UCLA MATH 31A: Differential and Integral Calculus

MATH 31A is UCLA's first calculus course — limits, derivatives, optimization, and the basics of integration — required across engineering, physical science, economics, and life-science-adjacent tracks. Most STEM freshmen take it (or skip it via AP credit) in fall quarter.

Fennie is independent and not affiliated with UCLA. This is an unofficial study guide.

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What makes it hard

Ten weeks means roughly a topic per week with no recovery time, and the curved exams are more conceptual than high school calculus. The common trap is treating it as an AP repeat: the early material is familiar, students coast, and then the midterm difficulty resets expectations the hard way.

What you'll cover

• Limits and continuity
• Differentiation rules and implicit differentiation
• Linear approximation and related rates
• Optimization and curve sketching
• The definite integral and Fundamental Theorem of Calculus

The MATH 31A study guide

How to study for UCLA MATH 31A, step by step.

1
Refuse the AP-repeat mindset
MATH 31A's early material feels familiar, which is exactly the trap — students coast, then the first midterm resets expectations. Work problems seriously from week one even when the topic looks like review.
2
Get your instructor's past exams early
Exam style varies by professor at UCLA, and past exams from your specific instructor are the best predictor available. Collect them in the first two weeks, not the night before the midterm.
3
Overweight the word problems
Optimization and related rates separate the grade bands far more than mechanical differentiation. Practice translating words into equations as its own skill, several problems at a time.
4
Self-test weekly — the first midterm is week four
On a ten-week quarter the first exam arrives fast. A short timed quiz on each week's material catches drift while there's still time to fix it.
5
Let Fennie compress-proof the quarter
Upload the MATH 31A syllabus and Fennie's Daily Plans build a day-by-day problem schedule paced to both midterms and the final, with weekly quizzes generated from your actual course materials. Free to start.
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How Fennie helps with MATH 31A

Daily Plans compress-proof your quarter: upload the MATH 31A syllabus and get a day-by-day problem schedule paced to both midterms and the final. Chat through any solution step that feels like magic, and run generated quizzes weekly so the first midterm isn't the first time you're tested.

FAQ

Is MATH 31A hard at UCLA?

The content overlaps AP Calculus AB, but the exams are harder and curved, and the quarter pace is unforgiving. Students who keep doing problems weekly find it very manageable; students who coast on AP familiarity often get caught at the first midterm.

Should I skip MATH 31A with AP credit?

A strong AP Calculus score lets many students start in 31B or beyond. If your AB foundation is solid, skipping is reasonable — but 31B's integration and series material punishes weak fundamentals, so be honest about your readiness.

How do I study for MATH 31A midterms?

Past exams from your instructor are the best resource — exam style varies by professor. Work them timed, and prioritize optimization and related-rates word problems, which separate grades more than mechanical differentiation.

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More UCLA courses

MATH 31B — Integration and Infinite Series

MATH 31B covers integration techniques, applications, improper integrals, and infinite sequences and series including Taylor series. It's the second course in UCLA's main calculus sequence and a prerequisite for the multivariable courses that follow.

MATH 32A — Calculus of Several Variables

MATH 32A introduces multivariable calculus: vectors, vector-valued functions, partial derivatives, gradients, and optimization in several variables. It's required for engineering, physics, math, and CS-adjacent tracks, typically taken in the first year after the 31 sequence.

MATH 32B — Calculus of Several Variables

MATH 32B completes UCLA's multivariable calculus sequence: multiple integrals, change of variables, line and surface integrals, and the vector-calculus theorems of Green, Stokes, and Gauss. It follows 32A and is required across engineering and the physical sciences.

MATH 33A — Linear Algebra and Applications

MATH 33A is UCLA's linear algebra course: systems of equations, matrices, vector spaces, linear transformations, orthogonality, least squares, eigenvalues, and eigenvectors. It's required across engineering, math, and CS, and the linear algebra it teaches underpins machine learning and upper-division applied coursework.