Top 200 Vocabulary Word List for AP Calculus BC | Must Know Words for Common Examination Instructions

The 200-word examination instruction list

AP Calculus BC asks you to calculate, interpret and justify. For a convergence question, name a test and check its conditions. For a motion or area question, identify the parameter, bounds and units before selecting an expression.

Start with the list, or jump to exam requirements, worked examples, the Fencing Method or practice. The list includes command words and subject vocabulary that helps you read the task.

Essential instructions

No.	Instruction or term	Meaning or action
1	Identify	Name or pick out the item the question specifies.
2	State	Express the required point directly and clearly.
3	Describe	Report the relevant characteristics, events or pattern.
4	Explain	Connect the result to its cause, mechanism or reasoning.
5	Compare	Set the items side by side on the same relevant feature.
6	Interpret	Say what a statement, result or display means in context.
7	Justify	Give evidence for the claim and explain how that evidence supports, qualifies or defends it.
8	Evaluate	Weigh relevant evidence or criteria before reaching a judgment.
9	Determine	Work from the given information to establish an answer.
10	Calculate	Use numbers or other given information to work out a result.
11	Estimate	Find an approximate answer using the available clues.
12	Sketch	Show the required shape, relationships and labels.
13	Construct	Produce the specified representation using the information given.
14	Label	Attach the correct name to the indicated part.
15	Select	Choose only the number of options requested.
16	Check	Test the answer against the original instruction or condition.
17	Analyze	Break the information into relevant parts and explain how they relate.
18	Hybrid digital	Check which questions are displayed digitally and which answers go in a paper booklet.
19	Multiple-choice question	Select the answer that satisfies the full stem and any limiting words.
20	Free-response question	Produce your own answer and complete each labelled task.
21	Stimulus	Use the figure, extract or other material attached to the question.
22	Answer each subpart	Treat each lettered instruction as a separate job.

