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

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 vs the courses around it

Review the previous quarter in the MATH 125 guide, and see where multivariable calculus finishes in the MATH 224 guide.

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

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

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 lists them.
• Visualize in 3D. Sketch planes and level curves by hand. The planes in space lesson builds that habit.
• Test yourself with timed practice exams, and review 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.