# UC Davis MAT 21A Calculus: Topics, Study Guide and Practice

> A study guide for UC Davis MAT 21A: limits, continuity, derivatives, graphing, extrema, differentials and L'Hôpital's rule, mapped to free practice.

Source: https://practicecalculus.com/courses/uc-davis-mat-21a
Updated: 2026-11-03

## At a glance

- **Course code:** MAT 021A (honors version: MAT 021AH)
- **Units:** 4
- **Prerequisites:** Two years of high school algebra, plane geometry, plane trigonometry and analytic geometry, plus the Mathematics Placement Requirement
- **Covers:** Functions, limits and continuity, the derivative, differentiating algebraic and transcendental functions, motion, growth, graphing, extrema, differentials and L'Hôpital's rule
- **Credit note:** Only 2 units if MAT 016A or 017A was completed
- **Next course:** MAT 021B (Calculus)
- **Practice here:** [Calculus 1](/learn/calculus-1) through applications of derivatives

## Key takeaways

- MAT 21A is the first quarter of the four-course UC Davis calculus sequence (21A, 21B, 21C, 21D), worth 4 units.
- It is entirely differential calculus: limits, derivatives and their applications, ending with differentials and L'Hôpital's rule.
- Integration waits until MAT 21B, so 21A covers roughly the first two thirds of a typical Calculus 1 course.
- MAT 21AH is a more intensive honors version for students who place well above the 21A minimum.

**MAT 21A** is the first course in the main UC Davis calculus sequence, the starting point for students who need a full calculus background. It is the start of a four-quarter path (21A, 21B, 21C, 21D) that ends with vector analysis. This guide maps every MAT 21A topic to free practice so you can learn each idea by solving problems with instant feedback.

## What UC Davis MAT 21A covers

According to the UC Davis General Catalog, MAT 21A covers functions, limits and continuity, then the slope of a curve and the derivative. Students differentiate algebraic and transcendental functions (exponentials, logarithms and trig functions) and apply derivatives to motion, natural growth, graphing and finding extrema. The course ends with differentials and L'Hôpital's rule.

Compared with a standard semester-long **Calculus 1** course, MAT 21A is the differential half. There is no integration in the 21A description: antiderivatives, the definite integral and the Fundamental Theorem all appear in MAT 21B.

## MAT 21A topic map

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

| MAT 21A topic | Practice | Read |
| --- | --- | --- |
| Functions | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Polynomial functions](/functions/polynomial-functions), [rational functions](/functions/rational-functions) |
| Limits | [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 | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity) |
| Slope and the derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiating algebraic functions | [The power rule](/learn/calculus-1/derivative-rules/power-rule) | [Power rule](/blog/power-rule), [product and quotient rules](/blog/product-and-quotient-rules) |
| Differentiating transcendental functions | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Exponential and log derivatives](/blog/derivatives-of-exponential-and-logarithmic-functions), [trig derivatives](/blog/derivatives-of-trig-functions) |
| Chain rule | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule) |
| Motion and natural growth | [Instantaneous rate of change](/learn/calculus-1/the-derivative/instantaneous-rate-of-change) | [Natural exponential function](/functions/natural-exponential-function) |
| Graphing | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Extrema | [Absolute extrema on an interval](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Optimization problems](/blog/optimization-problems), [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Differentials | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) |
| L'Hôpital's rule | [L'Hôpital's rule](/learn/calculus-1/applications-of-derivatives/lhopitals-rule) | [L'Hôpital's rule](/blog/lhopitals-rule) |

## The UC Davis calculus sequence

| Course | Focus | Prerequisite |
| --- | --- | --- |
| MAT 21A | Limits, derivatives and their applications | Placement requirement |
| [MAT 21B](/courses/uc-davis-mat-21b) | Integrals, techniques and applications | 21A or 21AH with C- or better |
| [MAT 21C](/courses/uc-davis-mat-21c) | Series, vectors, partial derivatives | 21B or 21BH with C- or better |
| [MAT 21D](/courses/uc-davis-mat-21d) | Multiple, line and surface integrals | 21C or 21CH with C- or better |

Each course has an honors version (21AH, 21BH, 21CH) described as a more intensive treatment of the same material. Students in the life sciences often take the MAT 17 series instead; MAT 17A with a B or better also leads into 21B. Continue with the [MAT 21B guide](/courses/uc-davis-mat-21b) when you finish.

## Example problems at the MAT 21A level

**1. A quotient rule derivative.** Differentiate $f(x) = \frac{x^2}{x + 1}$.

$$f'(x) = \frac{2x(x + 1) - x^2 \cdot 1}{(x + 1)^2} = \frac{x^2 + 2x}{(x + 1)^2}$$

**2. Natural growth.** A population follows $P(t) = 500e^{0.2t}$. How fast is it growing at $t = 0$?

$P'(t) = 100e^{0.2t}$, so $P'(0) = 100$ individuals per unit of time.

**3. L'Hôpital's rule.** Evaluate $\lim_{x \to 0} \frac{e^x - 1}{x}$.

Substitution gives $\frac{0}{0}$, so differentiate the top and bottom:

$$\lim_{x \to 0} \frac{e^x - 1}{x} = \lim_{x \to 0} \frac{e^x}{1} = 1$$

## How to do well in UC Davis MAT 21A

- **Build the derivative rules into reflexes.** Ten weeks leaves little slack. Daily [practice problems](/practice) on the [product rule](/rules/product-rule), [quotient rule](/rules/quotient-rule) and chain rule make exams much calmer.
- **Do not skip the limit definition.** Understanding why the derivative is a limit makes graphing and differentials far easier to reason about.
- **Fix algebra and sign errors.** Our [common calculus mistakes](/blog/common-calculus-mistakes) guide covers the slips that cost the most points.
- **Study from one page of formulas.** The [calculus formula sheet](/formulas) lists every derivative rule you need for 21A.

> **Tip:** before using L'Hôpital's rule, always check that the limit really has the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. Applying it to anything else gives wrong answers.

## Frequently asked questions

### What is MAT 21A at UC Davis?

MAT 21A is the first course in UC Davis's main calculus sequence. It covers functions, limits and continuity, the derivative, differentiation of algebraic and transcendental functions, applications to motion, growth, graphing and extrema, differentials and L'Hôpital's rule.

### What comes after MAT 21A?

MAT 21B, which covers definite integrals, the Fundamental Theorem of Calculus, integration techniques and applications such as area, volume and arc length.

### Is MAT 21A hard?

It is a fast ten-week quarter, but the content is focused on derivatives only. Students who have solid algebra and trigonometry and who practice derivative rules daily tend to find it manageable.

### What is the difference between MAT 17A and MAT 21A?

MAT 17A, Calculus for Biology and Medicine, introduces differential calculus through biology and medicine applications. MAT 21A is the general calculus course used by most science and engineering majors. If you completed 17A, you receive only 2 units for 21A.

### How should I prepare for MAT 21A?

Review functions, exponent and log rules and the key trig values, then start limits. The Calculus 1 warm-up unit on PracticeCalculus covers exactly those skills.