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

> A study guide for MATH 221 at UW-Madison: limits, derivatives, integrals, analytic geometry and transcendental functions, mapped to free practice.

Source: https://practicecalculus.com/courses/uw-madison-math-221
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 221
- **Credits:** 5
- **Requisites:** MATH 114, or MATH 112 and 113, or placement into MATH 221 (MATH 211 and 213 do not count)
- **Covers:** Differential and integral calculus, plane analytic geometry, applications, transcendental functions
- **Next course:** MATH 222 (Calculus and Analytic Geometry 2)
- **Alternative:** MATH 211 (Survey of Calculus 1), mainly for prebusiness and social science students
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH 221 is the 5-credit first semester of Calculus and Analytic Geometry at UW-Madison.
- It introduces both differential and integral calculus, plus plane analytic geometry and transcendental functions.
- Students enter through MATH 114, MATH 112 and 113, or placement; MATH 211 does not satisfy the requisite.
- MATH 221 leads to MATH 222 and then MATH 234, the standard sequence for math, science and engineering.

**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

| MATH 221 topic | Practice | Read |
| --- | --- | --- |
| Functions and transcendental functions | [Exponentials and logarithms](/learn/calculus-1/functions-warmup/exponentials-and-logs) | [Natural exponential function](/functions/natural-exponential-function), [sine function](/functions/sine-function) |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit), [limit laws](/rules/limit-laws) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Intermediate Value Theorem](/theorems/intermediate-value-theorem) |
| The derivative and tangent lines | [Tangent lines](/learn/calculus-1/the-derivative/tangent-lines) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiation rules | [Product rule](/learn/calculus-1/derivative-rules/product-rule) | [Product and quotient rules](/blog/product-and-quotient-rules), [chain rule](/blog/chain-rule) |
| Implicit differentiation | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Implicit differentiation](/blog/implicit-differentiation) |
| Curve sketching | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Optimization and related rates | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems), [related rates](/blog/related-rates) |
| Mean Value Theorem | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean Value Theorem](/theorems/mean-value-theorem) |
| Riemann sums and the definite integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [Derivatives of integrals](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1), [FTC part 2](/theorems/fundamental-theorem-of-calculus-part-2) |
| Substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution) |
| Applications of the integral | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves](/blog/area-between-curves) |

## The UW-Madison calculus sequence

| Course | Title | Credits |
| --- | --- | --- |
| MATH 221 | Calculus and Analytic Geometry 1 | 5 |
| MATH 222 | Calculus and Analytic Geometry 2 | 4 |
| MATH 234 | Calculus: Functions of Several Variables | 4 |
| MATH 211 | Survey of Calculus 1 | 4 |

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](/courses/uw-madison-math-222), the [MATH 234 guide](/courses/uw-madison-math-234) and the [MATH 211 guide](/courses/uw-madison-math-211).

## 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 $1$:

$$y = -1 + 1(x - 1) = x - 2$$

**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.**

$$\int_0^{\pi} \sin x\,dx = \left[-\cos x\right]_0^{\pi} = -\cos\pi + \cos 0 = 1 + 1 = 2$$

## How to do well in MATH 221

- **Practice a little every day.** A 5-credit course moves fast, and rules like the [chain rule](/rules/chain-rule) stick only with repetition. Short daily sets in [Calculus 1](/learn/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](/learn/calculus-1/functions-warmup/key-trig-values) lesson early.
- **Learn from your mistakes.** Our guide to [common calculus mistakes](/blog/common-calculus-mistakes) covers the sign errors and misapplied rules that cost the most points.
- **Use one formula sheet.** The [calculus formula sheet](/formulas) puts every derivative and basic integral together.
- **Take timed practice exams** from [practice](/practice) before each midterm, and see [how to study calculus](/blog/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](/learn/calculus-1/limits/factoring-zero-over-zero) lesson shows the pattern that solves most early MATH 221 limit problems.

## Frequently asked questions

### What is MATH 221 at UW-Madison?

MATH 221, Calculus and Analytic Geometry 1, is the first course in the main calculus sequence at the University of Wisconsin-Madison. It introduces differential and integral calculus and plane analytic geometry, with applications and transcendental functions.

### What comes after MATH 221?

MATH 222, Calculus and Analytic Geometry 2, which covers integration techniques, improper integrals, first-order differential equations, series and vector geometry. After that comes MATH 234.

### What is the difference between MATH 221 and MATH 211?

MATH 211 is Survey of Calculus 1, intended mainly for prebusiness and some social science students. The Guide says students heading to advanced math, physics, engineering or most biological sciences should take MATH 221, and MATH 211 is not open to students with credit for MATH 221.

### Is MATH 221 hard?

It is a 5-credit course that covers both derivatives and integrals in one semester, so the pace is quick. Regular short practice sessions make a big difference compared with studying only before exams.

### How should I prepare for MATH 221?

Brush up on functions, trig values, exponents and logarithms, then start on limits. The functions warm-up and limits units on PracticeCalculus are free and follow the same order.