Calculus BC

Browse readings and complete checks by topic.

• Introducing Calculus: Can Change Occur at an Instant?Reading · Functions, graphs, and the language of change

  Explain instantaneous change as a limit of average rates, keeping amount and rate units separate.

• Defining Limits and Using Limit NotationReading · Limits and continuity

  Distinguish a nearby limit from the value at the target, using one-sided and two-sided notation.

• Estimating Limit Values from GraphsReading · Limits and continuity

  Read nearby graph behavior, holes, jumps, oscillation, and unbounded output without mistaking a picture for proof.

• Estimating Limit Values from TablesReading · Limits and continuity

  Estimate a limit from both sides of a table and explain what finite samples cannot establish.

• Determining Limits Using Algebraic Properties of LimitsReading · Limits and continuity

  Apply limit laws and composition only when their denominator and continuity conditions hold.

• Determining Limits Using Algebraic ManipulationReading · Limits and continuity

  Use factors, conjugates, and trigonometric identities to preserve nearby values while resolving a limit.

• Selecting Procedures for Determining LimitsReading · Limits and continuity

  Select a limit procedure from the expression and direction of approach before calculating.

• Determining Limits Using the Squeeze TheoremReading · Limits and continuity

  Justify a trapping inequality and matching outer limits in a squeeze argument.

• Connecting Multiple Representations of LimitsReading · Limits and continuity

  Translate one limit among a formula, graph, table, and verbal statement with consistent units.

• Exploring Types of DiscontinuitiesReading · Limits and continuity

  Identify the actual obstruction to continuity and decide whether one replacement value can repair it.

• Defining Continuity at a PointReading · Limits and continuity

  Check that the function value, finite limit, and equality of the two all hold at a point.

• Confirming Continuity over an IntervalReading · Limits and continuity

  Use the domain and one-sided endpoint conditions to state intervals of continuity.

• Removing DiscontinuitiesReading · Limits and continuity

  Repair a removable discontinuity or choose a piecewise parameter by matching both sides and the target value.

• Connecting Infinite Limits and Vertical AsymptotesReading · Limits and continuity

  Use one-sided sign analysis to distinguish vertical asymptotes from removable holes.

• Connecting Limits at Infinity and Horizontal AsymptotesReading · Limits and continuity

  Compare end behavior and relative growth using justified limits, rather than a finite plotting window.

• Working with the Intermediate Value Theorem (IVT)Reading · Limits and continuity

  Use continuity and bracketing to justify an intermediate output, without claiming a unique solution.

• Using L’Hospital’s Rule for Determining Limits of Indeterminate FormsReading · Differentiation tools

  Check an indeterminate quotient and theorem hypotheses before comparing numerator and denominator derivatives.

• Defining Average and Instantaneous Rates of Change at a PointReading · Derivatives as rates of change

  Build a finite instantaneous rate from either form of the limiting secant quotient.

• Defining the Derivative of a Function and Using Derivative NotationReading · Derivatives as rates of change

  Connect the derivative function, derivative notation, and a tangent line at a specified point.

• Estimating Derivatives of a Function at a PointReading · Derivatives as rates of change

  Estimate a derivative from nearby data or a tangent, and retain approximation and unit labels.

• Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not ExistReading · Derivatives as rates of change

  Check continuity and one-sided slopes to distinguish a smooth point, corner, cusp, or vertical tangent.

• Interpreting the Meaning of the Derivative in ContextReading · Derivatives as rates of change

  Interpret a signed derivative with the quantity, input value, and output-per-input units.

• Straight-Line Motion: Connecting Position, Velocity, and AccelerationReading · Modeling with derivatives

  Calculate and compare position, velocity, acceleration, direction, and changing speed using sign evidence.

• Rates of Change in Applied Contexts Other Than MotionReading · Modeling with derivatives

  Interpret marginal and other contextual rates, separating instantaneous change from a finite interval change.

• Connecting Position, Velocity, and Acceleration of Functions Using IntegralsReading · Motion, area, and volume

  Recover position and velocity with initial values, and separate displacement from distance using velocity signs.

• Applying the Power RuleReading · Differentiation tools

  Apply the power rule to integer, negative, and fractional powers on the valid real domain.

• Derivative Rules: Constant, Sum, Difference, and Constant MultipleReading · Differentiation tools

  Differentiate constants and linear combinations, preserving signs before point evaluation.

