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Free Calculus Flashcards: Limits, Derivatives & Integrals

Intermediate Level

📐Math

Start your calculus journey with limits, derivatives, and integrals explained simply.

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Calculus flashcards, 36 cards. Hand-written and source-checked. The full set is on this page below.

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Practice Calculus with 6 flashcards using spaced repetition.

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Learning Tips for Calculus

Master Algebra First

Strong algebra skills are prerequisite for calculus.

Understand Limits

Limits are the foundation of calculus - don't skip them.

Practice Differentiation Rules

Memorize derivative rules through repetition.

How to Use Flashcards for Calculus

Start with the Basics

Review the flashcard deck to familiarize yourself with key concepts and terminology.

Use Spaced Repetition

Our system schedules reviews at optimal intervals to maximize long-term retention.

Test Yourself

Use quiz mode to actively recall information and identify areas that need more practice.

Practice Daily

Consistent 15-20 minute daily sessions are more effective than long occasional study periods.

Free Calculus flashcards: 36-card starter set

This starter set collects the formulas and theorems that single-variable calculus courses use most: limit facts, differentiation rules, common derivatives and integrals, and the big theorems. It is a quick-recall set, not a replacement for working problems.

No signup needed. Progress saves in your browser, and in your account if you sign in. Reviewed 2026-10-17.

Limits and continuity (7)

When does a two-sided limit exist at a point?

When the left-hand and right-hand limits both exist and are equal.

What three conditions make f continuous at x = a?

f(a) is defined, the limit of f(x) as x approaches a exists, and the limit equals f(a).

State the Intermediate Value Theorem.

If f is continuous on [a, b] and N is any value between f(a) and f(b), there is at least one c in [a, b] with f(c) = N.

Value of lim (sin x)/x as x approaches 0

Value of lim (1 - cos x)/x as x approaches 0

State L'Hopital's rule.

If the limit of f/g gives 0/0 or infinity/infinity, then lim f/g = lim f'/g', provided the latter limit exists.

What does the squeeze theorem say?

If g(x) <= f(x) <= h(x) near a and lim g = lim h = L, then lim f = L.

Derivative rules and common derivatives (10)

Limit definition of the derivative

f'(x) = lim as h approaches 0 of [f(x + h) - f(x)] / h.

Power rule

d/dx (x^n) = n x^(n-1).

Product rule

(fg)' = f'g + fg'.

Quotient rule

(f/g)' = (f'g - fg') / g^2.

Chain rule

d/dx f(g(x)) = f'(g(x)) g'(x): differentiate the outer function, then multiply by the derivative of the inner function.

Derivatives of sin x, cos x and tan x

(sin x)' = cos x; (cos x)' = -sin x; (tan x)' = sec^2 x.

Derivatives of sec x, csc x and cot x

(sec x)' = sec x tan x; (csc x)' = -csc x cot x; (cot x)' = -csc^2 x.

Derivatives of e^x, a^x, ln x and log_a x

(e^x)' = e^x; (a^x)' = a^x ln a; (ln x)' = 1/x; (log_a x)' = 1/(x ln a).

Derivatives of arcsin x and arctan x

(arcsin x)' = 1/sqrt(1 - x^2); (arctan x)' = 1/(1 + x^2).

How does implicit differentiation work?

Differentiate both sides with respect to x, treating y as a function of x (so a chain-rule factor dy/dx appears), then solve for dy/dx.

Theorems and applications of derivatives (7)

State Rolle's theorem.

If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f'(c) = 0 for some c in (a, b).

State the Mean Value Theorem.

If f is continuous on [a, b] and differentiable on (a, b), there is c in (a, b) with f'(c) = [f(b) - f(a)] / (b - a): the tangent slope equals the secant slope.

State the Extreme Value Theorem.

A function continuous on a closed interval [a, b] attains an absolute maximum and an absolute minimum on that interval.

What is a critical point?

An x in the domain where f'(x) = 0 or f'(x) does not exist.

How does the sign of f' tell you where f increases or decreases?

If f' > 0 on an interval, f is increasing there. If f' < 0, f is decreasing.

What does the second derivative tell you?

If f'' > 0, f is concave up; if f'' < 0, concave down. An inflection point is where concavity changes.

Write the linear approximation of f near x = a.

L(x) = f(a) + f'(a)(x - a).

Integrals and the Fundamental Theorem (12)

Fundamental Theorem of Calculus, part 1

If f is continuous on [a, b] and F(x) = integral from a to x of f(t) dt, then F'(x) = f(x).

Fundamental Theorem of Calculus, part 2

If f is continuous on [a, b] and F is any antiderivative of f, the integral of f from a to b equals F(b) - F(a).

Power rule for integrals

Integral of x^n dx = x^(n+1)/(n+1) + C, for n not equal to -1.

Integrals of 1/x and e^x

Integral of 1/x dx = ln|x| + C; integral of e^x dx = e^x + C.

