UIUC MATH 221 Calculus I: Topics, Study Guide and Practice

UIUC MATH 221 is Calculus I at the University of Illinois Urbana-Champaign, the first course in the MATH 221, 231, 241 calculus sequence. It is aimed at students who already have some calculus background, for example from a year of high school calculus. This guide maps every MATH 221 topic to free practice, so you can learn each idea by solving problems with instant feedback.

What UIUC MATH 221 covers

The Illinois course catalog lists MATH 221 as a 4-hour course covering analytic geometry, differentiation and integration techniques, and applications such as curve sketching. It continues into antidifferentiation, the Riemann integral and the Fundamental Theorem of Calculus, and works with exponential and trigonometric functions throughout. It counts toward the Quantitative Reasoning I general education requirement.

That is a standard Calculus 1 course: limits and derivatives first, then the definite integral. Since the course assumes some prior exposure, the early material on limits tends to move faster than in a first-time course.

MATH 221 topic map

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

MATH 221 vs MATH 220 and the rest of the sequence

MATH 220 and MATH 221 share the same topic list, and credit is not given for both. The catalog points students with previous calculus experience toward MATH 221. Either one leads to MATH 231 and then MATH 241. If you are starting calculus for the first time, see the MATH 220 guide.

Example problems at the MATH 221 level

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

2. Curve sketching. For f(x) = x^3 - 3x, find where f is increasing and its inflection point.

f'(x) = 3x^2 - 3 = 3(x - 1)(x + 1), so f increases on (-\infty, -1) and (1, \infty) and decreases on (-1, 1). Since f''(x) = 6x changes sign at x = 0, the inflection point is (0, 0).

3. The Fundamental Theorem.

How to do well in UIUC MATH 221

• Do not coast on high school calculus. The course assumes you have seen derivatives, so gaps show up fast. Rebuild the basics with limits practice in the first weeks.
• Make differentiation automatic. Drill the power rule, product rule and chain rule until you can apply them without thinking.
• Keep the formulas in one place. The calculus formula sheet has every derivative and integral rule on one page.
• Learn from errors. Read common calculus mistakes before each exam, then check yourself with timed practice exams.
• Plan your exam review. Our guide on how to prepare for a calculus exam walks through a week-by-week plan.

Tip: curve sketching questions combine everything. Practice reading sign charts for f' and f'' with the critical points lesson before the exam.