• Derivatives of cos x, sin x, e^x, and ln xReading · Differentiation tools

  Use familiar trigonometric, exponential, and logarithmic derivatives, including disguised derivative limits.

• The Product RuleReading · Differentiation tools

  Account for both changing factors in a product, whether values come from formulas or data.

• The Quotient RuleReading · Differentiation tools

  Choose a quotient rule or equivalent rewrite and preserve the original denominator restrictions.

• Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant FunctionsReading · Differentiation tools

  Derive reciprocal-trigonometric rules from identities and track signs, radians, and domains.

• The Chain RuleReading · Differentiation tools

  Identify composition layers, evaluate each rate at the correct input, and include every inside factor.

• Implicit DifferentiationReading · Differentiation tools

  Differentiate an implicit relation locally, checking the point and any division used to find its slope.

• Differentiating Inverse FunctionsReading · Differentiation tools

  Find the corresponding original input before taking a nonzero inverse slope reciprocal.

• Differentiating Inverse Trigonometric FunctionsReading · Differentiation tools

  Derive inverse-trigonometric rates on stated branches and distinguish function from derivative domains.

• Selecting Procedures for Calculating DerivativesReading · Differentiation tools

  Choose and combine rules from the outer structure, rather than treating a mixed expression as one memorized pattern.

• Calculating Higher-Order DerivativesReading · Differentiation tools

  Differentiate rates again while preserving chain rules, implicit dependence, notation, and units.

• Using the Mean Value TheoremReading · Graph shape and local approximation

  Verify the Mean Value Theorem hypotheses before linking an average rate to an interior instantaneous rate.

• Extreme Value Theorem, Global Versus Local Extrema, and Critical PointsReading · Graph shape and local approximation

  Distinguish attained absolute extrema, local extrema, and interior critical numbers from endpoints.

• Determining Intervals on Which a Function Is Increasing or DecreasingReading · Graph shape and local approximation

  Use a complete derivative sign chart to justify increasing and decreasing intervals.

• Using the First Derivative Test to Determine Relative (Local) ExtremaReading · Graph shape and local approximation

  Use one-sided derivative sign changes to classify a local extremum, including a continuous corner.

• Using the Candidates Test to Determine Absolute (Global) ExtremaReading · Graph shape and local approximation

  Compare original-function values at all endpoints and interior critical numbers on a closed interval.

• Determining Concavity of Functions over Their DomainsReading · Graph shape and local approximation

  Use changing slope to determine concavity and verify an actual concavity change at a point of the graph.

• Using the Second Derivative Test to Determine ExtremaReading · Graph shape and local approximation

  Apply the second derivative test at stationary points and use a different argument when it is inconclusive.

• Sketching Graphs of Functions and Their DerivativesReading · Graph shape and local approximation

  Build a labeled calculus sketch from slope signs, concavity, domain behavior, and a function anchor.

• Connecting a Function, Its First Derivative, and Its Second DerivativeReading · Graph shape and local approximation

  Connect heights and slopes across a function and its first two derivatives without conflating their outputs.

• Approximating Values of a Function Using Local Linearity and LinearizationReading · Graph shape and local approximation

  Construct a nearby tangent approximation and use interval concavity to justify its error direction.

• Exploring Behaviors of Implicit RelationsReading · Graph shape and local approximation

  Analyze local implicit branches, tangent candidates, and second derivatives with point and domain checks.

• Introduction to Related RatesReading · Modeling with derivatives

  Distinguish fixed constraints from snapshots, then differentiate all changing quantities with respect to the shared clock.

• Solving Related Rates ProblemsReading · Modeling with derivatives

  Build a geometric relationship, differentiate before inserting a snapshot, and interpret the resulting signed rate.

• Introduction to Optimization ProblemsReading · Modeling with derivatives

  Name an objective, constraint, and feasible domain before choosing an optimization procedure.

• Solving Optimization ProblemsReading · Modeling with derivatives

  Justify an optimum over the whole feasible domain and report the requested setting, output, and units.

• Exploring Accumulations of ChangeReading · Accumulation and definite integrals

  Accumulate signed rate contributions and distinguish net change from total activity and final amount.

• Approximating Areas with Riemann SumsReading · Accumulation and definite integrals

  Use each interval width in rectangular or trapezoidal approximations and justify any error direction.