Subject terms and tasks

No.	Instruction or term	Meaning or action
23	Determine whether a series converges	Name a valid test, check its hypotheses and state its conclusion.
24	Nth-term test	If the terms fail to approach zero, the series diverges. A zero limit leaves convergence undecided.
25	Geometric series	Identify the first term and common ratio. An infinite geometric series converges when the ratio’s magnitude is less than one.
26	p-series	A series of terms 1/nᵖ converges exactly when p is greater than one.
27	Harmonic series	The series of reciprocals of positive integers diverges.
28	Telescoping series	Simplify the partial sums by cancelling terms and take their limit.
29	Partial sum	Add a finite initial segment of an infinite series.
30	Sequence limit	Determine the value that the terms approach, if a finite limit exists.
31	Integral test	Compare a series of positive terms to an improper integral of a continuous, positive, eventually decreasing function matching those terms.
32	Direct comparison test	Bound nonnegative terms above by a convergent series or below by a divergent one.
33	Limit comparison test	If positive terms have a ratio approaching a positive finite constant, the two series share convergence behaviour.
34	Alternating series test	Check that magnitudes eventually decrease and approach zero.
35	Absolute convergence	Show that the series of absolute values converges.
36	Conditional convergence	Show that the original series converges while the series of absolute values diverges.
37	Ratio test	Use the limiting ratio of successive term magnitudes. A limit below one gives absolute convergence, above one gives divergence, and one is inconclusive.
38	Alternating series error bound	Bound the remainder by the first omitted magnitude when the alternating series conditions hold.
39	Power series	A series of coefficients times successive powers of the difference from a centre.
40	Centre of a power series	Identify the value about which the powers are expanded.
41	Radius of convergence	Find how far inputs may lie from the centre before reaching the convergence boundary.
42	Interval of convergence	Give the full set of inputs for which the series converges, with endpoint inclusion determined separately.
43	Test the endpoints	Substitute each boundary input into the series and test the resulting numerical series.
44	Taylor polynomial	Construct a finite sum using derivatives evaluated at the centre.
45	Maclaurin polynomial	Construct a Taylor polynomial centred at zero.
46	Taylor series	Write the infinite series built from derivatives at a chosen centre.
47	Taylor coefficient	Divide the corresponding derivative at the centre by the factorial of its order.
48	Lagrange error bound	Use a bound on the next derivative over the relevant interval to bound the Taylor remainder.
49	Termwise differentiation	Differentiate power-series terms inside the radius of convergence and reconsider endpoints.
50	Termwise integration	Integrate power-series terms inside the radius of convergence and reconsider endpoints.
51	Improper integral	Rewrite an unbounded interval or singular endpoint using a limit.
52	Integration by parts	Use the product rule in reverse to exchange one integral for another.
53	Partial fraction decomposition	Rewrite a rational expression as simpler fractions when appropriate.
54	Parametric curve	Represent coordinates as separate functions of a shared parameter.
55	Parametric derivative	Divide dy/dt by dx/dt where the denominator is nonzero.
56	Parametric second derivative	Differentiate dy/dx with respect to the parameter and divide by dx/dt where valid.
57	Parametric speed	Take the magnitude of the velocity vector.
58	Arc length	Integrate speed over the specified parameter interval.
59	Horizontal tangent	For a regular parametric point, seek zero vertical velocity and nonzero horizontal velocity.
60	Vertical tangent	For a regular parametric point, seek zero horizontal velocity and nonzero vertical velocity.
61	Polar coordinates	Locate a point using a radial coordinate and an angle.
62	Polar conversion	Use x = r cos θ and y = r sin θ.
63	Polar area	Integrate one-half the square of the radius over the relevant angular interval.
64	Area between polar curves	Subtract the inner squared radius from the outer one before integrating, splitting intervals if the boundary changes.
65	Polar intersection	Check for coincident points, including points reached at different angles or at the origin.
66	Vector-valued function	Read or compute the components of a position, velocity or acceleration vector.
67	Velocity vector	Differentiate position component by component.
68	Acceleration vector	Differentiate velocity component by component.
69	Distance travelled	Integrate the magnitude of velocity over time.
70	Displacement vector	Subtract the initial position vector from the final one.
71	Euler’s method	Use successive tangent-line steps from an initial value to estimate a solution.
72	Logistic differential equation	Model a growth rate proportional to the population and its remaining capacity under the stated form.
73	Carrying capacity	Identify the positive equilibrium limiting population size in the usual logistic growth model.
74	Taylor remainder	The difference between the actual function value and its Taylor polynomial approximation.
75	State the test conclusion	Connect the conditions you verified to the convergence or divergence claimed.

