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

> A study guide for UIUC MATH 221 (Calculus I): derivatives, curve sketching, the Riemann integral and the FTC, how it compares to MATH 220, and practice.

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

## At a glance

- **Course code:** MATH 221
- **Credit hours:** 4
- **Prerequisites:** An adequate ALEKS placement score plus either a year of high school calculus or at least a 2 on the AP Calculus AB exam
- **Covers:** Analytic geometry, differentiation and integration techniques, curve sketching, antiderivatives, the Riemann integral, the Fundamental Theorem, exponential and trig functions
- **General education:** Quantitative Reasoning I
- **Next course:** MATH 231 (Calculus II)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- UIUC MATH 221 is a 4-hour Calculus I course for students who already have some calculus background, such as a year of high school calculus.
- It covers derivatives and their applications, antiderivatives, the Riemann integral and the Fundamental Theorem of Calculus.
- MATH 220 covers the same topics over 5 hours for students without prior calculus, and you cannot get credit for both.
- After MATH 221, the next course is MATH 231 (Calculus II).

**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 topic | Practice | Read |
| --- | --- | --- |
| Functions and analytic geometry | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Linear functions](/functions/linear-function), [quadratic functions](/functions/quadratic-function) |
| Limits and continuity | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits), [continuity](/blog/continuity) |
| The derivative | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiation techniques | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule), [product and quotient rules](/blog/product-and-quotient-rules) |
| Exponential and trig functions | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Derivatives of trig functions](/blog/derivatives-of-trig-functions), [exponential and log derivatives](/blog/derivatives-of-exponential-and-logarithmic-functions) |
| Curve sketching | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Other applications | [Optimization](/learn/calculus-1/applications-of-derivatives/optimization) | [Optimization problems](/blog/optimization-problems), [related rates](/blog/related-rates) |
| Antidifferentiation | [The reverse power rule](/learn/calculus-1/integrals/reverse-power-rule) | [Antiderivatives](/blog/antiderivatives) |
| The Riemann integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums](/blog/riemann-sums) |
| The Fundamental Theorem | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Integration techniques | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution) |

## MATH 221 vs MATH 220 and the rest of the sequence

| Course | Title | Credit hours | Prerequisite |
| --- | --- | --- | --- |
| MATH 220 | Calculus | 5 | ALEKS score showing MATH 115 (precalculus) knowledge |
| MATH 221 | Calculus I | 4 | ALEKS score plus some prior calculus |
| MATH 231 | Calculus II | 3 | MATH 220 or 221 |
| MATH 241 | Calculus III | 4 | MATH 231 |

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](/courses/uiuc-math-231) and then [MATH 241](/courses/uiuc-math-241). If you are starting calculus for the first time, see the [MATH 220 guide](/courses/uiuc-math-220).

## Example problems at the MATH 221 level

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

$$f'(x) = \frac{2x(x + 1) - x^2 \cdot 1}{(x + 1)^2} = \frac{x^2 + 2x}{(x + 1)^2}$$

**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.**

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

## 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](/learn/calculus-1/limits/limits-practice) in the first weeks.
- **Make differentiation automatic.** Drill the [power rule](/rules/power-rule), [product rule](/rules/product-rule) and [chain rule](/rules/chain-rule) until you can apply them without thinking.
- **Keep the formulas in one place.** The [calculus formula sheet](/formulas) has every derivative and integral rule on one page.
- **Learn from errors.** Read [common calculus mistakes](/blog/common-calculus-mistakes) before each exam, then check yourself with timed [practice exams](/practice).
- **Plan your exam review.** Our guide on [how to prepare for a calculus exam](/blog/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](/learn/calculus-1/applications-of-derivatives/critical-points) lesson before the exam.

## Frequently asked questions

### What is UIUC MATH 221?

MATH 221 is Calculus I at the University of Illinois Urbana-Champaign. It covers differentiation and integration techniques, curve sketching, antiderivatives, the Riemann integral, the Fundamental Theorem of Calculus, and exponential and trigonometric functions.

### What is the difference between MATH 220 and MATH 221 at UIUC?

Both courses cover the same Calculus I topics. MATH 221 is 4 credit hours and expects some prior calculus, such as a year of high school calculus or a 2 on the AP Calculus AB exam. MATH 220 is 5 credit hours and is meant for students starting calculus from precalculus. Credit is not given for both.

### What comes after MATH 221?

The next course in the sequence is MATH 231, Calculus II, which covers techniques of integration, conic sections, polar coordinates and infinite series. MATH 241, Calculus III, follows it.

### Is MATH 221 hard?

Because it assumes you have seen some calculus before, it can move quickly through limits and derivatives. Students who review the basics early and practice in short daily sessions usually find it manageable.

### How should I prepare for MATH 221?

Refresh algebra, exponent rules, logarithms and trig values, then review limits and basic derivatives. The free Calculus 1 path on PracticeCalculus follows the same order as the course.