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

> A study guide for UC Davis MAT 21C: sequences, series, Taylor expansions, vectors, partial derivatives and multivariable max and min problems.

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

## At a glance

- **Course code:** MAT 021C (honors version: MAT 021CH)
- **Units:** 4
- **Prerequisites:** MAT 021B or 021BH with a C- or better, MAT 016C, 017C or 019C with a C- or better, or MAT 017B with a B or better
- **Covers:** Sequences, series, convergence tests, Taylor expansions, vector algebra and calculus, scalar and vector fields, partial derivatives, total differentials, multivariable max and min problems
- **Next course:** MAT 021D (Vector Analysis)
- **Practice here:** [Calculus 2](/learn/calculus-2) series units plus the first units of [Calculus 3](/learn/calculus-3)

## Key takeaways

- MAT 21C is the third quarter of the UC Davis calculus sequence and bridges single-variable and multivariable calculus.
- The first part covers sequences, series, convergence tests and Taylor expansions.
- The second part covers vectors, scalar and vector fields, partial derivatives, total differentials and optimization in several variables.
- It leads to MAT 21D, Vector Analysis, which covers multiple, line and surface integrals.

**MAT 21C** is the third quarter of the UC Davis calculus sequence, and the one where single-variable calculus hands off to several variables. The first half finishes the theory of infinite series; the second half introduces vectors, fields and partial derivatives. This guide maps every MAT 21C topic to free practice so you can keep both halves sharp.

## What UC Davis MAT 21C covers

According to the UC Davis General Catalog, MAT 21C covers sequences, series, tests for convergence and Taylor expansions. It then moves to vector algebra and vector calculus, scalar and vector fields, partial derivatives and total differentials, and maximum and minimum problems in several variables, with physical applications.

Compared with a semester sequence, that is the series portion of **Calculus 2** followed by the first part of **Calculus 3**. Multiple integrals and the theorems of vector calculus are saved for MAT 21D.

## MAT 21C topic map

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

| MAT 21C topic | Practice | Read |
| --- | --- | --- |
| Sequences | [Sequences and their limits](/learn/calculus-2/sequences-and-series/sequences) | [Limits at infinity](/blog/limits-at-infinity) |
| Series and partial sums | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Geometric series](/blog/geometric-series), [geometric series formula](/rules/geometric-series-formula) |
| Convergence tests | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests), [ratio test](/rules/ratio-test), [nth-term test](/rules/nth-term-test) |
| Alternating series | [Alternating series](/learn/calculus-2/sequences-and-series/alternating-series) | [Alternating series test](/rules/alternating-series-test) |
| Power series | [Power series and radius of convergence](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series) |
| Taylor expansions | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| Vector algebra | [The dot product](/learn/calculus-3/vectors-and-space/dot-product) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vector calculus of curves | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Chain rule](/rules/chain-rule) |
| Scalar fields and partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives) |
| Vector fields | [Vector fields](/learn/calculus-3/vector-calculus/vector-fields) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Total differentials | [Tangent planes and linear approximation](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Max and min in several variables | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |

## MAT 21C between 21B and 21D

| Course | Single variable or several? | Key ideas |
| --- | --- | --- |
| [MAT 21B](/courses/uc-davis-mat-21b) | Single | Integration and its applications |
| MAT 21C | Both | Series, vectors, partial derivatives |
| [MAT 21D](/courses/uc-davis-mat-21d) | Several | Multiple, line and surface integrals |

The honors version, MAT 21CH, requires a B or better in 21B or 21BH. Students coming from the MAT 17 series can also enter 21C, as the prerequisites above show. Look back at the [MAT 21B guide](/courses/uc-davis-mat-21b) for integration review, or ahead to the [MAT 21D guide](/courses/uc-davis-mat-21d).

## Example problems at the MAT 21C level

**1. The ratio test.** Does $\sum_{n=1}^{\infty} \frac{n}{2^n}$ converge?

$$\lim_{n \to \infty} \frac{(n + 1)/2^{n+1}}{n/2^n} = \lim_{n \to \infty} \frac{n + 1}{2n} = \frac{1}{2} < 1$$

The limit is less than 1, so the series converges.

**2. A Taylor polynomial.** The degree 2 Maclaurin polynomial of $f(x) = e^x$ is

$$P_2(x) = 1 + x + \frac{x^2}{2}$$

so $e^{0.1} \approx 1 + 0.1 + 0.005 = 1.105$.

**3. A critical point in two variables.** For $f(x, y) = x^2 + y^2 - 4x + 6y$, set $f_x = 2x - 4 = 0$ and $f_y = 2y + 6 = 0$ to get $(2, -3)$. Since $f_{xx} = f_{yy} = 2$ and $f_{xy} = 0$, the discriminant is $4 > 0$ with $f_{xx} > 0$, so $(2, -3)$ is a local minimum with value $f(2, -3) = -13$.

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

- **Learn a test-choosing routine.** Start with the nth-term test, then look for a geometric or $p$-series, then try ratio or comparison. The [choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) lesson drills that order.
- **Memorize a few Maclaurin series.** Knowing $e^x$, $\sin x$, $\cos x$ and $\frac{1}{1 - x}$ by heart turns many Taylor problems into substitutions. All are on the [calculus formula sheet](/formulas).
- **Do not let the vector half sneak up on you.** Start the [Calculus 3 vectors unit](/learn/calculus-3/vectors-and-space/vectors-and-length) a week early so 3D geometry feels familiar.
- **Practice under time pressure** with [practice exams](/practice), and review the [common calculus mistakes](/blog/common-calculus-mistakes) before each midterm.

> **Tip:** a series whose terms do not go to zero always diverges, but terms going to zero proves nothing. The harmonic series $\sum \frac{1}{n}$ is the classic reminder.

## Frequently asked questions

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

MAT 21C is the third course in UC Davis's main calculus sequence. It covers sequences and series with convergence tests and Taylor expansions, then vector algebra and calculus, scalar and vector fields, partial derivatives, total differentials and maximum and minimum problems in several variables.

### Is MAT 21C hard?

It asks you to switch gears mid-quarter, from series to three-dimensional geometry, so it can feel like two courses in one. Keeping up with both halves through steady practice is the best way to avoid falling behind.

### What comes after MAT 21C?

MAT 21D, Vector Analysis, which covers multiple integrals in different coordinate systems, line and surface integrals, and the theorems of Green, Stokes and Gauss.

### Does MAT 21C cover double integrals?

No. Multiple integrals are listed under MAT 21D. MAT 21C stops at partial derivatives and optimization in several variables.

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

Review limits at infinity and improper integrals from 21A and 21B, since series tests rely on them. The geometric series and the nth-term test are good first lessons.