AP Calculus AB Study Guide: Limits, Derivatives, and Integrals for the May Exam

AP Calculus AB is highly predictable — it tests the same conceptual understanding (what a derivative means, what an integral measures) and procedural fluency year after year, so you can prepare by question type rather than by trying to cover everything. The single biggest score lever is justification on free response: don't just reach the answer, state the theorem (First or Second Derivative Test, the Fundamental Theorem, the Mean Value Theorem), verify its conditions, and explain your reasoning — while drilling core derivatives and integrals weekly for the no-calculator section.

AP Calculus AB is the most widely taken AP math exam and one of the most predictable in terms of content — the College Board tests the same core concepts year after year with different surface-level problems. Students who recognise the underlying question types can prepare systematically rather than trying to cover every possible scenario.

The key insight is that AP Calculus AB tests two distinct skills: conceptual understanding (what does a derivative mean? what does an integral measure?) and procedural fluency (can you compute derivatives and integrals quickly and accurately?). Both matter. Students strong in one but weak in the other consistently fall short of fives.

Limits and continuity: the foundation

Limits are tested primarily in multiple choice. The core skills: evaluating limits algebraically (factoring, rationalising, L'Hôpital's rule for 0/0 and ∞/∞ indeterminate forms), evaluating limits graphically (left-hand and right-hand limits, existence of limit when both sides agree), and the ε-δ definition (conceptual understanding only — you will not need to construct ε-δ proofs).

Continuity conditions: f is continuous at x = a if: (1) f(a) is defined; (2) lim(x→a) f(x) exists; (3) lim(x→a) f(x) = f(a). Know how to identify discontinuities (removable, jump, infinite) on a graph and from a piecewise function.

Intermediate Value Theorem (IVT): If f is continuous on [a, b] and k is between f(a) and f(b), there exists c in (a, b) where f(c) = k. This appears in free-response questions requiring you to prove a root exists.

Differentiation: techniques and applications

Core derivatives — memorise completely:

• Power rule: d/dx[x^n] = nx^(n−1)
• Product rule: d/dx[f·g] = f'g + fg'
• Quotient rule: d/dx[f/g] = (f'g − fg')/g²
• Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
• Trigonometric: sin→cos, cos→−sin, tan→sec², sec→sec·tan, csc→−csc·cot, cot→−csc²
• Inverse trig: arcsin→1/√(1−x²), arctan→1/(1+x²), arcsec→1/(|x|√(x²−1))
• Exponential/logarithmic: e^x→e^x, a^x→a^x·ln a, ln x→1/x, log_a x→1/(x·ln a)

Implicit differentiation: Differentiate both sides with respect to x, treating y as a function of x and applying the chain rule to every term containing y. Then solve algebraically for dy/dx.

Related rates: Identify all given rates (derivatives with respect to time), find the geometric relationship between the variables, differentiate implicitly with respect to time, substitute known values. The most common error is substituting values before differentiating.

Analytical applications (the most exam-heavy area):

• Increasing/decreasing: f is increasing where f'(x) > 0, decreasing where f'(x) < 0
• Relative extrema: First Derivative Test (sign change of f') or Second Derivative Test (f''(c) > 0 means local min)
• Absolute extrema on closed intervals: Candidates Test — evaluate f at critical points and endpoints, the largest/smallest value is the absolute max/min
• Concavity: f is concave up where f''(x) > 0, concave down where f''(x) < 0
• Inflection points: where f'' changes sign (note: f''(c) = 0 is necessary but not sufficient)
• Mean Value Theorem: if f is continuous on [a,b] and differentiable on (a,b), there exists c such that f'(c) = (f(b) − f(a))/(b − a). Always state the conditions when applying it in FRQ answers.

Integration: the technique-heavy half

Fundamental Theorem of Calculus, Part 1: d/dx[∫(a to x) f(t) dt] = f(x). Variation with chain rule: d/dx[∫(a to g(x)) f(t) dt] = f(g(x))·g'(x).

Fundamental Theorem of Calculus, Part 2: ∫(a to b) f(x) dx = F(b) − F(a) where F is any antiderivative of f.

U-substitution: The chain rule in reverse. Let u = g(x), du = g'(x) dx. Rewrite the integral entirely in terms of u, integrate, substitute back. For definite integrals with substitution: either change the limits to u-values or substitute back before evaluating.

Integration by parts: ∫u dv = uv − ∫v du. Choose u using LIATE (Logarithm, Inverse trig, Algebraic, Trigonometric, Exponential) — u is the first category that appears.

Area between curves: ∫(a to b) [f(x) − g(x)] dx where f(x) ≥ g(x) on [a,b]. If the curves intersect, split the integral at the intersection point. If the region is bounded by functions of y, integrate with respect to y.

Volumes of solids: Disk method (rotating f(x) around the x-axis): V = π∫(a to b) [f(x)]² dx. Washer method (region between f and g around x-axis): V = π∫(a to b) ([f(x)]² − [g(x)]²) dx.

Particle motion problems: Know the relationship: position s(t), velocity v(t) = s'(t), acceleration a(t) = v'(t). Displacement = ∫(a to b) v(t) dt. Total distance = ∫(a to b) |v(t)| dt (split at zeros of v).

Differential equations

Separable equations: Write dy/dx = f(x)g(y), separate variables, integrate both sides. Include the constant of integration +C, then use the initial condition to find C explicitly.

Slope fields: Match a slope field to its differential equation by checking the slopes at specific points. Be able to sketch solution curves through a given point on a slope field.

Euler's method: Approximate y(x + Δx) ≈ y(x) + Δx · f(x, y). Know how to perform several steps by hand.

Free-response strategy

On each FRQ: read the entire problem before starting, identify each part's type (accumulation, analysis, rates, etc.), write complete mathematical statements including theorem names, show all algebra steps, include units in every answer that has a real-world context, and leave nothing blank — partial credit is available for every part.

Use the Spaced Repetition Flashcard Tool for derivative and antiderivative rules, the Pomodoro Timer for timed FRQ practice (aim for 15 minutes per FRQ), and past College Board released exams for authentic practice material.

If you are also taking AP Physics 1 or AP Physics C: Mechanics, the calculus you learn here applies directly — see the AP Physics 1 study guide for how calculus shows up in physics contexts.