FREE AP Calculus AB Study Guide 2026: All 8 Units, Limits to Integrals

This free AP Calculus AB study guide teaches to the College Board’s current course — all eight units, in the order the exam is built, from limits to derivatives to integrals.^[1] AP Calculus AB covers roughly the first two semesters of college calculus and is scored from 1 to 5, with a 3 generally counting as passing for college credit.^[2]

It’s interactive, not a wall of text: every unit has a built-in checkpoint quiz, hover-able glossary terms, worked examples with full derivative and integral notation, labeled diagrams, and concept questions, so you learn by doing. Read it unit by unit, test yourself at each checkpoint, then round out your free prep with our practice questions and flashcards.

AP Calculus AB is one of the 17 AP exams — explore our AP study guides to compare and prep across the whole family.

AP Calculus AB Exam Snapshot

Multiple choice and free response are each worth 50% of your score.^[3] The free-response section rewards justification — you earn points for correct setup, notation, and reasoning, not only the final number. Spend your study time across all eight units, but weight it toward the heaviest:

College Board reports each unit’s share of the multiple-choice section as a range, so the exact mix shifts slightly each year.^[1] This guide teaches all eight units as eight study modules, in the order they build on one another.

1 · Limits & Continuity

About 10–12% of the exam. Limits are the foundation of the whole course — they make both the derivative and the integral rigorous.^[1] A limit is the value $f(x)$ approaches as $x$ nears a target, and it can exist even where the function does not.

Defining & Estimating Limits

A two-sided limit $\lim_{x \to a} f(x) = L$ exists only when the one-sided limits agree: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$. You can estimate a limit from a graph or table, or read it off directly when the function is continuous there.

Evaluating Limits Algebraically

Try direct substitution first. If it gives $\tfrac{0}{0}$, use algebra — factor and cancel, multiply by a conjugate, or simplify a complex fraction. For limits at infinity of a rational function, compare the degrees of numerator and denominator.

Continuity, IVT & Discontinuities

Continuity at $x = a$ needs three things: $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and they are equal. Discontinuities come in three flavors: removable (a hole), jump (one-sided limits disagree), and infinite (a vertical asymptote). The Intermediate Value Theorem uses continuity to guarantee a function hits every value between $f(a)$ and $f(b)$.

2 · Differentiation: Definition & Basic Rules

About 10–12% of the exam. The derivative is the instantaneous rate of change — the slope of the tangent line.^[1] Unit 2 builds it from the limit definition, then gives you the rules that make differentiation fast.

The Derivative as a Limit

The derivative is the limit of the difference quotient: $f'(a) = \lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$. It is the limiting slope of secant lines as the second point slides toward the first. Where this limit fails — a corner, cusp, or vertical tangent — the function is not differentiable, even if it is continuous.

Power, Constant & Sum Rules

The power rule is your workhorse: $\dfrac{d}{dx} x^n = n x^{n-1}$ for any real $n$. Rewrite roots and reciprocals as powers first — e.g. $\sqrt{x} = x^{1/2}$. Combine it with the constant-multiple and sum rules and the key transcendental derivatives:

Product & Quotient Rules

For a product, use the product rule $(fg)' = f'g + fg'$. For a quotient, use the quotient rule $\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$. The most common error is reversing the order in the quotient rule’s numerator.

3 · Differentiation: Composite, Implicit & Inverse

About 9–13% of the exam. This unit is the chain rule and everything it unlocks — composite functions, implicit differentiation, and the derivatives of inverse functions.^[1]

The Chain Rule

The chain rule differentiates a composite $f(g(x))$: $\dfrac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$ — outer derivative at the inner function, times the inner derivative. It is the most-used rule in the course, often nested inside a product or quotient.

Implicit Differentiation

When a curve is given by an equation not solved for $y$ — like $x^2 + y^2 = 25$ — differentiate both sides in $x$, attaching a $\dfrac{dy}{dx}$ to every $y$-term (chain rule), then solve for $\dfrac{dy}{dx}$.

Inverse & Inverse-Trig Derivatives

If $g$ is the inverse of $f$, then $g'(x) = \dfrac{1}{f'(g(x))}$. Two inverse-trig derivatives are tested most:

4 · Contextual Applications of Differentiation

About 10–15% of the exam. Derivatives in the real world: related rates, linear approximation, L’Hopital’s Rule, and straight-line motion.^[1]

Related Rates

Related rates connect the time rates of two linked quantities. Write the equation relating them, differentiate both sides with respect to $t$, then substitute known values after differentiating.

Linearization & L’Hopital’s Rule

Linearization uses the tangent line $L(x) = f(a) + f'(a)(x - a)$ to estimate $f$ near $a$. For a limit in the indeterminate form $\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$, L’Hopital’s Rule replaces it with $\lim \dfrac{f'(x)}{g'(x)}$.

