AP Calculus AB
review & study guide

Every unit of the course summarized, with each chapter linked to its lesson.

Reach your target score

No credit card required

This free AP Calculus AB review summarizes all 8 units of Achievable's AP Calculus AB course: what each covers, the points to remember, and its 54 chapters in order. Read a unit before its test, or work through the whole guide before exam day.

Each chapter links to its full lesson in Achievable, with the review quizzes and full-length practice exams.

Unit 1: Limits

9 chapters

Limit trend: A limit describes what value a function approaches as $x$ approaches a number, not necessarily what its value is at that point.
Existence condition: A limit exists only if both one-sided limits exist and match, and $f(a)$ does not need to be defined for the limit to exist.
Analytical methods: Always try direct substitution first, then use limit laws and composite function steps, with continuity at $x=L$ required for $\lim_{x \to a} f(g(x)) = f(L)$ .
Algebraic techniques: Indeterminate forms like $\frac{0}{0}$ require factoring, rationalizing, common denominators, or piecewise absolute value analysis before final substitution.
Infinity and continuity: Infinite limits create vertical asymptotes, limits at infinity describe horizontal asymptotes, special limits and the squeeze theorem apply, and continuity requires all three conditions.

In this unit

Tables and graphs
Free
Analytical limits
- Direct substitution
  Free
- Composite functions
  Free
Algebraic limits
- Indeterminate forms
  Free
- Absolute value functions
  Free
Limits and infinity
- Vertical asymptotes
- Horizontal asymptotes
Special limits
Continuity

Start this unit free

Unit 2: Derivative basics

7 chapters

Average and instantaneous rates: Average rate of change uses $\frac{f(b)-f(a)}{b-a}$ , while instantaneous rate of change is found by $f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$ or $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ .
AP recognition tricks: Identify which limit definition applies from $x \to a$ or $h \to 0$ , and use table values directly for average rate of change over $[a,b]$ .
Tangent and normal lines: A secant line represents the average rate of change, a tangent line represents the instantaneous rate of change, and $y - f(a) = f'(a)(x - a)$ .
Basic derivative rules: The power rule, constant multiple rule, and sum/difference rule apply to $x^n$ , $c \cdot f(x)$ , and $f(x) = g(x) \pm h(x)$ , and quotient and product rules handle products and quotients.
Special and validity rules: $\frac{d}{dx} (e^x) = e^x$ , $\frac{d}{dx} (\ln x) = \frac{1}{x}$ , differentiability implies continuity, and a discontinuity, sharp corner, cusp, or vertical tangent will not be differentiable there.

In this unit

Study this unit in Achievable

Unit 3: Advanced differentiation

6 chapters

Chain rule basics: Differentiate composite functions $f(g(x))$ by multiplying the derivative of the outer function by the derivative of the inner function.
Common rule combinations: Identify overall structure first, then use $(fg)' = f'g + fg'$ or $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$ with the chain rule for composite factors.
Implicit and higher derivatives: When $y$ is not explicitly defined as a function of $x$ , differentiate both sides with respect to $x$ , solve for $\frac{dy}{dx}$ , and differentiate again for $\dfrac{d^2y}{dx^2}$ .
Log and inverse rules: Logarithmic differentiation streamlines complex products, quotients, and variable exponents, and the slope of $f^{-1}(x)$ at $(a, b)$ is the reciprocal of the slope of $f(x)$ at $(b, a)$ .

In this unit

Study this unit in Achievable

Unit 4: Contextual uses

6 chapters

Derivative meaning: The sign of $f'(x)$ indicates direction: positive = increasing, negative = decreasing, zero = not changing, and AP FRQ interpretations need input value, noun/subject, direction, and numerical value with units.
Motion interpretation: Positive means right and negative means left, while position graphs give velocity from slope and velocity graphs give acceleration from slope, so speeding up or slowing down comes from comparing velocity and acceleration signs.
Related rates strategy: Define variables, relate them with equations, differentiate with respect to time, then substitute known values after differentiating and solve for the unknown rate.
Approximations and limits: $L(x) = f(a) + f'(a)(x - a)$ estimates nearby values, $f''(x) > 0$ gives an underestimate and $f''(x) < 0$ gives an overestimate, and L'Hopital's rule applies only to $\dfrac{0}{0}$ or $\pm \dfrac{\infty}{\infty}$ .

In this unit

Derivatives in context
Straight-line motion
- Position, velocity, & acceleration
- Graphs & tables
Related rates
Linear approximations
L'Hôpital's rule

Study this unit in Achievable

Achievable AP Calculus AB

Pass the AP Calculus AB exam on your first try with Achievable's interactive online exam preparation course. Includes everything you need: easy-to-understand online textbook, 380+ review questions, and 3+ full-length practice exams.

