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

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 between 21B and 21D

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 for integration review, or ahead to the MAT 21D guide.

Example problems at the MAT 21C level

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

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

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 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.
• Do not let the vector half sneak up on you. Start the Calculus 3 vectors unit a week early so 3D geometry feels familiar.
• Practice under time pressure with practice exams, and review the 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.