Straight-Line Motion

For a particle with position $s(t)$, velocity is $v(t) = s'(t)$ and acceleration is $a(t) = v'(t) = s''(t)$. The particle is at rest when $v = 0$, and it is speeding up when $v$ and $a$ share the same sign.

5 · Analytical Applications of Differentiation

About 15–18% of the exam — one of the two heaviest units. Using $f'$ and $f''$ to analyze graphs, the key theorems (MVT, EVT), and optimization.^[1]

MVT & the Extreme Value Theorem

The Mean Value Theorem says that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, some $c$ gives $f'(c) = \dfrac{f(b) - f(a)}{b - a}$ — the tangent is parallel to the secant. The Extreme Value Theorem guarantees an absolute max and min when $f$ is continuous on a closed interval.

First & Second Derivative Tests

At a critical point ($f' = 0$ or undefined), the First Derivative Test checks the sign of $f'$: a change $+ \to -$ is a relative max, $- \to +$ a relative min. The Second Derivative Test uses $f''$ at the point: $f'' > 0$ ⟹ min, $f'' < 0$ ⟹ max. An inflection point is where $f''$ changes sign.

Optimization & Curve Sketching

Optimization finds the largest or smallest value of a quantity. Write the quantity as a function of one variable, differentiate, find critical points, and test them — checking the endpoints too on a closed interval (the EVT guarantees the extreme is attained there).

6 · Integration & Accumulation of Change

About 17–20% of the exam — the single heaviest unit. Riemann sums, the definite integral as accumulated area, the Fundamental Theorem, and u-substitution.^[1]

Riemann Sums & Accumulation

A Riemann sum approximates area with rectangles whose heights come from left endpoints, right endpoints, or midpoints. As the rectangles shrink, the sum becomes the definite integral. An accumulation function $g(x) = \int_a^x f(t)\,dt$ stores total change up to $x$.

The Fundamental Theorem of Calculus

The Fundamental Theorem ties the two halves of calculus together. Part 1: $\dfrac{d}{dx} \int_a^x f(t)\,dt = f(x)$. Part 2 (evaluation): $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$.

Antiderivatives & u-Substitution

Reverse the basic derivative rules to find antiderivatives, then use u-substitution to reverse the chain rule. Three you should know:

7 · Differential Equations

About 6–12% of the exam. A differential equation relates a function to its derivative. On AB you visualize them with slope fields and solve the separable ones analytically.^[1]

Slope Fields

A slope field draws a short segment at each grid point with slope $\dfrac{dy}{dx} = f(x, y)$ there, picturing the family of solution curves. Segments are horizontal where $\dfrac{dy}{dx} = 0$ and steep where the slope is large.

Separable Equations & Growth

A separable equation can be rearranged so all $y$-terms (with $dy$) are on one side and all $x$-terms (with $dx$) on the other; integrate both sides and use an initial condition for $C$. The model $\dfrac{dy}{dt} = ky$ gives exponential growth/decay $y = y_0 e^{kt}$.

8 · Applications of Integration

About 10–15% of the exam. Putting the integral to work: average value, area between curves, accumulated change, and volumes of revolution.^[1]

Area Between Curves & Average Value

The area between $y = f(x)$ (top) and $y = g(x)$ (bottom) is $\int_a^b [f(x) - g(x)]\,dx$ — always top minus bottom. The average value of $f$ on $[a, b]$ is $\dfrac{1}{b - a}\int_a^b f(x)\,dx$. Total distance traveled by a particle is $\int |v(t)|\,dt$ (use the absolute value).

Volumes of Revolution

Revolve a region about an axis and integrate the cross-sectional area. Use the disk method $V = \pi \int R^2\,dx$ when the region touches the axis, and the washer method $V = \pi \int (R^2 - r^2)\,dx$when a gap leaves a hollow core. For a solid with known cross sections (squares, triangles), integrate that cross-section’s area instead.

How to Use This Study Guide

A study guide is a map, not the whole territory — use it alongside official College Board practice (AP Classroom and released free-response questions). The eight units build on one another, so work them in order; limits underpin derivatives, and derivatives underpin integrals. Because the free-response section rewards justification and notation, practice writing out your reasoning, not just the answer.

AP Calculus AB Glossary

Quick definitions for the terms and theorems you’ll see most across AP Calculus AB:

Accumulation function

A function defined by an integral with a variable upper limit, $g(x) = \int_a^x f(t)\,dt$; its derivative is $f(x)$.

Antiderivative

A function $F$ whose derivative is $f$. The indefinite integral $\int f(x)\,dx = F(x) + C$.

Average value

The average value of $f$ on $[a, b]$ is $\frac{1}{b - a} \int_a^b f(x)\,dx$.

Chain rule

The derivative of a composite $f(g(x))$ is $f'(g(x)) \cdot g'(x)$ — outer derivative times inner derivative.

Concavity

The way a curve bends: concave up where $f''(x) > 0$ (cups upward) and concave down where $f''(x) < 0$.

