# UCSD MATH 20A Calculus: Topics, Study Guide and Practice

> A study guide for UCSD MATH 20A: limits, derivatives, tangent lines, optimization and a first look at the integral, mapped to free practice problems.

Source: https://practicecalculus.com/courses/uc-san-diego-math-20a
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 20A
- **Units:** 4
- **Prerequisites:** A qualifying Math Placement Exam score, AP Precalculus 5, AP Calculus AB 3 (or the AB subscore on BC), MATH 4C, or MATH 10A
- **Covers:** Functions and graphs, limits, continuity, the derivative, tangent lines, optimization, algebraic, exponential, logarithmic and trig functions, and an introduction to the integral
- **Credit note:** Two units if taken after MATH 10A; no credit after MATH 10B or 10C
- **Next course:** MATH 20B (Calculus for Science and Engineering)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH 20A is the first quarter of the UC San Diego calculus sequence for science and engineering students, worth 4 units.
- It covers limits, continuity, derivatives and their applications, and ends with an introduction to the integral.
- MATH 10A is the separate Calculus I course for students outside science and engineering, and it limits credit for 20A.
- After MATH 20A comes MATH 20B, which is mostly integral calculus and series.

**MATH 20A** is the first course in UC San Diego's calculus sequence for science and engineering. As the title says, it is the version of Calculus I built for science and engineering students, who then continue to MATH 20B and MATH 20C. This guide maps every MATH 20A topic to free practice, so you can learn each idea by solving problems and getting instant feedback.

## What UCSD MATH 20A covers

The UC San Diego course catalog describes MATH 20A as the foundations of one-variable differential and integral calculus. The listed topics are functions and their graphs, continuity, limits, the derivative and the tangent line, and optimization problems, worked with algebraic, exponential, logarithmic and trigonometric functions. The course closes with an introduction to the integral.

That lines up closely with a standard **Calculus 1** course. The main difference is pace: UCSD uses ten-week quarters, so most of the integral material (Riemann sums, the Fundamental Theorem and substitution) is finished in MATH 20B rather than here.

## MATH 20A topic map

Each row links to a lesson you can practice and to a reference page or guide that explains the idea.

| MATH 20A topic | Practice | Read |
| --- | --- | --- |
| Functions and graphs | [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), [how to evaluate limits](/blog/how-to-evaluate-limits) |
| 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) |
| Tangent lines | [Tangent lines](/learn/calculus-1/the-derivative/tangent-lines) | [Linear approximation](/blog/linear-approximation) |
| Differentiation rules | [The power rule](/learn/calculus-1/derivative-rules/power-rule) | [Power rule](/rules/power-rule), [product rule](/rules/product-rule), [chain rule](/rules/chain-rule) |
| Exponential, log and trig derivatives | [Sine, cosine and tangent](/learn/calculus-1/derivative-rules/trig-derivatives) | [Derivatives of trig functions](/blog/derivatives-of-trig-functions), [exponential and log derivatives](/blog/derivatives-of-exponential-and-logarithmic-functions) |
| Optimization | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems), [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Graph shape from derivatives | [Increasing and decreasing](/learn/calculus-1/applications-of-derivatives/increasing-and-decreasing) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Introduction to the integral | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives](/blog/antiderivatives), [Riemann sums](/blog/riemann-sums) |

## MATH 20A in the UCSD calculus sequence

| Course | Focus | Units |
| --- | --- | --- |
| MATH 10A | Calculus I for students outside science and engineering | 4 |
| MATH 20A | Limits, derivatives and applications, intro to the integral | 4 |
| [MATH 20B](/courses/uc-san-diego-math-20b) | Integration methods, series, polar coordinates | 4 |
| [MATH 20C](/courses/uc-san-diego-math-20c) | Vectors, partial derivatives, double integrals | 4 |

MATH 20A is the entry point for the science and engineering track. MATH 10A also opens the door to 20B, but if you take 20A after 10A you only earn two units for it, so it pays to pick the right course the first time. Continue with the [MATH 20B guide](/courses/uc-san-diego-math-20b) when you finish.

## Example problems at the MATH 20A level

**1. A limit that needs factoring.** Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.

Direct substitution gives $\frac{0}{0}$, so factor first:

$$\lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3} (x + 3) = 6$$

**2. A tangent line.** Find the tangent line to $f(x) = e^x + x^2$ at $x = 0$.

Here $f(0) = 1$ and $f'(x) = e^x + 2x$, so $f'(0) = 1$. The tangent line is

$$y = 1 + 1(x - 0) = x + 1$$

**3. An optimization problem.** A rectangle has perimeter 40. What dimensions give the largest area?

With sides $x$ and $20 - x$, the area is $A(x) = x(20 - x) = 20x - x^2$. Then $A'(x) = 20 - 2x = 0$ gives $x = 10$, and $A''(x) = -2 < 0$ confirms a maximum. The best rectangle is a $10 \times 10$ square with area $100$.

## How to do well in UCSD MATH 20A

- **Keep pace with the quarter.** Ten weeks goes fast. Short daily sessions on [practice problems](/practice) beat a long weekend catch-up.
- **Master the derivative rules early.** The [chain rule](/rules/chain-rule) shows up in almost every problem after week four, so drill it until it is automatic.
- **Watch for algebra slips.** Factoring, signs and exponent rules cost more points than calculus ideas. Read our list of [common calculus mistakes](/blog/common-calculus-mistakes).
- **Keep a formula sheet handy** while you study. The [calculus formula sheet](/formulas) puts every derivative on one page.
- **Prepare for midterms deliberately.** Our guide on [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam) walks through a one-week plan.

> **Tip:** optimization problems feel hard because of the setup, not the calculus. Always write the quantity to maximize as a function of one variable before you differentiate.

## Frequently asked questions

### What is MATH 20A at UCSD?

MATH 20A is UC San Diego's first-quarter calculus course for science and engineering majors. It builds the foundations of one-variable calculus: functions, limits, continuity, derivatives, tangent lines and optimization, followed by a short introduction to the integral.

### What comes after MATH 20A?

The next course is MATH 20B, which covers integration methods, infinite series, polar coordinates and complex exponentials. After that, MATH 20C covers multivariable calculus.

### Is MATH 20A hard?

UCSD runs on ten-week quarters, so the course moves quickly from limits to applications of derivatives. Students with solid algebra and trigonometry who practice a little every day usually keep up much more easily than those who cram before midterms.

### What is the difference between MATH 10A and MATH 20A?

Both are a first course in calculus. MATH 20A is the version for science and engineering students and leads into 20B and 20C. MATH 10A is a Calculus I course aimed at other majors. If you take 20A after 10A you receive only two units for 20A.

### How should I prepare for MATH 20A?

Review function notation, exponent and log rules, and key trig values, then start on limits. The Calculus 1 warm-up and limits units on PracticeCalculus cover exactly those skills with instant feedback.