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.
UCLA Updated 3 min read
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
On this page
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.
Math 31A topic map
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 or the Math 32A guide, and later Math 32B.
Example problems at the Math 31A level
1. A limit at infinity.
2. An inverse trig derivative with the chain rule.
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 and mixed 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 lists them all.
- Prepare for midterms with timed practice exams and the common calculus mistakes list.
Practice it now
What is a limit?
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