How to Pass Calculus 1 in College: A Practice Plan for Every Unit

Here is how to pass Calculus 1 from the middle of the semester: fix the algebra and trig skills your current unit depends on, solve problems with the solutions covered instead of rereading worked examples, fold earlier units into every week's practice, and log each mistake until it stops happening. Before every exam, take old exams under a timer.

That advice is easy to nod along to and hard to turn into a Tuesday night, so this guide makes it concrete: what a first college calculus course covers (checked against three universities' course pages and the free OpenStax textbook), why it trips people up, a unit-by-unit plan with a free place to practice every topic, and what to do on exam day or if you've fallen behind.

Exam in a few days? Go straight to how to take a Calculus 1 exam and catching up.

What Calculus 1 covers

First-semester calculus looks much the same from school to school. The course descriptions at the University of Houston, the University of Illinois Chicago and Iowa State run through the same core, and so does the free OpenStax Calculus Volume 1, which says it follows the scope and sequence of most general calculus courses:

1. Limits and continuity: what a function approaches, the limit laws, limits at infinity, continuity.
2. The derivative: defined as a limit of slopes and read as an instantaneous rate of change.
3. Differentiation rules: the power, product, quotient and chain rules; derivatives of trig, exponential and logarithmic functions; implicit differentiation.
4. Applications of derivatives: related rates, linear approximation, maxima and minima, the Mean Value Theorem, curve sketching, optimization and L'Hôpital's rule.
5. An introduction to integrals: antiderivatives, Riemann sums, the definite integral, the Fundamental Theorem of Calculus and substitution.

The order varies. OpenStax opens with a full chapter reviewing functions, including trig, inverse, exponential and log functions, before it reaches limits, and it saves integration techniques such as integration by parts for Volume 2. Your syllabus decides what your exams cover.

College Board lists AP Calculus AB's college equivalent as "a first-semester college calculus course" (College Board). AP courses cover introductory college-level material, so Lykke's free AP Calculus AB course, which works through College Board's framework one topic at a time, makes a good practice bank for a college Calculus 1 class. The plan below links its sections.

Why Calculus 1 is hard (and what to fix first)

Calculus adds a handful of new ideas, such as limits, instantaneous rates and accumulation, on top of everything from algebra and precalculus. Four things make it harder than it looks:

• The course assumes your algebra and trig. Iowa State's Calculus I lists two years of high school algebra, a year of geometry and a semester of trigonometry (or a placement score) as its prerequisite. College Board's recommended preparation for AP Calculus AB asks you to know the properties and graphs of linear, polynomial, rational, exponential, logarithmic, trig and inverse trig functions.
• Preparation predicts performance. In a study of 6,207 students taught by 216 professors at 133 randomly selected US colleges and universities, mastery of algebra, geometry and precalculus had, on average, more than double the impact on college calculus performance of having taken calculus in high school (Sadler & Sonnert, 2018).
• Much of the work is algebra. Paul Dawkins, who teaches Calculus I at Lamar University, writes in his free Algebra/Trig Review that in many calculus problems, 90% or more of the work is algebra. A correct derivative followed by a factoring slip is still a wrong answer.
• The pace doesn't let up. Iowa State's course, for example, has three evening midterms, a cumulative final, weekly quizzes in recitation and three online homework sets due most weeks. Each unit is built on the one before it, so a gap from September is still there in November.

Check your prerequisites this week

Dawkins wrote his Algebra/Trig Review as a self-test for his own calculus students. Spend 30 minutes on it, note which kinds of problems you miss, then fix only what your current unit uses:

How to study for Calculus 1

Our midterm study guide covers the research behind recall, spacing and mixed practice. In a calculus class, those ideas turn into four habits.

Solve problems with the solution covered

A worked solution looks obvious while you're reading it; the question is whether you can produce one on a blank page. Carleton College's TRIO program recommends covering the answer to each example, working it yourself, and uncovering the solution a little at a time only when you're stuck (Carleton TRIO). A 2007 practice guide from the U.S. Department of Education's Institute of Education Sciences recommends the same rhythm: interleave worked examples with problems students solve on their own, a recommendation it rates as backed by moderate evidence (IES). So read one example, close it, and do a similar problem from scratch before you read the next.