Calculations and representations

No.	Instruction or term	Meaning or action
76	Average rate of change	Divide the change in output by the change in input over the stated interval.
77	Instantaneous rate of change	Evaluate the derivative at the specified input and report units when relevant.
78	Average value of a function	Divide the definite integral by the length of the interval.
79	Rate of accumulation	Identify the rate at which the accumulated quantity changes.
80	Net change	Integrate the signed rate over the stated interval.
81	Total distance	Integrate speed or add the magnitudes of displacement on intervals with constant direction.
82	Displacement	Find the signed change in position.
83	Speed increasing	Compare the signs of velocity and acceleration where both are nonzero.
84	Tangent line equation	Use the derivative as slope and the point of tangency in a line equation.
85	Linear approximation	Use the tangent line near the chosen input to estimate a function value.
86	Estimate a derivative from a table	Use a suitable nearby difference quotient and show the values used.
87	Estimate an integral from a table	Apply the requested sum with the actual subinterval widths.
88	Left Riemann sum	Use the left endpoint in each subinterval.
89	Right Riemann sum	Use the right endpoint in each subinterval.
90	Midpoint Riemann sum	Use the midpoint of each subinterval.
91	Trapezoidal sum	Multiply each interval width by the mean of its endpoint heights and add.
92	Fundamental Theorem of Calculus	Connect definite integrals, antiderivatives and derivatives of accumulation functions.
93	Evaluate a limit	Find the value approached by the function from the indicated direction or directions.
94	One-sided limit	Approach the given point from the side specified.
95	Justify continuity	Check the function value and the existence and equality of the limit there.
96	Justify differentiability	Use the derivative definition or an applicable theorem in the given context.
97	Intermediate Value Theorem	Check continuity and establish that the requested output lies between endpoint values.
98	Mean Value Theorem	Check continuity on the closed interval and differentiability on its interior before equating rates.
99	Candidates for an absolute extremum	Check eligible endpoints and interior critical points in the stated domain.
100	Critical point	Identify an interior domain point where the derivative is zero or undefined.
101	Relative extremum	Use local behaviour such as a change in sign of the first derivative.
102	Point of inflection	Establish an actual change in concavity.
103	Concave up	Identify where the derivative is increasing or use a positive second derivative when applicable.
104	Separate variables	Put expressions in the two variables on their respective sides before integrating.
105	Initial condition	Use the given input and output to select a particular solution.
106	Slope field	Match the local slopes to the differential equation and its solution behaviour.
107	Area between curves	Integrate the upper function minus the lower function with respect to the horizontal variable when appropriate.
108	Cross-sectional volume	Integrate the area of a cross section over the stated axis.
109	Horizontal slice	Use right minus left and integrate with respect to the vertical variable when appropriate.
110	Piecewise definition	Use the appropriate rule on each interval and split the calculation when needed.
111	Interpret in context	State what the value means for the named quantity and include suitable units.
112	Justify using the derivative	Use derivative values or signs to explain the specific conclusion.
113	Signed area	Account for regions above and below the axis with the appropriate signs.
114	Chain rule	Differentiate the outer function at the inner input and multiply by the inner derivative.
115	Product rule	Differentiate each factor in turn and add the two products.
116	Quotient rule	Use the denominator times the derivative of the numerator minus the reverse product, divided by the denominator squared.
117	Implicit differentiation	Differentiate the relation with respect to the given variable while accounting for the dependent variable.
118	Derivative of an inverse	Use the reciprocal of the original derivative at the corresponding input when the conditions apply.
119	Related rates	Differentiate the relationship with respect to time and substitute values for the same moment.
120	Linearisation point	Use a nearby input where the function and derivative can be evaluated.
121	Derivative from first principles	Use the appropriate limit of a difference quotient.
122	Discontinuity	Identify how the function value or limit fails the continuity test.
123	Removable discontinuity	Check whether assigning the limiting value would make the function continuous.
124	Vertical asymptote	Use unbounded behaviour from the relevant side.
125	Horizontal asymptote	Check the limit as the input increases or decreases without bound.
126	Antiderivative	Find a function whose derivative gives the stated integrand.
127	Constant of integration	Include an arbitrary constant in an indefinite antiderivative.
128	Substitution in an integral	Change the variable and update the differential and bounds as needed.
129	Definite integral	Evaluate the signed accumulation over the specified interval.
130	Washers	Subtract the inner disk area from the outer disk area before integrating.
131	Disk method	Integrate π times the squared radius along the stated axis.
132	Optimization	Identify the quantity to optimise and check the candidates allowed by the constraints.
133	Absolute maximum value	Report the largest output on the domain, rather than only where it occurs.