• Riemann Sums, Summation Notation, and Definite Integral NotationReading · Accumulation and definite integrals

  Translate partitions and sampled inputs between Riemann sums and definite integral notation.

• The Fundamental Theorem of Calculus and Accumulation FunctionsReading · Accumulation and definite integrals

  Differentiate an accumulation function using continuity, moving endpoints, and the chain rule.

• Interpreting the Behavior of Accumulation Functions Involving AreaReading · Accumulation and definite integrals

  Use the integrand signs and trends to analyze an accumulation function and compare its candidate values.

• Applying Properties of Definite IntegralsReading · Accumulation and definite integrals

  Use interval addition, reversal, linearity, symmetry, and bounded discontinuity properties before calculating.

• The Fundamental Theorem of Calculus and Definite IntegralsReading · Accumulation and definite integrals

  Check an antiderivative and continuity, then evaluate signed accumulation as an endpoint difference.

• Finding Antiderivatives and Indefinite Integrals: Basic Rules and NotationReading · Antiderivatives and integration tools

  Reverse familiar derivative rules to find antiderivative families on their valid intervals.

• Integrating Using SubstitutionReading · Antiderivatives and integration tools

  Match an inside expression and its differential, changing both bounds or returning to the original variable.

• Integrating Functions Using Long Division and Completing the SquareReading · Antiderivatives and integration tools

  Use long division or completing the square to expose a basic integral without changing its domain.

• Integrating Using Integration by PartsReading · Antiderivatives and integration tools

  Reverse the product rule with a justified parts choice, compatible bounds, and a derivative check.

• Integrating Using Linear Partial FractionsReading · Antiderivatives and integration tools

  Decompose distinct nonrepeating linear factors, verify the algebra, and preserve excluded inputs.

• Evaluating Improper IntegralsReading · Antiderivatives and integration tools

  Define improper integrals using separate finite one-sided limits, rather than cancellation across a pole.

• Selecting Techniques for AntidifferentiationReading · Antiderivatives and integration tools

  Combine the shared AB method-selection lesson with its genuine BC addition: parts, distinct linear partial fractions, and interval limits.

• Modeling Situations with Differential EquationsReading · Differential equations and growth

  Translate a rate statement into a differential model with variables, signs, units, and an initial condition.

• Verifying Solutions for Differential EquationsReading · Differential equations and growth

  Differentiate and substitute throughout the domain, checking an initial condition separately.

• Sketching Slope FieldsReading · Differential equations and growth

  Compute and draw local slope-field directions from both coordinates, leaving undefined locations unfilled.

• Reasoning Using Slope FieldsReading · Differential equations and growth

  Reason about a solution family, equilibrium, direction, and concavity from a slope field and equation.

• Approximating Solutions Using Euler’s MethodReading · Differential equations and growth

  Build Euler estimates step by step from the current point and distinguish an approximation from an exact solution.

• Finding General Solutions Using Separation of VariablesReading · Differential equations and growth

  Separate variable factors, check lost equilibria, and verify the resulting solution family.

• Finding Particular Solutions Using Initial Conditions and Separation of VariablesReading · Differential equations and growth

  Use an initial condition to choose a solution, branch, and connected interval of validity.

• Exponential Models with Differential EquationsReading · Differential equations and growth

  Fit and interpret exponential growth or decay models, separating a continuous relative rate from interval percent change.

• Logistic Models with Differential EquationsReading · Differential equations and growth

  Use a logistic sign chart and initial population to distinguish capacity, equilibrium, and future growth-rate behavior.

• Finding the Average Value of a Function on an IntervalReading · Motion, area, and volume

  Use an integral average and its units, rather than a secant rate or an unweighted mean of unequal-time samples.

• Using Accumulation Functions and Definite Integrals in Applied ContextsReading · Motion, area, and volume

  Write initial amount plus accumulated net rate, then justify inventory values, extrema, and targets in context.

• Finding the Area Between Curves Expressed as Functions of xReading · Motion, area, and volume

  Find intersections and integrate a justified top-minus-bottom vertical strip.

• Finding the Area Between Curves Expressed as Functions of yReading · Motion, area, and volume

  Choose horizontal strips with correct branches, right-minus-left boundaries, and matching y-bounds.

• Finding the Area Between Curves That Intersect at More Than Two PointsReading · Motion, area, and volume

  Split at relevant crossings so positive area does not disappear through signed cancellation.