Mix old units into every week

Homework is sorted by section, so you always know which rule a problem wants. Exams aren't sorted. Duane Kouba of UC Davis's math department warns that knowing how to do a problem because you know which section it came from "could give you a false sense of security," and recommends preparing with new, randomly mixed problems (Kouba). Once a week, do a short mixed set, two problems from each earlier unit, and name the method before you start each one. The AP Calculus AB section on selecting procedures for derivatives drills that exact decision.

Keep an error log

Carleton's TRIO page also suggests keeping track of problems that gave you trouble, even the ones you eventually solved. For calculus, give each miss one row and a cause:

Redo every logged problem from a blank page two or three days later, and again in the week before the exam. When one cause keeps showing up, such as algebra slips or dropped notation, that's your review target.

Use office hours and classmates well

Bring a specific problem to office hours with your attempt written out, and ask where your reasoning broke rather than how to do it. Kouba suggests finding one or two classmates to work with regularly, after you've made an honest attempt on your own. Check your course page for a help room too: Iowa State lists help hours for its Calculus I, and UIC's course page links a learning center with TA office hours.

How to pass Calculus 1: a unit-by-unit practice plan

Work through the rows in your syllabus's order. Every link is free: the AP Calculus AB sections each come with flashcards, multiple-choice questions and a free-response question, and the demos let you drag or slide an input and watch the idea respond.

A week in any unit looks like this:

1. After each lecture, work the matching section's examples with the solutions covered, then take its quiz.
2. Do the homework with your notes closed the first time through, and log every miss.
3. Once a week, do a mixed set from earlier units and redo the error-log problems that are due.
4. When a rule feels like symbol-pushing, open the demo. The Chain Rule Onion, for example, shows the inside rate and the outside rate multiplying into the total rate.
5. Two weeks before each exam, switch to the 14-day plan and use your instructor's old exams as the practice tests.

How to take a Calculus 1 exam

Before exam day, find out three things from your syllabus or instructor: whether a calculator is allowed, whether you get a formula sheet, and whether answers have to be simplified. Then practice under those rules.

During the exam:

1. Budget time by points. Skim the whole exam, then give each problem time in proportion to what it's worth, so a stubborn 5-point limit can't eat the time for a 15-point optimization problem.
2. Write the setup before the arithmetic. The equation relating the variables, the function you're maximizing, the limit you're evaluating: a grader can give partial credit only for work that's on the page.
3. Keep the notation true. Write "lim" on every line until you actually substitute, and end every indefinite integral with + C. Dawkins lists dropped limit notation and dropped constants of integration among the errors that show up in calculus classes.
4. Use radians. Calculus is done in radians, and the difference is not small: cos(10) is about −0.84 in radians but about 0.98 in degrees (Dawkins). If a calculator is allowed, check its mode before you start.
5. Justify with the theorem. Name it and check its conditions: the Mean Value Theorem needs a function that is continuous on [a, b] and differentiable on (a, b). The AP Calculus AB section on using the Mean Value Theorem shows what a complete justification looks like.
6. Answer applied problems in a sentence, with units. Write "the water level is rising at 0.3 ft/min," not just "0.3."
7. Check by a second route. Differentiate your antiderivative, plug a nearby number into a limit, or compare the sign of your derivative with the shape of the graph.

For timed practice beyond your old exams, the AP Calculus AB course's practice sets, such as AP Practice 3, are multi-part free-response problems in the AP style.