134	Use a function model	Translate the stated scenario into a function and use only its allowed inputs.
135	Mathematical Reasoning	Use the supplied conditions and mathematical relationships to establish the requested conclusion.
136	Read the axes	Identify the quantities, units and intervals on a graph before using a value.
137	Gradient from a graph	Calculate change in the vertical quantity per change in the horizontal quantity.
138	Calculate a rate	Divide the relevant change or quantity by the appropriate time or other denominator.
139	Calculate the percentage change	Divide the change by the starting value and multiply by 100.
140	Unit rate	Calculate the amount corresponding to one unit of the other quantity.
141	Initial value	The value at the stated starting point.
142	Final value	The value at the stated end point.
143	Ratio	Compare two quantities by division in the order specified.
144	Relative change	Divide the difference by the initial value when that baseline is meaningful.
145	Direction of change	State whether the quantity increases, decreases or stays unchanged.
146	Magnitude	The size of a quantity, considered separately from direction.
147	Unit conversion	Use a conversion factor to change the units while keeping the quantity the same.
148	Accumulation function	Define a function by integrating a rate from a fixed lower limit to the input.
149	Position from velocity	Add net displacement to the initial position.
150	Speed from components	Take the square root of the sum of the squared velocity components.
151	Concavity from a derivative	Use whether the first derivative is increasing or decreasing to infer concavity.
152	Integral accumulation with a variable upper limit	Differentiate using the Fundamental Theorem of Calculus and the chain rule when needed.
153	Estimate from the graph	Use the plotted trend to obtain a sensible approximate value.
154	Interpret the slope	Give the predicted change in response per one unit increase in the explanatory variable.
155	Interpret the intercept	Explain the predicted response at zero if zero makes sense in context.
156	Line of best fit	Draw a line representing the overall pattern of plotted data.
157	Scatterplot	Describe form, direction, strength and unusual points when relevant.
158	Regression prediction	Use the fitted relationship to estimate the response for a stated input.
159	Extrapolation	Recognise when the requested prediction is outside the observed input range.
160	Difference versus association	Distinguish a numerical group contrast from an inference about the relationship.
161	Use the scale of the display	Check axis intervals and group labels before comparing.
162	Initial condition for a differential equation	Use the specified point to determine the solution’s constant or starting value.
163	Direction field	Use short line segments to show the derivative given at selected points.
164	Choose an appropriate display	Match the display to the variable type and comparison.
165	Exact value	Leave the answer in a form that preserves its full value.
166	Give units	Attach the required measurement unit to the final result.
167	Find the exact value	Retain radicals, fractions, logarithms or π in exact form where appropriate.
168	Set up an equation	Translate the given conditions into a mathematical relationship.
169	Check the solution	Substitute the result into the original equation to verify it.
170	In terms of	Use the specified quantity or symbol in the final expression.
171	Solve	Find the values that satisfy the equation or inequality.
172	Substitute	Replace a variable with the given value or expression.
173	Decimal places	Give the requested number of digits to the right of the decimal point.
174	Significant figures	Give the requested number of meaningful digits.
175	Round	Adjust a number to the requested precision using the next digit.
176	Plot	Place the supplied points on the correct axes and scale.
177	Coordinate	Give the position using the specified ordered values.
178	Closed interval	Include both endpoints when the interval is closed.
179	Open interval	Exclude both endpoints when the interval is open at both ends.
180	Range of values	Find all possible outputs under the stated restrictions.
181	Boundary condition	Identify a requirement at an endpoint or boundary.
182	Rate of change	Find the change in one quantity per unit change in another.
183	Interpolate	Estimate within the range covered by the data.
184	Extrapolate	Estimate outside the observed range using an assumed pattern.
185	Residual	Find the observed value minus the model’s predicted value.
186	Separate the variables	Move terms into expressions involving one variable each before integration, checking excluded equilibria.
187	Baseline scenario	Use the stated reference case against which an intervention is compared.
188	Per capita	Divide the total by the number of people represented.
189	Percentage point	Find the simple difference between two percentage values.
190	Equilibrium solution	Find a constant function for which the differential equation gives zero derivative.
191	Choose the next Euler step	Use the slope at the current point and multiply by the step size to update the dependent variable.
192	Error tolerance	Compare a justified approximation bound with the largest allowable error.
193	Integrate a known series	Integrate its terms within the convergence radius and use any initial value to determine the constant.
194	Differentiate a known series	Differentiate term by term inside the convergence radius and test endpoints if requested.
195	Evaluate an improper tail	Take the limit of an integral as its upper bound tends to infinity.
196	Parametric interval	Restrict calculations to the stated parameter values and account for the direction of traversal.

