AP Calculus AB — Cheatsheet

Formulas & relationships

On the exam

• For piecewise functions, always compute the two one-sided limits separately using the correct branch, then compare. Writing "lim⁻ = lim⁺ = L, therefore the limit exists" is exactly the justification AP readers want to see.
• Reach for the Squeeze Theorem whenever you see a bounded factor (like sin or cos of something) multiplied by a factor going to 0. Bound the trig part between −1 and 1, multiply through, and let the outer limits pinch the answer.
• A horizontal asymptote describes end behavior *only* — a graph is allowed to cross its horizontal asymptote at finite x-values. Do not claim a curve "can never touch" y = L; that restriction applies to vertical asymptotes, not horizontal ones.
• A full-credit IVT argument has three moving parts: state that f is continuous on [a, b], compute both endpoint values to show the target lies between them, then conclude "by the IVT" a c exists. Skipping the continuity sentence loses the point even when the arithmetic is right.
• A table **suggests** a limit; it never establishes one. If a free-response question asks you to justify a limit, use algebra or a theorem — never "the table shows it approaches 5." Conversely, if a question says *estimate* or *approximate*, the table is exactly what it wants.
• When a problem says "using the definition of the derivative", you must show the limit of the difference quotient — the shortcut rules earn no credit on that specific prompt. Look for the h → 0 phrasing as your cue.
• Watch the sign and the new exponent carefully with negative powers. d/dx[x⁻²] = −2x⁻³ — the coefficient is negative *and* the exponent becomes more negative. Rushing this step is a frequent multiple-choice trap.
• Before reaching for the quotient rule, ask whether the denominator is a single term you can divide out. Rewriting (x³ + x)/x as x² + 1 turns a quotient-rule problem into a one-line power-rule problem and removes a chance to slip on signs.
• Think of the chain rule as peeling an onion: differentiate the outermost layer first, keep the inside intact, then multiply by the derivative of the inside, repeating for nested compositions. Track every layer and you will never drop a factor.
• Table problems test whether you know the **rules**, not whether you can differentiate a formula — there is no formula. Write the general rule first, then substitute. Students who substitute first almost always drop a term of the product rule or reverse the subtraction in the quotient rule.
• To find a tangent slope on an implicit curve you need both coordinates, because dy/dx typically depends on x *and* y. Differentiate, solve for dy/dx, then substitute the full point — not just the x-value.
• The formula g′(a) = 1/f′(g(a)) shows up on the AP exam most often in table form: you are handed values of f and f′ and asked for the inverse’s derivative. Identify the matching point from the table, then take the reciprocal.
• On motion problems, read the question’s verb carefully: "how fast" usually means velocity (first derivative), while "how is the velocity changing" means acceleration (second derivative). Matching the derivative order to the phrasing is worth easy points.
• When several rules stack, name the outermost operation before computing anything. "This is a product, whose second factor is a composite" gives you a plan and keeps the chain-rule factors from getting lost in a long expression.
• AP related-rates responses must show the differentiated equation with the d/dt factors before any numbers appear. Readers look for that general relationship; a correct final number with no setup earns little credit.
• To justify over- versus underestimate, cite concavity explicitly: "f″ < 0 on the interval, so the graph is concave down and the tangent line overestimates." A bare "overestimate" with no reason to concavity does not earn the justification point.
• Always write the phrase "form is 0/0" (or ∞/∞) before applying L’Hôpital on free response. Stating the indeterminate form is part of a valid justification; jumping straight to differentiating the pieces can cost the point even with a correct answer.
• For "speeding up or slowing down", always compare the signs of v and a at that instant. Same sign ⇒ speeding up; opposite signs ⇒ slowing down. The sign of acceleration by itself tells you nothing about speed.
• When you interpret a **negative** derivative, say "decreasing at a rate of 3.2 units per minute" rather than "increasing at −3.2." Both are mathematically true; only the first reads as an interpretation. And never write "decreasing at a rate of −3.2," which says the opposite of what you mean.
• To use the MVT on the exam, first state that f is continuous on [a, b] and differentiable on (a, b), then compute the average rate and set f′(c) equal to it. Solving for c without confirming the hypotheses loses justification credit.
• A critical point is only a *candidate* for an extremum — you must show a sign change in f′ to confirm it. Where f′ = 0 but does not change sign (as at x = 0 for y = x³), there is no local max or min.
• When the Second Derivative Test returns f″(c) = 0, do not guess — state that the test is inconclusive and switch to the First Derivative Test, examining the sign change of f′ around c to classify the point.
• Optimization free-response points hinge on justification: after finding a critical point, state a reason it gives the extreme value (sign of f′ or f″), and on a closed domain compare it against the endpoint values. A number with no justification rarely earns full credit.
• The most common way to lose this question is to **forget the endpoints**. A critical point is a candidate, not an answer, and on many problems the absolute extremum sits at an endpoint where f′ is not zero at all. Write all candidates in a table before comparing.
• Justifications that earn nothing, seen constantly: "because f′ = 0 there" (true of maxima, minima and neither); "because the graph turns around" (describes rather than justifies); "because it is the highest point" (restates the conclusion). Each of these accompanies correct arithmetic and still scores zero on the justification point.
• AP problems often give data in a table and ask for a specific Riemann sum. Read carefully whether left, right, or midpoint endpoints are required, and note that subintervals may have *unequal* widths — compute each rectangle’s width separately then.
• Verify any antiderivative by differentiating it back — if F′ returns the integrand, F is correct. This quick check catches reverse-power-rule slips (like forgetting to divide by the new exponent) before they cost points.
• On the exam, converting the limits to u-values is usually faster and less error-prone than back-substituting. Write the new limits explicitly next to the integral so you evaluate the u-antiderivative at the correct endpoints.
• For "how much is there at time b" problems, use final value = initial value + ∫[a to b] rate dt. Integrating the rate gives only the *change*; you must add the starting amount to get the total. This structure appears on nearly every AP integral free-response.
• If a question shows an integral and asks for a **derivative**, reach for Part 2 and do not integrate. Students who evaluate the integral first can still reach the right answer on simple integrands, but they lose the time and they fail entirely when the integrand has no elementary antiderivative — which the exam includes deliberately.
• When asked to sketch a solution curve on a given slope field, start at the initial point and keep your curve tangent to the nearby segments, never crossing them at odd angles. Follow the flow smoothly in both directions from the starting point.
• Free-response separable-equation problems award points for the separated equation, the antiderivatives of both sides, and finding C from the initial condition. Show each stage explicitly — a correct final formula with no work in between leaves points on the table.
• For half-life or doubling-time questions, set y = y₀/2 (or 2y₀) in y = y₀·e^(kt); the y₀ cancels, leaving an equation you solve with a natural log. The half-life depends only on k, never on the starting amount.
• When a problem gives a data point to find k, substitute it into y = y₀·e^(kt), isolate the exponential, and take a natural log — do not try to guess k. Carry enough decimal places through so the final rounded answer stays accurate.
• Before integrating, sketch the two curves and their intersections to see which is on top and whether they cross inside the interval. That quick picture prevents both limit errors and top/bottom mix-ups that graders penalize.
• Identify the cross-section shape before writing the integral: disks and washers bring a π (they are circles), but squares, triangles, and rectangles do not. Slapping a π on a non-circular cross section is a classic AP volume error.
• When a free-response question asks for the "average value" of a quantity, use the integral formula and divide by the interval length. If it asks for the "average rate of change", use the endpoint difference over the length instead — the two prompts want different computations.
• Watch the wording: "displacement" or "position change" means ∫v dt, while "total distance traveled" means ∫|v| dt with the interval split at sign changes. And "where is the particle" always needs the initial position added to the integral.
• Always determine **which curve is on top** by testing a point inside the interval, not by looking at the equations. Reversing top and bottom produces the negative of the right answer, and an area cannot be negative — so if your result comes out negative, that is the error.

