UW-Madison MATH 221 Calculus 1: Topics, Study Guide and Practice

MATH 221 is the first semester of Calculus and Analytic Geometry at the University of Wisconsin-Madison, and the starting point for students in math, physics, engineering, computer science and most biological sciences. At 5 credits it is one of the larger first-year courses on campus. This guide maps every MATH 221 topic to free practice, so you can learn each idea by solving problems with instant feedback.

What MATH 221 covers

The UW-Madison Guide describes MATH 221 as an introduction to differential and integral calculus together with plane analytic geometry, including applications and transcendental functions such as exponentials, logarithms and trig functions.

That matches a standard Calculus 1 course: limits, derivatives and their applications, then the definite integral and the Fundamental Theorem of Calculus. The "analytic geometry" in the title refers to working with curves in the plane, which shows up in tangent lines, curve sketching and area.

MATH 221 topic map

The UW-Madison calculus sequence

MATH 222 requires MATH 221 (or MATH 217), and MATH 234 requires MATH 222. MATH 211 is a separate, shorter track for prebusiness and some social science students, and it does not lead into MATH 222. See the MATH 222 guide, the MATH 234 guide and the MATH 211 guide.

Example problems at the MATH 221 level

1. A tangent line. Find the tangent line to y = x^3 - 2x at x = 1. The point is (1, -1) and y' = 3x^2 - 2, so the slope is 1:

2. Implicit differentiation. For x^2 + y^2 = 25, differentiate both sides: 2x + 2y\,y' = 0, so y' = -\frac{x}{y}. At (3, 4) the slope is -\frac{3}{4}.

3. A definite integral with the Fundamental Theorem.

How to do well in MATH 221

• Practice a little every day. A 5-credit course moves fast, and rules like the chain rule stick only with repetition. Short daily sets in Calculus 1 beat long weekend sessions.
• Know your trig and log facts cold. Transcendental functions are in the course title for a reason. Review the key trig values lesson early.
• Learn from your mistakes. Our guide to common calculus mistakes covers the sign errors and misapplied rules that cost the most points.
• Use one formula sheet. The calculus formula sheet puts every derivative and basic integral together.
• Take timed practice exams from practice before each midterm, and see how to study calculus for a weekly routine.

Tip: when a limit gives \frac{0}{0}, try factoring and canceling first. The factor and cancel lesson shows the pattern that solves most early MATH 221 limit problems.