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

The 200-word examination instruction list

CLEP Calculus asks you to interpret limits, differentiate and use integrals. Determine whether the task asks for a value at a point, a rate at an instant or a change over an interval before setting up the calculation.

Start with the list or jump to exam requirements, worked examples, the Fencing Method or practice. The entries help you recognise what a question asks and which details matter.

Essential instructions

No.	Instruction or term	Meaning or action
1	Calculate	Use numbers or other given information to work out a result.
2	Determine	Work from the given information to establish an answer.
3	Solve	Find the values that satisfy the equation or inequality.
4	Simplify	Rewrite the expression in an equivalent simpler form.
5	Evaluate	Weigh relevant evidence or criteria before reaching a judgment.
6	Identify	Name or pick out the item the question specifies.
7	Estimate	Find an approximate answer using the available clues.
8	Interpret	Say what a statement, result or display means in context.
9	Compare	Set the items side by side on the same relevant feature.
10	Find all solutions	Include every value that meets the stated equation and domain.
11	Give an exact value	Use a form that preserves the full mathematical value.
12	Check the solution	Substitute the result into the original equation to verify it.
13	Find the exact value	Retain radicals, fractions, logarithms or π in exact form where appropriate.
14	Set up an equation	Translate the given conditions into a mathematical relationship.
15	Use the formula	Substitute the appropriate values with consistent units.
16	Read the axes	Identify the quantities, units and intervals on a graph before using a value.

Limits, derivatives and integrals

No.	Instruction or term	Meaning or action
17	Read a limit from a graph	Follow outputs as the input approaches the specified value from the relevant side.
18	Compare one-sided limits	A two-sided limit exists as a finite value if both sides approach that same finite value.
19	Check continuity at a point	Verify that the function value exists and equals the limit there.
20	Removable discontinuity	A hole where a finite limit exists but the value is absent or different.
21	Difference quotient	A ratio of output change to input change used in defining the derivative.
22	Secant slope	Use the difference quotient between two distinct points.
23	Tangent slope	Use the derivative at the specified input.
24	Instantaneous rate in context	State the derivative’s value and what its sign and units mean.
25	Differentiate a composite	Apply the chain rule and multiply by the inner derivative.
26	Differentiate an implicit equation	Differentiate both sides with respect to the independent variable and solve for the requested derivative.
27	Use the derivative sign	Use positive and negative derivative intervals to infer where the function increases and decreases.
28	Find absolute extrema	Evaluate the endpoints and relevant interior critical points on the stated interval.
29	Distinguish critical point and extremum	Check how the function behaves around the critical point before classifying it.
30	Test concavity	Use whether the first derivative increases or decreases.
31	Locate an inflection point	Verify a change in concavity at the candidate.
32	Optimise with a constraint	Express the objective in terms of feasible inputs before finding its optimum.
33	Related rates setup	Write an equation between changing quantities before differentiating with respect to time.
34	Tangent-line approximation	Use a function value and derivative at a nearby input to estimate an output.
35	Initial value from an antiderivative	Use the stated value to determine an additive constant.
36	Definite integral from a graph	Combine signed areas over the interval.
37	Accumulation function derivative	Apply the Fundamental Theorem of Calculus and the chain rule when the upper bound varies.
38	Average value on an interval	Divide the definite integral by the interval length.
39	Area between curves	Integrate top minus bottom or right minus left, splitting the interval if the order changes.
40	Volume by cross-sections	Integrate the area of the specified cross section along the relevant axis.
41	Disk method	Integrate π times the square of the radius of each disk.
42	Washer method	Integrate π times the difference of squared outer and inner radii.
43	Integration by substitution	Change variables to recognise a derivative pattern and handle bounds consistently.
44	Riemann sum	Estimate accumulation from a sum of representative heights times widths.
45	Net change theorem	Use a definite integral of a rate to find the change in its accumulated quantity.
46	Displacement from velocity	Integrate signed velocity over the time interval.
47	Total distance from velocity	Integrate speed, accounting for sign changes in velocity.
48	Check units of an integral	Multiply integrand units by the units of the integration variable.