Evidence and other exam wording

No.	Instruction or term	Meaning or action
197	Contrast	Show how the items differ on a shared point of comparison.
198	Infer	Use clues to reach a conclusion the material implies.
199	Predict	Use available evidence to say what is likely to happen.
200	Suggest	Offer a plausible answer suited to the information given.

What the AP Calculus BC exam requires

College Board lists the regular 2027 exam for May 10. The exam is hybrid digital. Answer multiple-choice questions and read free-response prompts in Bluebook, then handwrite the free responses in the supplied booklet.

Exam component	Format and weighting
I · Multiple choice	42 questions in 100 minutes, worth 50%. Part A has 29 questions in 62 minutes without a calculator. Part B has 13 questions in 38 minutes with a graphing calculator required.
II · Free response	6 questions in 90 minutes, worth 50%. Part A has 2 questions in 30 minutes with a graphing calculator required. Part B has 4 questions in 60 minutes without a calculator.
Calculator policy	Use a permitted graphing calculator or the built-in Desmos graphing calculator on the designated parts. Practise the calculator-free parts separately.

AP is a U.S.-based exam programme taken domestically and internationally. Confirm registration and local reporting time with your school or test centre. Use the official materials for your exact subject and exam year.

Read the task before choosing a method

Use an older released question to practise the mathematics, then use the 2027 section timings for a mock. When using a graphing calculator, record the mathematical setup that produces your answer, such as the integral or equation being evaluated.

Worked examples for common exam instructions

Determine and justify convergence

Prompt. Does the sum from n = 1 to infinity of 1/[n(n+1)] converge? Find its sum if it does.

Answer. Write each term as 1/n − 1/(n+1). The partial sum through N is 1 − 1/(N+1), which tends to 1. Therefore the series converges to 1.

Check both endpoints

Prompt. Find the interval of convergence of the sum from n = 1 to infinity of (x−2)ⁿ/n.

Answer. The ratio test gives convergence for |x−2| < 1. At x = 1 the alternating harmonic series converges. At x = 3 the harmonic series diverges. The interval is [1, 3).

Calculate distance from a parametric model

Prompt. For x(t) = 3t and y(t) = 4t on 0 ≤ t ≤ 2, find the distance travelled.

Answer. Velocity is (3, 4), so speed is 5. Integrating this constant speed over the two time units gives 10 distance units.

Set up a polar area integral

Prompt. Set up the area inside r = 2 on the angular interval from 0 to π/2.

Answer. The region is a quarter disk. Its area is the integral from 0 to π/2 of (1/2)(2²) dθ. If asked to evaluate it, the result is π square units.

Use the Fencing Method to build an answer

Begin with one accurate answer. Add a detail that fixes it to the question, then add the rule or mechanism that justifies it. Transfer the same structure to a fresh case.

Start. The series converges.

Add context. The series converges by the alternating series test.

Add reasoning. The signs alternate and the positive magnitudes decrease to zero, so the alternating series test gives convergence. Compare the absolute series separately if the prompt asks for classification.

Transfer. Try an alternating series whose terms fail to approach zero and explain which test settles it.

Practise with the 200 terms

Choose one calculation, one interpretation and one justification. Underline the part of the answer that fulfils each verb. Repeat with a different representation, such as a table instead of an equation.

Check the test before deciding

Try it. For the sum of 1/n², the ratio test gives a limit of one. Can you determine convergence?

Check. Yes. The ratio result is inconclusive, but this is a p-series with p = 2, so it converges.

Frequently asked questions

When should I write the setup?

Show the expression, equation or model that produces the result when the prompt calls for work or reasoning. Use the requested precision and state units when the quantities have them.

Official sources and next reading

College Board AP Calculus BC exam information and released questions · AP calculator policy · 2027 AP exam schedule

Related AP vocabulary guide · eduKateSG vocabulary learning guide · Learn the Fencing Method