Easy-to-understand online textbook

380+ chapter quizzes

3+ practice exams

Start studying free View the course

Screenshot of the Achievable AP Calculus AB course dashboard

Unit 5: Analytical uses

7 chapters

Mean value theorem: If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$ , then $\exists\, c \in (a,b)$ such that $f'(c) = \dfrac{f(b)-f(a)}{b-a}$ .
Rolle and EVT: Rolle's theorem is the special case $f(a)=f(b)$ and guarantees $\exists\, c \in (a,b)$ such that $f'(c)=0$ , while EVT says a continuous function on $[a,b]$ attains absolute maximum and minimum there.
First derivative test: Find critical points by setting $f'(x)=0$ or where $f'(x)$ is undefined, then use the sign of $f'(x)$ to decide whether a critical point is a relative maximum or minimum.
Second derivative test: For extrema, the test applies only when $f(c)$ is defined, $f'(c)=0$ , and $f''(c)\neq 0$ , and inflection points occur where $f''(x)$ changes sign.
Graphs and optimization: $f'(x)$ gives slope and increasing or decreasing behavior, $f''(x)$ gives concavity, curve sketching combines domain, intercepts, asymptotes, critical points, and inflection points, and optimization uses objective and constraint equations to find max or min values.

In this unit

Important theorems
1st derivative test
- Relative extrema
- On critical points
2nd derivative test
- Concavity
- Inflection points
Graphs & curve sketching
Optimization

Study this unit in Achievable

Unit 6: Integration

9 chapters

Accumulated change: Accumulated change equals area under a rate function graph, with units as (vertical units) × (horizontal units), and it works for both constant and variable rates.
Signed area rules: Area above $x$ -axis is positive, area below $x$ -axis is negative, and net change (displacement) is the sum of signed areas.
Riemann sums: Approximate area under a curve by summing areas of rectangles or trapezoids after dividing $[a, b]$ into $n$ subintervals of width $\Delta x = \frac{b-a}{n}$ .
Definite integrals: $\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{k=1}^{n} \Delta x \cdot f(x_k)$ , and it equals the exact net signed area under $f(x)$ from $a$ to $b$ .
Accumulation functions: $A(x) = \int_a^x f(t) dt$ , $F'(x) = f(x)$ , and $h'(x) = f(g(x)) \cdot g'(x)$ for $\int_a^{g(x)} f(t) dt$ .

In this unit

Study this unit in Achievable

Unit 7: Differential equations

4 chapters

Differential equations: A differential equation relates an unknown function to one or more of its derivatives, and a solution satisfies the equation when substituted in.
Slope fields: Slope fields show solution behavior for differential equations by plotting the slope at each point $(x, y)$ with the specified slope.
Separation method: For first-order differential equations of the form $\frac{dy}{dx} = f(x)g(y)$ , isolate $y$ -terms with $dy$ and $x$ -terms with $dx$ , then integrate both sides.
Exponential models: Exponential differential equations are modeled by $\frac{dy}{dt} = ky$ , with general solution $y = Ce^{kt}$ and $k>0$ for growth, $k<0$ for decay.

In this unit

Study this unit in Achievable

Unit 8: Applications of integrals

6 chapters

Average value: $f_\text{avg} = \frac{1}{b-a} \int_a^b f(x) \, dx$ , and its units always match units of $f(x)$ , so check units to choose between average value and average rate of change.
Motion integrals: Displacement is $\int_a^b v(t)dt$ , total distance is $\int_a^b |v(t)|dt$ , and position satisfies $s(t) = s(a) + \int_a^t v(x)dx$ .
Area between curves: Area is the definite integral of top minus bottom for vertical slices or right minus left for horizontal slices, and split the integral at intersection points when roles switch.
Volume solids: Use cross sections, the disk method $V = \pi \int_a^b [R(x)]^2 \, dx$ or $V = \pi \int_c^d [R(y)]^2 \, dy$ , and the washer method $V = \pi \int_a^b [R(x)^2 - r(x)^2]dx$ .

In this unit

Study this unit in Achievable

AP Calculus AB exam at a glance

Questions: 42 multiple-choice questions, 6 free-response questions
Time: 3 hours 10 minutes
Passing score: 3 (on scale of 1-5)
Exam fee: $99

AP Calculus AB review & study guide

Unit 1: Limits

In this unit

Unit 2: Derivative basics

In this unit

Unit 3: Advanced differentiation

In this unit

Unit 4: Contextual uses

In this unit

Unit 5: Analytical uses

In this unit

Unit 6: Integration

In this unit

Unit 7: Differential equations

In this unit

Unit 8: Applications of integrals

In this unit

AP Calculus AB exam at a glance

More on the AP Calculus AB exam

AP Calculus AB
review & study guide