Continuity

A function $f$ is continuous at $x = a$ when $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and the limit equals $f(a)$.

Critical point

A point where $f'(x) = 0$ or $f'(x)$ is undefined — a candidate for a relative maximum or minimum.

Definite integral

$\int_a^b f(x)\,dx$ — the signed area between $f$ and the $x$-axis from $a$ to $b$; a limit of Riemann sums.

Derivative

The instantaneous rate of change of $f$, defined as $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ — the slope of the tangent line at $x = a$.

Difference quotient

The average rate of change $\frac{f(a+h) - f(a)}{h}$ — the slope of a secant line; its limit as $h \to 0$ is the derivative.

Disk method

Finds a volume of revolution with no gap to the axis: $V = \pi \int R^2\,dx$, where $R$ is the radius of each slice.

Exponential growth

Change modeled by $\frac{dy}{dt} = ky$, whose solution is $y = y_0 e^{kt}$; the rate of change is proportional to the current amount.

Extreme Value Theorem

A function continuous on a closed, bounded interval attains both an absolute maximum and an absolute minimum on that interval.

Fundamental Theorem of Calculus

Links derivatives and integrals: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$ (Part 1) and $\int_a^b f(x)\,dx = F(b) - F(a)$ (Part 2).

Implicit differentiation

Finding $\frac{dy}{dx}$ from an equation not solved for $y$ by differentiating both sides in $x$ and treating $y$ as a function of $x$.

Inflection point

A point where a graph's concavity changes — where $f''(x)$ changes sign.

Intermediate Value Theorem

If $f$ is continuous on $[a, b]$, it takes every value between $f(a)$ and $f(b)$; opposite signs guarantee a root in between.

L'Hopital's Rule

For a limit in the indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, the limit equals $\lim \frac{f'(x)}{g'(x)}$.

Limit

The single value a function approaches as the input approaches a target. Written $\lim_{x \to a} f(x)$, it can exist even when $f(a)$ is undefined.

Linearization

Using the tangent line at $x = a$, $L(x) = f(a) + f'(a)(x - a)$, to estimate $f$ near $a$.

Mean Value Theorem

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, some $c$ gives $f'(c) = \frac{f(b) - f(a)}{b - a}$.

One-sided limit

The value a function approaches from only the left ($x \to a^-$) or only the right ($x \to a^+$). A two-sided limit exists only when both one-sided limits agree.

Power rule

If $f(x) = x^n$, then $f'(x) = n x^{n-1}$ for any real exponent $n$.

Product rule

$(f \cdot g)' = f' g + f g'$ — used to differentiate a product of two functions of $x$.

Quotient rule

$\left( \dfrac{f}{g} \right)' = \dfrac{f' g - f g'}{g^2}$ — used to differentiate a quotient of two functions.

Related rates

A problem that links the time rates of two quantities sharing an equation; you differentiate the equation with respect to $t$ and solve for the unknown rate.

Removable discontinuity

A 'hole' in a graph where the limit exists but does not equal the function's value (or the value is undefined), often from a common factor that cancels.

Riemann sum

An approximation of area using rectangles whose heights come from the function at left endpoints, right endpoints, or midpoints of subintervals.

Separable differential equation

A differential equation that can be written so all $y$-terms (with $dy$) are on one side and all $x$-terms (with $dx$) on the other, then integrated.

Slope field

A grid of short segments, each with slope $\frac{dy}{dx} = f(x, y)$ at that point, picturing the solution curves of a differential equation.

Squeeze theorem

If $g(x) \le f(x) \le h(x)$ near $a$ and $g$ and $h$ share the same limit $L$ at $a$, then $f$ also has limit $L$.

u-substitution

An integration technique that reverses the chain rule: substitute $u$ for an inner function so the integral becomes a basic antiderivative.

Washer method

Finds a volume of revolution with a gap to the axis: $V = \pi \int (R^2 - r^2)\,dx$, outer radius $R$, inner radius $r$.

Free AP Calculus AB Study Materials & Resources

Everything you need to prepare for AP Calculus AB is free here — no paywall, no sign-up. This guide is the foundation; pair it with the rest of our free AP Calculus AB study materials for active recall, timed practice, and last-minute review:

• AP Calculus AB Practice Test — exam-style questions across all eight units, with worked explanations.
• AP Calculus AB Flashcards — active-recall decks for the derivative rules, integral forms, and theorems you must know cold.

AP Calculus AB Study Guide FAQ

References

1. 1.College Board. “AP Calculus AB and BC Course and Exam Description.” College Board. ↑
2. 2.College Board. “AP Calculus AB – AP Students.” College Board. ↑
3. 3.College Board. “AP Calculus AB Exam – AP Students.” College Board. ↑
4. 4.College Board. “AP Calculus AB and BC Course at a Glance.” College Board. ↑

Sources for the concept answers

Every answer in the AP Calculus AB concept questions above is drawn from an official primary source:

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