• Volumes with Cross Sections: Squares and RectanglesReading · Motion, area, and volume

  Turn a stated square or rectangular cross section into an area function before integrating volume.

• Volumes with Cross Sections: Triangles and SemicirclesReading · Motion, area, and volume

  Use the specified triangle, semicircle, or circular geometry, distinguishing a supplied radius from a diameter.

• Volume with Disc Method: Revolving Around the x- or y-AxisReading · Motion, area, and volume

  Choose perpendicular disc slices and measure a radius from a coordinate axis.

• Volume with Disc Method: Revolving Around Other AxesReading · Motion, area, and volume

  Measure disc radii as distances from the stated shifted axis, checking whether the region reaches it.

• Volume with Washer Method: Revolving Around the x- or y-AxisReading · Motion, area, and volume

  Identify nonnegative outer and inner radii and subtract their squared disc areas.

• Volume with Washer Method: Revolving Around Other AxesReading · Motion, area, and volume

  Measure both washer radii from the same shifted axis and keep the slice variable and bounds consistent.

• The Arc Length of a Smooth, Planar Curve and Distance TraveledReading · Motion, area, and volume

  Build functional-curve arc length from small distances, checking smoothness and splitting at corners.

• Defining and Differentiating Parametric EquationsReading · Parametric curves and vector motion

  Use one parameter for point, direction, and tangent rates, checking both tangent conditions.

• Second Derivatives of Parametric EquationsReading · Parametric curves and vector motion

  Differentiate the tangent slope and divide by the horizontal rate again to analyze local concavity.

• Finding Arc Lengths of Curves Given by Parametric EquationsReading · Parametric curves and vector motion

  Accumulate parameter speed and distinguish a single curve trace from repeated traversals.

• Defining and Differentiating Vector-Valued FunctionsReading · Parametric curves and vector motion

  Differentiate vector components, separating vector velocity and acceleration from scalar speed.

• Integrating Vector-Valued FunctionsReading · Parametric curves and vector motion

  Integrate components with separate constants and the appropriate initial velocity and position.

• Solving Motion Problems Using Parametric and Vector-Valued FunctionsReading · Parametric curves and vector motion

  Compare vector displacement and scalar distance, and use velocity and acceleration together for changing speed.

• Defining Polar Coordinates and Differentiating in Polar FormReading · Polar coordinates and regions

  Interpret directed polar radii and use coordinate rates for tangent and second-derivative reasoning.

• Find the Area of a Polar Region or the Area Bounded by a Single Polar CurveReading · Polar coordinates and regions

  Choose a once-traced polar region and accumulate sectors with the one-half squared-radius factor.

• Finding the Area of the Region Bounded by Two Polar CurvesReading · Polar coordinates and regions

  Identify the intended common-ray region, changing boundaries, and correct squared-radius differences.

• Defining Convergent and Divergent Infinite SeriesReading · Sequences, series, and convergence

  Distinguish a term sequence from partial sums and use their finite limit to define a series sum.

• Working with Geometric SeriesReading · Sequences, series, and convergence

  Use the actual first included term and ratio, verifying convergence before applying a geometric sum.

• The nth Term Test for DivergenceReading · Sequences, series, and convergence

  Use a nonzero or missing term limit to prove divergence and recognize that a zero limit is inconclusive.

• Integral Test for ConvergenceReading · Sequences, series, and convergence

  Verify a positive, continuous, decreasing tail before using an improper integral to decide convergence.

• Harmonic Series and p-SeriesReading · Sequences, series, and convergence

  Use harmonic and p-series benchmarks with the correct exponent and strict convergence threshold.

• Comparison Tests for ConvergenceReading · Sequences, series, and convergence

  Choose a positive comparison with the useful inequality direction or a finite positive ratio limit.

• Alternating Series Test for ConvergenceReading · Sequences, series, and convergence

  Check eventually decreasing magnitudes and a zero limit before using alternating-series convergence.

• Ratio Test for ConvergenceReading · Sequences, series, and convergence

  Use absolute successive-term ratios and switch tests when the limit is one or does not exist.

• Determining Absolute or Conditional ConvergenceReading · Sequences, series, and convergence

  Test both signed and magnitude sums to distinguish absolute, conditional, and divergent behavior.

