# UIUC MATH 220 Calculus: Topics, Study Guide and Practice

> A study guide for UIUC MATH 220, the 5-hour first calculus course: limits, derivatives, curve sketching and the FTC, how it differs from MATH 221, and practice.

Source: https://practicecalculus.com/courses/uiuc-math-220
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 220
- **Credit hours:** 5
- **Prerequisites:** An adequate ALEKS placement score showing knowledge of MATH 115 (Preparation for Calculus) topics
- **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 220 is a 5-hour first calculus course for students coming from precalculus.
- It covers the same topics as MATH 221: derivatives, curve sketching, antiderivatives, the Riemann integral and the Fundamental Theorem.
- Credit is not given for both MATH 220 and MATH 221, and students with prior calculus are pointed to MATH 221.
- MATH 220 leads directly to MATH 231, Calculus II.

**UIUC MATH 220** is the University of Illinois Urbana-Champaign's 5-hour first course in calculus, for students who are starting calculus after precalculus. It covers the same Calculus I topics as MATH 221 and leads into the same next course, MATH 231. This guide maps every MATH 220 topic to free practice, starting from the precalculus skills the course builds on.

## What UIUC MATH 220 covers

The Illinois catalog lists MATH 220 as a first calculus course. Its topics are differentiation and integration techniques, applications such as curve sketching, antidifferentiation, the Riemann integral, the Fundamental Theorem of Calculus, and exponential and trigonometric functions. The prerequisite is an ALEKS placement score that shows you know the material of MATH 115, Preparation for Calculus. It counts toward Quantitative Reasoning I.

This is a standard **Calculus 1** course. Because MATH 220 assumes no earlier calculus, a strong precalculus foundation matters more here than in MATH 221.

## MATH 220 topic map

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

| MATH 220 topic | Practice | Read |
| --- | --- | --- |
| Precalculus foundations | [Exponent rules](/learn/calculus-1/functions-warmup/exponent-rules) | [Exponential functions](/functions/exponential-functions), [logarithmic functions](/functions/logarithmic-functions) |
| Trig values | [Key trig values](/learn/calculus-1/functions-warmup/key-trig-values) | [Sine function](/functions/sine-function), [cosine function](/functions/cosine-function) |
| Limits | [Direct substitution](/learn/calculus-1/limits/direct-substitution) | [What is a limit?](/blog/what-is-a-limit), [limit laws](/rules/limit-laws) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity), [Intermediate Value Theorem](/theorems/intermediate-value-theorem) |
| The derivative | [Tangent lines](/learn/calculus-1/the-derivative/tangent-lines) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Differentiation techniques | [The power rule](/learn/calculus-1/derivative-rules/power-rule) | [Power rule](/blog/power-rule), [chain rule](/blog/chain-rule) |
| Exponential and trig derivatives | [Sine, cosine and tangent](/learn/calculus-1/derivative-rules/trig-derivatives) | [Derivatives of trig functions](/blog/derivatives-of-trig-functions) |
| Curve sketching | [Increasing and decreasing](/learn/calculus-1/applications-of-derivatives/increasing-and-decreasing) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Antidifferentiation | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives](/blog/antiderivatives) |
| The Riemann integral | [The definite integral as area](/learn/calculus-1/integrals/signed-area) | [Riemann sums](/blog/riemann-sums) |
| The Fundamental Theorem | [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) |

## MATH 220 vs MATH 221

| | MATH 220 | MATH 221 |
| --- | --- | --- |
| Title | Calculus | Calculus I |
| Credit hours | 5 | 4 |
| Prerequisite | ALEKS score showing precalculus (MATH 115) knowledge | ALEKS score plus a year of high school calculus or an AP Calculus AB score of at least 2 |
| Topics | Same list | Same list |
| Next course | MATH 231 | MATH 231 |

The catalog advises students with previous calculus experience to consider MATH 221, and credit is not given for both. Either way, the next step is [MATH 231](/courses/uiuc-math-231). See the [MATH 221 guide](/courses/uiuc-math-221) for the 4-hour version.

## Example problems at the MATH 220 level

**1. A limit by factoring.**

$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x + 2) = 4$$

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

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

**3. An antiderivative and a definite integral.**

$$\int_1^2 (3x^2 + 1)\,dx = \left[x^3 + x\right]_1^2 = (8 + 2) - (1 + 1) = 8$$

## How to do well in UIUC MATH 220

- **Shore up precalculus in week one.** Algebra, exponents and trig values show up in every problem. The [functions warm-up](/learn/calculus-1/functions-warmup/warmup-practice) is a quick check.
- **Understand limits before rules.** The derivative is defined as a limit, so spend real time on [limits practice](/learn/calculus-1/limits/limits-practice) before racing ahead to shortcuts.
- **Use one formula sheet all term.** The [calculus formula sheet](/formulas) collects every rule you will need.
- **Study the classic slips.** Our list of [common calculus mistakes](/blog/common-calculus-mistakes) covers the errors that cost the most points, and [how to study calculus](/blog/how-to-study-calculus) has a weekly routine.
- **Rehearse exams.** Take timed [practice exams](/practice) a week before each test.

> **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 drills exactly that move.

## Frequently asked questions

### What is UIUC MATH 220?

MATH 220 is a first calculus course at the University of Illinois Urbana-Champaign. It is worth 5 credit hours and covers differentiation and integration, curve sketching, antiderivatives, the Riemann integral and the Fundamental Theorem of Calculus.

### Should I take MATH 220 or MATH 221?

The topics are the same. MATH 220 is for students without previous calculus and needs an ALEKS score showing precalculus readiness. MATH 221 expects some prior calculus, such as a year in high school or a 2 on the AP Calculus AB exam. You cannot get credit for both.

### Is MATH 220 hard?

It is a full Calculus I course, so it asks for steady work every week. Because it starts from precalculus, students who keep their algebra and trig sharp and practice in short daily sessions usually do well.

### What comes after MATH 220?

MATH 220 satisfies the prerequisite for MATH 231, Calculus II, which is followed by MATH 241, Calculus III.

### How can I prepare for MATH 220?

Make sure functions, exponent rules, logarithms and trig values are solid, since the course starts from precalculus. Then try the first limits lessons on PracticeCalculus to get a head start.