Techniques and representations

No.	Instruction or term	Meaning or action
49	L’Hôpital’s rule	For a qualifying quotient with a 0/0 or ∞/∞ form, evaluate the limit of the quotient of derivatives when the rule’s conditions hold.
50	Exponential growth differential equation	Solve y′ = ky with y = Ceᵏˣ and use an initial condition to find C.
51	Average rate of change	Divide the change in output by the change in input over the stated interval.
52	Instantaneous rate of change	Evaluate the derivative at the specified input and report units when relevant.
53	Average value of a function	Divide the definite integral by the length of the interval.
54	Rate of accumulation	Identify the rate at which the accumulated quantity changes.
55	Net change	Integrate the signed rate over the stated interval.
56	Total distance	Integrate speed or add the magnitudes of displacement on intervals with constant direction.
57	Displacement	Find the signed change in position.
58	Speed increasing	Compare the signs of velocity and acceleration where both are nonzero.
59	Tangent line equation	Use the derivative as slope and the point of tangency in a line equation.
60	Linear approximation	Use the tangent line near the chosen input to estimate a function value.
61	Estimate a derivative from a table	Use a suitable nearby difference quotient and show the values used.
62	Estimate an integral from a table	Apply the requested sum with the actual subinterval widths.
63	Left Riemann sum	Use the left endpoint in each subinterval.
64	Right Riemann sum	Use the right endpoint in each subinterval.
65	Midpoint Riemann sum	Use the midpoint of each subinterval.
66	Trapezoidal sum	Multiply each interval width by the mean of its endpoint heights and add.
67	Fundamental Theorem of Calculus	Connect definite integrals, antiderivatives and derivatives of accumulation functions.
68	Evaluate a limit	Find the value approached by the function from the indicated direction or directions.
69	One-sided limit	Approach the given point from the side specified.
70	Justify continuity	Check the function value and the existence and equality of the limit there.
71	Justify differentiability	Use the derivative definition or an applicable theorem in the given context.
72	Intermediate Value Theorem	Check continuity and establish that the requested output lies between endpoint values.
73	Mean Value Theorem	Check continuity on the closed interval and differentiability on its interior before equating rates.
74	Candidates for an absolute extremum	Check eligible endpoints and interior critical points in the stated domain.
75	Critical point	Identify an interior domain point where the derivative is zero or undefined.
76	Relative extremum	Use local behaviour such as a change in sign of the first derivative.
77	Point of inflection	Establish an actual change in concavity.
78	Concave up	Identify where the derivative is increasing or use a positive second derivative when applicable.
79	Separate variables	Put expressions in the two variables on their respective sides before integrating.
80	Initial condition	Use the given input and output to select a particular solution.
81	Cross-sectional volume	Integrate the area of a cross section over the stated axis.
82	Horizontal slice	Use right minus left and integrate with respect to the vertical variable when appropriate.
83	Piecewise definition	Use the appropriate rule on each interval and split the calculation when needed.
84	Interpret in context	State what the value means for the named quantity and include suitable units.
85	Justify using the derivative	Use derivative values or signs to explain the specific conclusion.
86	Signed area	Account for regions above and below the axis with the appropriate signs.
87	Chain rule	Differentiate the outer function at the inner input and multiply by the inner derivative.
88	Product rule	Differentiate each factor in turn and add the two products.
89	Quotient rule	Use the denominator times the derivative of the numerator minus the reverse product, divided by the denominator squared.
90	Implicit differentiation	Differentiate the relation with respect to the given variable while accounting for the dependent variable.
91	Derivative of an inverse	Use the reciprocal of the original derivative at the corresponding input when the conditions apply.
92	Related rates	Differentiate the relationship with respect to time and substitute values for the same moment.
93	Linearisation point	Use a nearby input where the function and derivative can be evaluated.
94	Derivative from first principles	Use the appropriate limit of a difference quotient.
95	Discontinuity	Identify how the function value or limit fails the continuity test.
96	Vertical asymptote	Use unbounded behaviour from the relevant side.
97	Horizontal asymptote	Check the limit as the input increases or decreases without bound.
98	Antiderivative	Find a function whose derivative gives the stated integrand.
99	Constant of integration	Include an arbitrary constant in an indefinite antiderivative.
100	Substitution in an integral	Change the variable and update the differential and bounds as needed.
101	Definite integral	Evaluate the signed accumulation over the specified interval.
102	Washers	Subtract the inner disk area from the outer disk area before integrating.
103	Optimization	Identify the quantity to optimise and check the candidates allowed by the constraints.
104	Absolute maximum value	Report the largest output on the domain, rather than only where it occurs.
105	Use a function model	Translate the stated scenario into a function and use only its allowed inputs.
106	Mathematical Reasoning	Use the supplied conditions and mathematical relationships to establish the requested conclusion.
107	Identify the function family	Use the pattern in the equation, table or graph to select an appropriate family.
108	Compare representations	Match equivalent features across equations, graphs, tables and verbal descriptions.
109	Polynomial function	A finite sum of terms with nonnegative integer powers of the input.
110	Leading term	Use the highest-degree term to determine polynomial end behaviour.
111	End behaviour	Describe how output behaves as input grows without bound in either direction.
112	Multiplicity of a zero	Count how many times the corresponding factor appears.
113	Rational function	A quotient of polynomials defined where its denominator is nonzero.
114	Equal input increments	Compare outputs at evenly spaced inputs to identify a model pattern.
115	Constant first differences	Evidence of a linear pattern at equal input increments.
116	Constant second differences	Evidence of a quadratic pattern at equal input increments when the second difference is nonzero.
117	Constant ratio	Evidence of an exponential pattern when successive outputs at equal input steps are nonzero.
118	Growth factor	Multiply by this number for each stated input increment in an exponential model.
119	Decay factor	A positive multiplier below one for each stated input increment.
120	Doubling time	The time needed for an exponentially growing amount to double.