Integrals of sin x, cos x and sec^2 x

Integral of sin x = -cos x + C; of cos x = sin x + C; of sec^2 x = tan x + C.

Integrals that give inverse trig functions

Integral of 1/sqrt(1 - x^2) dx = arcsin x + C; integral of 1/(1 + x^2) dx = arctan x + C.

Integral of ln x

Integral of ln x dx = x ln x - x + C (use integration by parts).

Integration by parts formula

Integral of u dv = uv - integral of v du.

What is u-substitution?

The reverse of the chain rule: choose u = g(x), replace g'(x) dx with du, integrate in u, then substitute back (or change the limits for definite integrals).

Average value of a function on [a, b]

f_avg = [1 / (b - a)] x integral from a to b of f(x) dx. The Mean Value Theorem for Integrals says f takes this value at some c in [a, b] if f is continuous.

Disk method volume formula

V = pi x integral from a to b of [R(x)]^2 dx for revolution about the x-axis. For washers, subtract the inner radius squared.

For which p does the integral from 1 to infinity of 1/x^p converge?

For p > 1. It diverges for p less than or equal to 1.

What this set covers

Topics covered

Limits and continuity, derivative rules and common derivatives, applications and theorems, integration formulas, and the Fundamental Theorem of Calculus.

Best used for

Memorising formulas and statements of theorems so that you can focus on technique during practice.

Not covered

Worked problem technique such as partial fractions and trig substitution, and most of multivariable calculus. Add cards from your course notes for those.

How to study this set

Calculus formulas need to be automatic so that your attention is free for setting up problems.

1
Day 1
Write each derivative and integral formula from memory, then check. Say theorem hypotheses aloud (continuous? differentiable?).
2
Days 2-7
Review missed cards daily and do three or four derivative or integral problems from each section after each session.
3
Week 2
Test the theorem cards by sketching a graph that shows why each hypothesis matters (for example, a function where MVT fails).
4
Final week
Interleave derivatives and integrals and ask yourself which method applies before computing anything.

Spaced repetition schedules each card to come back just before you would forget it, so reviews get shorter as your recall gets stronger. Cards you miss come back sooner.

Common mistakes to avoid

Forgetting the chain rule

When the inside is not just x, multiply by the derivative of the inside. This is the most common lost-point error.

Dropping +C or the absolute value in ln|x|

Indefinite integrals need +C, and the antiderivative of 1/x is ln|x|, not ln x.

Applying a theorem without checking hypotheses

IVT and MVT need continuity (and differentiability for MVT). State why they hold before using them.

Mixing up the sign in derivatives of co-functions

The derivatives of cos, cot and csc all carry negative signs; the integral of sin carries one.

Make your own from your notes

A starter set is a head start, not your whole course. Your class notes, slides and textbook are what your teacher actually tests, so turn them into cards too.

Put the problem type on the front (for example, integral of x e^x) and the method plus first step on the back.
Make theorem cards with the hypotheses on the front and the conclusion on the back.
Avoid cards with full long solutions; keep the key idea and practise the algebra elsewhere.
Create "spot the mistake" cards from graded homework.

Generate cards from a topic or notes Turn a PDF into flashcards

Calculus flashcards: questions

What is the difference between the MVT and Rolle's theorem?

Rolle's theorem is the special case of the Mean Value Theorem where f(a) = f(b), so the guaranteed derivative value is zero. MVT guarantees a point where the derivative equals the average rate of change.

What does the Fundamental Theorem of Calculus connect?

It connects differentiation and integration. Part 1 says the derivative of an accumulation function is the integrand, and part 2 lets you evaluate a definite integral from an antiderivative.

Should I memorise integral formulas or derive them?

Memorise the basic derivatives and integrals so they are instant. Derivations help you understand them, but speed on tests comes from recall.

Do these cards work for AP Calculus AB/BC and college Calculus I-II?

Most of the cards apply to both, since derivative rules and the main theorems are standard. Series and some integration techniques vary by course, so add your own.

How many calculus flashcards do I need?

A focused set of formulas and theorems, typically fewer than 100 cards, is plenty. Spend most of your time on problems.

Sources and review

These cards are an independent study aid, not official exam content, and are not affiliated with or endorsed by any exam body or publisher. They were written by hand and checked against the sources below on 2026-10-17. Always confirm details with your course materials and the current official documents.

OpenStax: Calculus Volume 1 — General reference for the standard derivative rules, theorems and integrals on the cards (cards written from standard course content, not checked line by line).
OpenStax: Calculus Volume 2 — Reference for integration by parts, disk method and improper integrals (cards written from standard course knowledge).

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Frequently Asked Questions

Calculus builds on algebra. With proper foundation and flashcard practice, it's manageable.

Our intro flashcards cover limits, basic derivatives, and integration fundamentals.

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