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University of Illinois Urbana-Champaign

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.

University of Illinois Urbana-Champaign Updated 3 min read

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
On this page
  1. What UIUC MATH 220 covers
  2. MATH 220 topic map
  3. MATH 220 vs MATH 221
  4. Example problems at the MATH 220 level
  5. How to do well in UIUC MATH 220

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 Exponential functions, logarithmic functions
Trig values Key trig values Sine function, cosine function
Limits Direct substitution What is a limit?, limit laws
Continuity Continuity Continuity, Intermediate Value Theorem
The derivative Tangent lines Limit definition of the derivative
Differentiation techniques The power rule Power rule, chain rule
Exponential and trig derivatives Sine, cosine and tangent Derivatives of trig functions
Curve sketching Increasing and decreasing Curve sketching with derivatives
Antidifferentiation Undoing derivatives Antiderivatives
The Riemann integral The definite integral as area Riemann sums
The Fundamental Theorem Derivatives of integrals FTC part 1, FTC 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. See the MATH 221 guide 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 is a quick check.
  • Understand limits before rules. The derivative is defined as a limit, so spend real time on limits practice before racing ahead to shortcuts.
  • Use one formula sheet all term. The calculus formula sheet collects every rule you will need.
  • Study the classic slips. Our list of common calculus mistakes covers the errors that cost the most points, and how to study calculus has a weekly routine.
  • Rehearse exams. Take timed practice exams a week before each test.

Tip: when a limit gives \frac{0}{0}, try factoring and canceling first. The factor and cancel lesson drills exactly that move.

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Functions and notation

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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.

Written by the PracticeCalculus team

We build interactive calculus lessons and write every page to be clear and correct. Spotted something off? Tell us and we will fix it.

Related university courses

UIUC MATH 221 Calculus 1 Calculus I 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. UIUC MATH 231 Calculus 2 Calculus II A study guide for UIUC MATH 231 (Calculus II): integration techniques, conic sections, polar coordinates and infinite series, with practice for every topic. UIUC MATH 241 Calculus 3 Calculus III A study guide for UIUC MATH 241 (Calculus III): partial derivatives, multiple integrals, line and surface integrals and vector calculus, with practice.