Common Calculus 1 mistakes and how to fix them

Dawkins points out that many of the errors made in calculus classes aren't really calculus errors, and several of the eight below are plain algebra or trig. Check for all of them on every exam:

1. Differentiating a product factor by factor. The derivative of fg is f'g + fg', not f'g'. Check it: x² · x³ = x⁵, whose derivative is 5x⁴, while 2x · 3x² = 6x³. 2.8 The product rule explains why f'g' fails; quotients need the quotient rule for the same reason.
2. Dropping the inside derivative. The derivative of sin(x²) is cos(x²) · 2x. Every layer of a composite contributes a factor (3.1 The chain rule).
3. Substituting too early in related rates. See the worked example below.
4. Treating 0/0 as an answer. 0/0 is an indeterminate form: it means more work is needed, not that the limit is 0 or doesn't exist. Factor, multiply by a conjugate (1.6) or, once you've learned it, use L'Hôpital's rule.
5. Canceling terms instead of factors. (x² − 9)/(x − 3) simplifies to x + 3 (for x ≠ 3) because x − 3 is a factor of the top. In (x + 3)/x, nothing cancels.
6. Distributing over a sum when you can't. (a + b)² is a² + 2ab + b², and √(a + b) is not √a + √b.
7. Working in degrees. Radians throughout, on paper and on the calculator.
8. Forgetting endpoints and + C. On a closed interval, the absolute maximum or minimum can sit at an endpoint, so compare endpoints with critical points (5.5 The candidates test). Every indefinite integral ends in + C.

Worked example: the related-rates trap

A 10-foot ladder leans against a wall, and its base slides away from the wall at 2 ft/s. How fast is the top sliding down when the base is 6 feet out?

• The base x and the height y satisfy x² + y² = 100.
• Differentiate with respect to time first: 2x · (dx/dt) + 2y · (dy/dt) = 0.
• Now substitute x = 6, y = 8 and dx/dt = 2: 24 + 16 · (dy/dt) = 0, so dy/dt = −1.5. The top slides down at 1.5 ft/s.

Substitute x = 6 before differentiating and the equation becomes 36 + y² = 100, which differentiates to 2y · (dy/dt) = 0: a ladder whose top never moves. OpenStax's related-rates strategy warns about exactly this, because a quantity you've replaced with a number behaves as a constant. Lykke's free falling-ladder demo runs the same ladder, so you can drag the base and watch dy/dt change.

How to catch up in Calculus 1 if you're behind

Catching up is a triage problem, not a reason to reread from chapter 1. Work backward from the next exam:

1. List what the next exam covers, and treat that list as your whole plan until the exam. Older units come in only where these topics need them.
2. Do the grade math. Find each component's weight in the syllabus, check for policies such as a dropped quiz or a final that can replace a low exam score, and work out what you need on the remaining work.
3. Rebuild each topic with one worked example and three problems. Read the example, close it, solve three similar problems, and log what you miss. Skip rewriting notes.
4. Backfill prerequisites only when they block you. If a limit stalls on factoring, spend 20 minutes on factoring and go back to the limit. Don't restart algebra.
5. Get help this week, not next. Take your error log to office hours, your department's help room or your campus tutoring center.
6. If the numbers still don't work, ask early. Talk to your instructor and your academic advisor before your school's withdrawal deadline. If you receive financial aid, ask the financial aid office how dropping the course would affect it.

Free Calculus 1 resources

• OpenStax Calculus Volume 1: a free textbook, online or as a PDF, with an answer key. Its chapters follow the core list above, so it works as a second explanation of any section in your own book.
• MIT OpenCourseWare 18.01SC: MIT's single-variable calculus, built for independent study, with lecture and recitation videos, worked examples, and problem sets and exams with solutions. It continues past a typical Calculus 1 into integration techniques and series, so use its first three units.
• Khan Academy Calculus 1: free videos, articles and practice questions, unit by unit.
• Paul's Online Notes (Lamar University): besides the Algebra/Trig Review and the error lists above, the site has full Calculus I notes with practice problems.
• Your own department: old exams from your course are the closest rehearsal for the real one, so ask whether they're posted. Iowa State, for example, puts practice exams for its Calculus I online.
• Lykke's free courses: the AP Calculus AB course for practice by topic, and College Algebra and Precalculus for prerequisite review. Explore has the rest of the ready-made courses.