# UW MATH 126 Calculus III: Topics, Study Guide and Practice

> A study guide for University of Washington MATH 126: 3D vectors, vector functions, partial derivatives and Taylor series, with free practice.

Source: https://practicecalculus.com/courses/university-of-washington-math-126
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 126
- **Credits:** 5 (one quarter)
- **Offered:** Autumn, winter, spring and summer
- **Prerequisites:** Minimum grade of 2.0 in MATH 125 or MATH 135, or a score of 4 on the AP Calculus BC exam
- **Covers:** Vector geometry in three dimensions, vector functions, multivariate differential calculus, Taylor polynomials and Taylor series
- **Next course:** MATH 224 (Advanced Multivariable Calculus)
- **Practice here:** [Calculus 3](/learn/calculus-3) plus [Taylor series in Calculus 2](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials)

## Key takeaways

- MATH 126 is the third quarter of the UW calculus sequence, worth 5 credits and offered every quarter.
- It introduces vector geometry in 3D, vector functions and multivariable differential calculus.
- It also covers Taylor polynomials and Taylor series, which many schools teach in Calculus 2.
- Multivariable integration and vector calculus come next, in MATH 224.

**UW MATH 126**, Calculus with Analytic Geometry III, is the third quarter of calculus at the University of Washington. It is a 5-credit course offered every quarter that mixes the start of multivariable calculus with Taylor series. This guide maps every MATH 126 topic to practice with instant feedback.

## What UW MATH 126 covers

The UW course catalog lists four themes: an introduction to vector geometry in three dimensions, vector functions, multivariate differential calculus, and Taylor polynomials and Taylor series. You need a 2.0 in MATH 125 or MATH 135, or a 4 on the AP Calculus BC exam.

This is an unusual mix compared with a semester-system school. The vector and partial derivative material is roughly the first half of a standard **Calculus 3** course, while Taylor polynomials and series usually sit at the end of **Calculus 2**. Multiple integrals and vector calculus come later, in MATH 224.

## MATH 126 topic map

| MATH 126 topic | Practice | Read |
| --- | --- | --- |
| Points and distance in 3D | [Points and distance in space](/learn/calculus-3/vectors-and-space/points-and-distance) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vectors, dot and cross products | [The dot product](/learn/calculus-3/vectors-and-space/dot-product), [the cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Dot and cross product](/blog/vectors-dot-and-cross-product) |
| Vector functions and motion | [Curves and motion](/learn/calculus-3/vectors-and-space/curves-and-motion) | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) |
| Functions of several variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives), [Clairaut's theorem](/theorems/clairauts-theorem) |
| Tangent planes and linear approximation | [Tangent planes](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Gradient and directional derivatives | [Gradient and directional derivatives](/learn/calculus-3/partial-derivatives/gradient-and-directional-derivatives) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Critical points in two variables | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Taylor polynomials | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor's theorem](/theorems/taylors-theorem) |
| Error estimates | [Estimating with Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) | [Taylor series](/blog/taylor-series) |
| Taylor series | [Common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [power series](/blog/power-series) |

## MATH 126 vs the courses around it

| Course | Credits | Covers |
| --- | --- | --- |
| MATH 125 | 5 | Integral calculus (prerequisite) |
| MATH 126 | 5 | 3D vectors, vector functions, partial derivatives, Taylor series |
| MATH 136 | 5 | Honors alternative, overlaps with MATH 126 |
| MATH 224 | 4 | Multiple integrals, line and surface integrals, Green, Stokes, Divergence |

Review the previous quarter in the [MATH 125 guide](/courses/university-of-washington-math-125), and see where multivariable calculus finishes in the [MATH 224 guide](/courses/university-of-washington-math-224).

## Example problems at the MATH 126 level

**1. Angle between vectors.** For $\mathbf{u} = \langle 1, 1, 0 \rangle$ and $\mathbf{v} = \langle 1, 0, 1 \rangle$, $\mathbf{u} \cdot \mathbf{v} = 1$ and $|\mathbf{u}| = |\mathbf{v}| = \sqrt{2}$, so

$$\cos\theta = \frac{1}{\sqrt{2}\cdot\sqrt{2}} = \frac{1}{2}, \qquad \theta = \frac{\pi}{3}$$

**2. A Taylor polynomial with error.** The degree 2 Maclaurin polynomial of $\cos x$ is $T_2(x) = 1 - \frac{x^2}{2}$, so $\cos(0.2) \approx 1 - 0.02 = 0.98$. The next nonzero term is $\frac{x^4}{24}$, about $0.000067$ at $x = 0.2$, so the error is tiny.

**3. A second partial derivative.** For $f(x, y) = x^3 y^2$, $f_x = 3x^2 y^2$ and

$$f_{xy} = 6x^2 y$$

which equals $f_{yx}$, as Clairaut's theorem predicts.

## How to do well in UW MATH 126

- **Keep two threads going.** Practice vectors and partial derivatives each week even while the course is on Taylor series, and vice versa.
- **Memorize the core Maclaurin series.** $e^x$, $\sin x$, $\cos x$ and $\frac{1}{1-x}$ generate most others. The [formula sheet](/formulas) lists them.
- **Visualize in 3D.** Sketch planes and level curves by hand. The [planes in space lesson](/learn/calculus-3/vectors-and-space/planes-in-space) builds that habit.
- **Test yourself** with timed [practice exams](/practice), and review [common calculus mistakes](/blog/common-calculus-mistakes) before the final.

> **Tip:** a quick check for a partial derivative: hold the other variable at a number, like $y = 1$, and differentiate the one-variable function you get.

## Frequently asked questions

### What is MATH 126 at the University of Washington?

MATH 126, Calculus with Analytic Geometry III, is the third quarter of UW's calculus sequence. It covers vector geometry in three dimensions, vector functions, multivariable differential calculus, and Taylor polynomials and Taylor series.

### What are the prerequisites for MATH 126?

A minimum grade of 2.0 in MATH 125 or MATH 135, or a score of 4 on the AP Calculus BC exam.

### Does MATH 126 include multiple integrals?

The catalog description lists multivariate differential calculus, not integration. Double and triple integrals, line and surface integrals, and the theorems of Green and Stokes are in MATH 224, the fourth quarter.

### Is MATH 126 hard?

It combines two different kinds of material in one quarter: 3D geometry and partial derivatives, plus Taylor series. Students who keep both threads going with regular practice tend to do best.

### What is the difference between MATH 126 and MATH 136?

MATH 136 is the third quarter of the accelerated honors sequence (MATH 134, 135, 136), which covers the material of MATH 124 through 126 plus MATH 207 and 208. The catalog lists it as overlapping with MATH 126.