires.