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What the exam asks for

• Estimate limits from graphs and tables while separating nearby behavior from the function value at the point.
• Choose algebraic limit methods that fit the expression, including equivalent forms and the squeeze theorem.
• Distinguish removable, jump, and unbounded behavior; use continuity conditions to support conclusions.
• Interpret limits at infinity, vertical and horizontal asymptotes, and the conclusion guaranteed by the IVT.

Common slips to watch for

• Treating a plotted point or table entry as the limit without checking nearby values.
• Assuming a two-sided limit exists when one-sided behavior disagrees, or overlooking an infinite limit.
• Claiming continuity from a limit alone without checking the function value, or using the IVT without its continuity condition.

What the exam asks for

• Connect average rates over intervals to the derivative as a limiting difference quotient.
• Use derivative notation and the derivative at a point to determine tangent-line equations.
• Apply basic derivative rules accurately, including product, quotient, and trigonometric-function rules.
• Read graphical or tabular slope information as an estimate of a derivative and distinguish differentiability from continuity.

Common slips to watch for

• Reversing the order of subtraction in a difference quotient or using an interval length inconsistent with it.
• Leaving out a factor in the product or quotient rule, or misapplying a trigonometric derivative sign.
• Assuming continuity guarantees differentiability at a point, or confusing the derivative value with the function value.

What the exam asks for

• Recognize nested functions and apply the chain rule through every composition layer.
• Differentiate implicit relations while treating dependent variables appropriately and solving for the requested derivative.
• Use inverse-function derivative relationships and inverse trigonometric derivative rules with their domain conditions.
• Select a suitable rule from the expression’s structure and calculate successive derivatives when needed.

Common slips to watch for

• Applying the outer derivative but omitting the derivative of the inner function.
• Differentiating an implicit term without including the dependent variable’s derivative, or failing to isolate the requested derivative.
• Using an inverse-function or inverse-trigonometric formula outside its applicable conditions, or stopping before the requested derivative order.

What the exam asks for

• Interpret derivatives as instantaneous rates in motion and other applied settings, including the meaning and units of the rate.
• Relate position, velocity, and acceleration through successive derivatives and distinguish velocity from speed.
• Set up related-rate equations by linking quantities, differentiating with respect to a common variable, and using known values at the appropriate stage.
• Use tangent-line linearization near a specified point and apply L’Hospital’s Rule only to eligible indeterminate forms.

Common slips to watch for

• Reporting a rate without units or describing a signed velocity as speed.
• Substituting changing quantities too early in a related-rates equation or differentiating them with respect to different variables.
• Using a tangent-line estimate far from its point of tangency, or applying L’Hospital’s Rule before confirming an eligible indeterminate form.

What the exam asks for

• Use derivative signs to establish increasing and decreasing intervals, local extrema, concavity, and inflection points.
• Justify global extrema with endpoint and critical-point comparisons, and distinguish local from absolute conclusions.
• Connect function, first-derivative, and second-derivative graphs to explain behavior and sketch plausible curves.
• Set up optimization quantities and constraints, then interpret the resulting optimum in its applied context.
• Use derivative evidence for implicit relations and apply the Mean Value and Extreme Value Theorems with their conditions.

Common slips to watch for

• Claiming an extremum from a critical point without checking derivative sign changes or comparing endpoints.
• Confusing a zero of the second derivative with an inflection point without verifying a concavity change.
• Applying a theorem without checking its continuity or differentiability conditions on the stated interval.
• Reporting an optimization value without relating it to the original variable, constraints, or context.

What the exam asks for

• Interpret definite integrals as accumulated change, including signed area and the contextual meaning of a rate graph.
• Approximate integrals with Riemann or trapezoidal sums and determine whether the estimate overstates or understates the value.
• Move between sums, limits, integral notation, and accumulation functions; use the Fundamental Theorem to differentiate or evaluate integrals.
• Apply integral properties and geometric area to simplify evaluations, and choose suitable antiderivatives or substitutions.
• Track bounds, signs, and constants when rearranging integrands or changing variables.

Common slips to watch for

• Treating all area below an axis as positive accumulated change, or confusing net change with total distance.
• Using inconsistent sample points or widths in a sum, or reversing an overestimate and underestimate.
• Applying the Fundamental Theorem with mismatched bounds or differentiating an accumulation function without accounting for its inner function.
• Forgetting the arbitrary constant in an indefinite integral or failing to adjust bounds after substitution.

What the exam asks for

• Translate verbal rates of change into differential equations with variables and units clearly identified.
• Read and sketch slope fields by matching local slope patterns, then infer qualitative solution behavior.
• Separate variables and integrate correctly to obtain general solutions, then use initial data to determine particular solutions.
• Interpret exponential growth and decay models, including what the solution and parameters mean in context.
• Verify proposed solutions by differentiation and substitution into the original equation.

Common slips to watch for

• Reversing dependent and independent variables or omitting a required factor when separating variables.
• Treating a slope-field segment as a solution curve rather than using the slopes to guide a smooth curve.
• Losing the constant of integration or failing to apply the initial condition to the general solution.
• Checking a proposed solution only at one point instead of substituting it into the differential equation.

What the exam asks for

• Calculate average value and interpret definite integrals as net change in motion or other applied rate settings.
• Set up area integrals with the correct upper-minus-lower or right-minus-left boundary, splitting where curves cross.
• Build volume integrals from cross-sectional area or from disk and washer radii measured from the axis of rotation.
• Choose bounds and integration variables that match the region, including horizontal slices and rotation about shifted axes.
• Interpret results with appropriate units and distinguish a modeled quantity from its rate of change.

Common slips to watch for

• Integrating speed when the task asks for displacement, or ignoring direction changes in motion.
• Subtracting curves in the wrong order or failing to split an area integral where the boundary changes.
• Using a diameter as a radius, omitting a square in cross-sectional area, or reversing inner and outer washer radii.
• Measuring radii from the coordinate axis instead of the stated axis of rotation, or using bounds for the wrong variable.