How to get a 5

• On calculator-active FRQs, write the setup integral/equation before using the calculator, and store answers to 3 decimals — bare answers lose points.
• Justify with the specific theorem or sign analysis: extrema cite f' sign change, concavity cites f'', and cite continuity/differentiability for MVT/IVT.
• Include units and interpret in context; remember total distance uses ∫|v| dt while displacement uses ∫v dt — a very common trap.
• Know FTC accumulation functions cold: g(x) = ∫ₐˣ f(t)dt problems appear yearly — g' = f and g'' = f' drive the reasoning.
• On FRQs, always show the setup — the integral, derivative equation, or limit — before computing; setup points are awarded even when arithmetic goes wrong, and a bare answer with no work can earn nothing.
• Justifications must cite calculus, not the graph: say “f′ changes from positive to negative at x = 2, so f has a relative maximum” rather than “the graph goes up then down.”
• Keep units and context in every applied answer (e.g. “the water level rises at 3/(4π) meters per minute”); interpretation points require both the number and its meaning.
• When the calculator is allowed, use it for definite integrals and root-finding, but write the expression you evaluated — an unexplained decimal earns no credit.
• Memorize the difference between displacement (∫v dt) and total distance (∫|v| dt), and between average value and average rate of change — the exam tests these pairings every year.
• Do not simplify past what is asked: an unsimplified exact answer like π(8 − 16/3) is full credit, while an arithmetic slip during unnecessary simplification loses the answer point.

Key terms