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

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.

The UC Davis calculus sequence

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 when you finish.

Example problems at the MAT 21A level

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

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:

How to do well in UC Davis MAT 21A

• Build the derivative rules into reflexes. Ten weeks leaves little slack. Daily practice problems on the product rule, 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 guide covers the slips that cost the most points.
• Study from one page of formulas. The calculus formula sheet 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.