Other examination vocabulary

No.	Instruction or term	Meaning or action
121	Necessary condition	Identify what must hold whenever the stated outcome holds.
122	Sufficient condition	Identify what guarantees the stated outcome.
123	Necessary and sufficient	Establish the implication in both directions.
124	If	Identify the hypothesis and what it guarantees in the stated rule.
125	Only if	Identify the condition that must hold when the stated outcome holds.
126	If and only if	Check that each statement implies the other.
127	Converse	Swap the hypothesis and conclusion and test the resulting statement separately.
128	Contrapositive	Reverse the implication and negate both statements to produce an equivalent implication.
129	Negation	State precisely what would make the original proposition false.
130	Implication	Read which statement guarantees the other.
131	Equivalence	Check both directions or transform each statement into the other.
132	For all	Check that the statement holds across every value in the specified domain.
133	There exists	Find at least one permitted value or case that works.
134	Exactly one	Establish that there is a solution and that a second one cannot occur.
135	At least one	Find one permitted example to establish existence.
136	At most one	Show there cannot be two or more valid cases.
137	Counterexample	Give one permitted case where the proposed general statement fails.
138	Direct proof	Use valid steps from the assumptions to establish the conclusion.
139	Invalid step	Find the specific transformation or inference that does not follow.
140	Extraneous solution	Identify a value introduced by a manipulation that fails the original condition.
141	Lost solution	Check whether dividing or simplifying excluded a value that could work.
142	Domain restriction	Keep only the inputs for which the original expression or situation is defined.
143	Range of values	Find all possible outputs under the stated restrictions.
144	Excluded endpoint	Use a strict boundary where the endpoint itself is not allowed.
145	Included endpoint	Use a non-strict boundary where the endpoint is allowed.
146	Parameter	Use the fixed or adjustable quantity that defines a family of cases.
147	Existence	Show that at least one object or value meets the conditions.
148	Uniqueness	Show that only one permitted object or value meets the conditions.
149	Assume without loss of generality	Check that the cases are symmetric before simplifying to one representative case.
150	Equivalent transformation	Use an operation that preserves the solution set under the stated conditions.
151	Squaring both sides	Check signs and verify candidate answers in the original statement afterwards.
152	Divide by a variable	Check whether the divisor can be zero before taking the step.
153	Change of sign	Use the sign changes and stated assumptions relevant to the inference.
154	Check all cases	Test each permitted case against the original question.
155	Quantifier	Check whether the claim applies to all, some, none or exactly one of the allowed cases.
156	Disjunction	Read the alternatives and check whether the wording permits both to hold.
157	Conjunction	Check that both required statements hold together.
158	Mutually exclusive cases	Use cases that cannot occur together when the problem calls for a partition.
159	Exhaustive cases	Check that the cases cover every allowed possibility.
160	Open interval	Exclude both endpoints when the interval is open at both ends.
161	Closed interval	Include both endpoints when the interval is closed.
162	Piecewise function	Apply the rule for the part of the domain containing the input.
163	Monotonic	Check whether the function moves in one direction across the stated interval.
164	Increasing function	Compare outputs for ordered inputs and distinguish strict increase where required.
165	Decreasing function	Check whether outputs fall as inputs rise within the stated domain.
166	Discriminant	Use the sign of the quadratic discriminant to reason about real roots.
167	Repeated root	Identify a root with multiplicity greater than one.
168	Distinct roots	Count different values of the roots rather than multiplicities.
169	Intersection points	Find the coordinates at which the stated graphs share the same point.
170	Tangent condition	Use the relevant contact or gradient relationship to establish tangency.
171	Gradient of the tangent	Find the derivative at the named point when that method applies.
172	Sign chart	Determine where the expression is positive, negative or zero.
173	Absolute value	Use the distance from zero or separate cases based on the expression’s sign.
174	Solve the inequality	Find the set of values that make it true and keep the correct inequality direction.
175	Identify a graph transformation	Determine how the expression changes the position or shape of the graph.
176	Error interval	Give the range of possible original values using the correct endpoint signs.
177	Estimate from a graph	Use the scale and plotted information to obtain a sensible approximate value.
178	Explain a limitation	Identify the assumption or data issue that restricts the conclusion.
179	Give units	Attach the required measurement unit to the final result.
180	Change to	Express the value in the requested form or unit.
181	Do not use a graphical method	Use an algebraic method or interpretation as the task directs.
182	Integer solutions	Include only the whole-number values, including negative values or zero where allowed.
183	Repeated percentage change	Apply each period’s multiplier to the result from the previous period.
184	Optimise over the domain	Check the allowed values before deciding which output is largest or smallest.
185	Determine the graph shift	Identify the direction and size of the translation from the new expression.
186	Calculate a rate	Divide the relevant change or quantity by the appropriate time or other denominator.
187	To scale	Use the stated relationship between drawing distances and actual distances.
188	Not to scale	Use the stated measurements and relationships when calculating.
189	Exact value	Leave the answer in a form that preserves its full value.
190	Estimate from the graph	Use the plotted trend to obtain a sensible approximate value.
191	Illustrate	Make an idea clear with a relevant example or diagram.
192	Demonstrate	Show how a fact, process or result can be established.
193	Identify the feature of the graph	Use the graph or equation to decide what the named point represents.
194	Find the area	Find a non-negative area and account for changes in sign or intersecting curves.
195	Resolve a ratio	Divide the total into parts using the stated ratio.
196	Initial amount plus net change	Add the accumulated signed change to the amount present at the start.
197	Rounding interval	Use the stated precision to find the range of values that could produce the rounded result.
198	Rate of change	Find the change in one quantity per unit change in another.
199	One possible value	Supply a single value that satisfies the given conditions.
200	Equation	Use or form a statement that two expressions have equal value.

