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What the exam asks for

• Estimate limits by comparing left- and right-hand graphical or tabular behavior, including when the two-sided limit fails to exist.
• Choose algebraic limit properties, equivalent-expression manipulation, or the squeeze theorem to fit the expression.
• Check continuity through the function value, the limit, and their agreement; use these conditions to identify removable discontinuities.
• Use infinite and end-behavior limits to describe asymptotes, and apply the Intermediate Value Theorem to justify an interval conclusion.

Common slips to watch for

• Treating a function's value at a point as proof of its limit, or assuming a limit exists because nearby values look close on only one side.
• Canceling a factor without checking the original domain or whether the limit is being evaluated at the canceled point.
• Claiming continuity from a graph's appearance without checking the function value and two-sided limit.
• Using the Intermediate Value Theorem without establishing continuity on the whole interval or without verifying the target value lies between endpoint values.

What the exam asks for

• Move between average rates, difference quotients, and the derivative as an instantaneous rate at a point.
• Use derivative notation consistently and connect derivative values to tangent-line slopes in symbolic, graphical, and numerical settings.
• Apply basic derivative rules accurately to sums, products, quotients, powers, and the listed elementary functions.
• Distinguish differentiability from continuity, explaining why a derivative can fail to exist even when the function is continuous.

Common slips to watch for

• Confusing an average rate over an interval with the instantaneous rate at its endpoint.
• Dropping a factor or misapplying the product or quotient rule, especially when both numerator and denominator vary.
• Using a derivative formula outside its domain, or confusing the derivative of sine with that of cosine.
• Assuming continuity guarantees differentiability at corners, cusps, or other points where a single tangent slope does not exist.

What the exam asks for

• Apply the chain rule through nested functions, identifying the outer and inner derivatives before combining them.
• Differentiate implicit relationships by treating dependent variables as functions of the independent variable and solving for the requested derivative.
• Use inverse-function relationships to find inverse derivatives at corresponding input-output points, including inverse trigonometric derivatives.
• Select an efficient rule sequence for a given expression and repeatedly differentiate when a higher-order derivative is required.

Common slips to watch for

• Forgetting to multiply by the derivative of the inner function in a composite expression.
• Differentiating an implicit dependent-variable term as though it were an independent variable, omitting its derivative.
• Evaluating an inverse derivative at the original input instead of the corresponding point on the inverse.
• Applying an inverse-trigonometric derivative formula without its required inner derivative or domain conditions.

What the exam asks for

• Interpret derivatives in applied settings with units that state the changing quantity per unit of the independent variable.
• For straight-line motion, connect position, velocity, and acceleration and distinguish signed velocity from speed.
• Set up related-rates equations by relating quantities first, then differentiate with respect to their common time variable and use the given instant's values.
• Use tangent-line linearization for local estimates and judge overestimation or underestimation; apply L’Hospital’s Rule only to indeterminate forms.

Common slips to watch for

• Reporting a rate without units, or reversing the dependent and independent quantities in the units.
• Substituting changing quantities before differentiating a related-rates equation, or mixing rates evaluated at different instants.
• Confusing speed with signed velocity or acceleration with a change in position.
• Using a linear approximation far from its base point, misjudging its error direction, or applying L’Hospital’s Rule to a non-indeterminate form.

What the exam asks for

• Use derivative signs, critical points, endpoints, and continuity or differentiability conditions to justify behavior and extrema.
• Connect graphs of a function and its derivatives to intervals of increase, decrease, concavity, and inflection.
• Set up optimization from a contextual constraint, identify the feasible interval, and interpret the resulting extremum.
• For implicit relations, use derivative information to locate and classify points where the derivative is zero or undefined.

Common slips to watch for

• Claiming a global extremum from a local test without checking endpoints and all critical points.
• Treating a point where a derivative is zero or undefined as automatically an extremum or inflection point.
• Applying the Mean Value or Extreme Value Theorem without verifying its required conditions.
• Optimizing an expression without stating the feasible domain or interpreting the result in context.

What the exam asks for

• Interpret signed area as net accumulated change, tracking the sign of the rate and the units of the result.
• Translate between sums, definite integrals, and accumulation functions; use the Fundamental Theorem to connect an integrand to accumulated change.
• Estimate integrals from graphs, tables, or partitions, and distinguish approximation from exact evaluation.
• Choose and execute an antiderivative method that fits the integrand, including substitution, parts, partial fractions, and limits for improper integrals.

Common slips to watch for

• Confusing signed area or net change with total geometric area or total amount accumulated.
• Dropping the constant of integration for indefinite integrals or failing to change limits after substitution.
• Using a Riemann sum with inconsistent sample points, widths, or indexing.
• Evaluating an improper integral as an ordinary antiderivative without checking the limiting behavior.

What the exam asks for

• Translate descriptions of rates into differential equations and verify proposed solutions by substitution and differentiation.
• Read slope fields for local direction, equilibrium behavior, and plausible solution curves.
• Apply Euler steps or separation of variables carefully, using initial values and respecting domain restrictions.
• Interpret exponential and logistic models through their parameters, initial conditions, carrying capacity, and rate of change.

Common slips to watch for

• Reversing dependent and independent variables when translating a rate description.
• Treating a slope field as a set of solution curves rather than local tangent directions.
• Losing the arbitrary constant during separation or applying an initial condition before finding the general form.
• Ignoring domain restrictions or misidentifying logistic carrying capacity and fastest growth.

What the exam asks for

• Set up definite integrals that match the geometry or context, including correct bounds and units.
• Distinguish displacement from total distance by accounting for velocity sign changes; interpret accumulated change in context.
• For area and volume, identify which curve is on top or outside and split integrals where the ordering or cross-section changes.
• Build cross-sectional or revolution integrands from the appropriate shape formula, radius, or washer dimensions.

Common slips to watch for

• Using signed displacement when the task asks for total distance, or omitting units from an interpretation.
• Reversing top-minus-bottom or outer-minus-inner dimensions, producing a negative area or incorrect volume.
• Failing to split the integral at intersections or at changes in the region's boundary.
• Using a cross-section formula or radius measured from the wrong axis of rotation.

What the exam asks for

• Connect parametric and polar representations to slopes, second derivatives, and curve behavior.
• Set up definite integrals for parametric arc length and polar areas, choosing appropriate limits.
• Differentiate and integrate vector components to relate position, velocity, acceleration, and displacement.
• Interpret planar motion by distinguishing velocity, speed, acceleration, displacement, and total distance.

Common slips to watch for

• Using the reciprocal or omitting the chain-rule factor when finding a parametric slope or second derivative.
• Confusing speed with a signed velocity component, or displacement with total distance traveled.
• Using the wrong angular interval or subtracting polar radii in the wrong order for an area between curves.
• Omitting an initial-condition constant when integrating a vector rate, or leaving an answer without suitable units.

What the exam asks for

• Choose convergence tests by checking their hypotheses, then distinguish convergence from divergence and absolute from conditional convergence.
• Use geometric, harmonic, and p-series behavior as reference points for comparison and limit comparison arguments.
• Build Taylor and Maclaurin polynomials from derivatives or known series, and use them to approximate nearby values.
• Find power-series radii and intervals by testing both endpoints; use alternating-series and Lagrange bounds to assess approximation error.

Common slips to watch for

• Applying a convergence test without verifying its conditions, or treating a failed test as proof of convergence.
• Checking only the radius of convergence and forgetting that each endpoint needs its own test.
• Misclassifying conditional convergence by testing the original alternating series but not its absolute-value series.
• Using the wrong derivative order or center in a Taylor polynomial, or omitting the required maximum-derivative bound.