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Exam Prep

AP Calculus AB: study sheet

8 units, with a key formula and what each covers, from this hub's syllabus notes. Not yet checked by a teacher: check it against your own notes.

The exam

Section I: 45 multiple-choice questions in 1 hr 45 min (Part A: 30 questions, 60 min, no calculator; Part B: 15 questions, 45 min, graphing calculator). Section II: 6 free-response questions in 1 hr 30 min (Part A: 2 questions, 30 min, calculator; Part B: 4 questions, 60 min, no calculator). Source: College Board, AP Calculus AB and BC Course and Exam Description.

Scored 1–5. Multiple choice and free response each count for 50% of the exam score. Unit weightings are the College Board's published AB ranges.

  1. 1. Unit 1: Limits and Continuity

    10–12% of exam score

    lim⁡x→cf(x)=f(c)  ⟺  f is continuous at c\lim_{x \to c} f(x) = f(c) \iff f \text{ is continuous at } c

    • Estimating limits from graphs and tables

    • Algebraic limits and the squeeze theorem

    • Limits at infinity and asymptotes

    • Continuity and the Intermediate Value Theorem

  2. 2. Unit 2: Differentiation: Definition and Fundamental Properties

    10–12% of exam score

    f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

    • Derivative as a limit

    • Differentiability and continuity

    • Power, product and quotient rules

    • Derivatives of sin, cos, eˣ and ln x

  3. 3. Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

    9–13% of exam score

    ddxf(g(x))=f′(g(x)) g′(x)\frac{d}{dx} f(g(x)) = f'(g(x))\,g'(x)

    • Chain rule

    • Implicit differentiation

    • Inverse and inverse trig derivatives

    • Higher-order derivatives

  4. 4. Unit 4: Contextual Applications of Differentiation

    10–15% of exam score

    L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x - a)

    • Rates of change in context

    • Motion along a line

    • Related rates

    • Linearization and L'Hôpital's rule

  5. 5. Unit 5: Analytical Applications of Differentiation

    15–18% of exam score

    f′(c)=f(b)−f(a)b−a for some c∈(a,b)f'(c) = \frac{f(b) - f(a)}{b - a} \text{ for some } c \in (a, b)

    • Mean Value and Extreme Value Theorems

    • First and second derivative tests

    • Concavity and points of inflection

    • Optimization

  6. 6. Unit 6: Integration and Accumulation of Change

    17–20% of exam score

    ∫abf(x) dx=F(b)−F(a),F′=f\int_a^b f(x)\,dx = F(b) - F(a), \quad F' = f

    • Riemann sums and the trapezoidal rule

    • Accumulation functions

    • Fundamental Theorem of Calculus

    • Antiderivatives and u-substitution

  7. 7. Unit 7: Differential Equations

    6–12% of exam score

    dydt=ky  ⟹  y=y0ekt\frac{dy}{dt} = ky \implies y = y_0 e^{kt}

    • Verifying solutions

    • Slope fields

    • Separation of variables

    • Exponential models

  8. 8. Unit 8: Applications of Integration

    10–15% of exam score

    V=π∫ab(R(x)2−r(x)2)dxV = \pi \int_a^b \left(R(x)^2 - r(x)^2\right) dx

    • Average value of a function

    • Position, velocity and total distance

    • Area between curves

    • Volumes by cross-sections, discs and washers