What CLEP Calculus requires

Exam component	Format and scope
Section 1	About 27 questions in 50 minutes. This section is calculator-free.
Section 2	About 17 questions in 40 minutes. The integrated TI-84 Plus CE graphing calculator is available.
Content emphasis	Limits about 10%, differential calculus 50% and integral calculus 40%.

CLEP is a U.S.-based college-credit examination programme. The receiving institution decides which exams and scores it accepts and how credit applies. Candidates outside the United States should confirm testing options with their test centre and acceptance with their institution.

Read the task before choosing a method

Use the sign of a derivative to interpret behaviour and the sign of a definite integral to interpret net accumulation. When using the graphing calculator, identify the input, bounds and requested quantity before entering an expression.

Worked examples for common instructions

Evaluate a removable limit

Prompt. Find the limit of (x² − 4)/(x − 2) as x approaches 2.

Answer. For x ≠ 2, the quotient equals x + 2. The limit is 4 even though the original expression is undefined at 2.

Differentiate a composite

Prompt. Find the derivative of (3x + 1)⁴.

Answer. The chain rule gives 4(3x + 1)³ × 3 = 12(3x + 1)³.

Compare displacement and distance

Prompt. Velocity is −3 m/s for two seconds and +2 m/s for three seconds. Find both quantities.

Answer. Displacement is −6 + 6 = 0 m. Distance is 6 + 6 = 12 m.

Use accumulation and the chain rule

Prompt. Let F(x) be the integral of cos t from 0 to x². Find F′(x).

Answer. Evaluate the integrand at x² and multiply by the derivative of x². The answer is 2x cos(x²).

Use the Fencing Method to build an answer

Start with one accurate result. Add the condition, unit or interval that fixes its meaning. Then give the reason that connects the result to the problem. Apply that reasoning again in a different example.

Start. The function has a local maximum.

Add context. The derivative changes from positive to negative at x = 2.

Add reasoning. The derivative sign shows that the function increases before x = 2 and decreases after it, so the point is a local maximum.

Transfer. Try a critical point where the derivative is positive on both sides and classify it.

Practise with the 200 terms

Choose one term about a calculation, one about interpretation and one about checking. Solve a short task that uses all three. Check your answer against the original conditions before selecting or entering it.

Distinguish value and rate

Try it. If h(4) = 12 metres and h′(4) = −2 metres per second, what does each number tell you?

Check. At 4 seconds the height is 12 metres and the instantaneous height is falling at 2 metres per second.

Frequently asked questions

How should I decide whether to use the calculator?

Read the section’s calculator rule first. Where the calculator is available, decide whether a graph or numerical estimate answers the question. If the prompt asks for an exact result, use the algebraic relationship and check the requested format.

Official sources and next reading

College Board CLEP Calculus exam description and sample resources

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