• Alternating Series Error BoundReading · Sequences, series, and convergence

  Use the first omitted alternating magnitude to give a justified error bound and interval for the sum.

• Radius and Interval of Convergence of Power SeriesReading · Power series and Taylor approximation

  Determine radius and open convergence interval, then test each endpoint of the original series separately.

• Finding Taylor Polynomial Approximations of FunctionsReading · Power series and Taylor approximation

  Build a center-specific Taylor polynomial from derivative values and label its value as an approximation.

• Lagrange Error BoundReading · Power series and Taylor approximation

  Bound the next derivative across the whole interval before stating a Lagrange remainder guarantee.

• Finding Taylor or Maclaurin Series for a FunctionReading · Power series and Taylor approximation

  Build Taylor or Maclaurin series and justify function representation with a vanishing remainder.

• Representing Functions as Power SeriesReading · Power series and Taylor approximation

  Derive function representations by valid series operations, fixing constants and rechecking endpoints.

• Explain a cart’s instantaneous rateCheck understanding · Functions, graphs, and the language of change

  Form and simplify a variable-interval rate, then take its limit with motion units.

  Checks instantaneous change; it does not assess the whole prerequisite review.

  Separate worked reasoning and criteria →
• Connect four representations of a limitCheck understanding · Limits and continuity

  Relate a removable-hole formula to nearby values, a graph feature, limit notation, and a verbal conclusion.

  Separate worked reasoning and criteria →
• Test all three continuity conditionsCheck understanding · Limits and continuity

  Identify the actual target value, the two-sided limit, and whether they agree.

  Separate worked reasoning and criteria →
• Justify a root without locating itCheck understanding · Limits and continuity

  State continuity and bracket zero with endpoint values to justify existence in the open interval.

  Separate worked reasoning and criteria →
• Estimate a rate from position samplesCheck understanding · Derivatives as rates of change

  Use a centered difference and explain that the result is an estimate in meters per second.

  This numerical estimate does not establish the exact derivative or assess every derivative representation.

  Separate worked reasoning and criteria →
• Explain a continuous cornerCheck understanding · Derivatives as rates of change

  Compare the value and limit, then use unequal one-sided slopes to decide differentiability.

  Separate worked reasoning and criteria →
• Interpret a decreasing amount’s rateCheck understanding · Derivatives as rates of change

  Name the instant, sign, quantity, and rate units, and explain why the amount itself remains unknown.

  Separate worked reasoning and criteria →
• Choose rules for a mixed derivativeCheck understanding · Differentiation tools

  Identify and apply the power, product, sine, and chain rules with every changing factor included.

  Separate worked reasoning and criteria →
• Find an inverse rate from a tableCheck understanding · Differentiation tools

  Choose the original input matching the inverse input before taking the reciprocal slope.

  Separate worked reasoning and criteria →
• Justify repeated L’Hospital stepsCheck understanding · Differentiation tools

  Identify the indeterminate form and verify the conditions before each numerator-and-denominator differentiation.

  Separate worked reasoning and criteria →
• Reject an invalid L’Hospital stepCheck understanding · Differentiation tools

  Explain why a one-over-zero form fails the hypothesis and determine the one-sided unbounded behavior.

  Checks when the rule is invalid; it does not assess a qualifying indeterminate quotient.

  Separate worked reasoning and criteria →
• Connect an average rate to an interior rateCheck understanding · Graph shape and local approximation

  Verify both MVT hypotheses and find the matching derivative inside the open interval.

  Separate worked reasoning and criteria →
• Build a function sketch from its derivativeCheck understanding · Graph shape and local approximation

  Use derivative height and slope separately for extrema, concavity, and an anchored inflection point.

  Separate worked reasoning and criteria →
• Estimate a nearby value and its error directionCheck understanding · Graph shape and local approximation

  Build a tangent approximation from supplied data and use interval concavity to justify an underestimate.

  Separate worked reasoning and criteria →
• Model a rising depth in a conical tankCheck understanding · Modeling with derivatives

  Use similar triangles to link changing water dimensions, differentiate before substituting, and interpret the depth rate.

  Separate worked reasoning and criteria →
• Justify the least-material cylinderCheck understanding · Modeling with derivatives

  Use the volume constraint, positive-radius domain, and complete derivative signs to justify dimensions and minimum area.

  Separate worked reasoning and criteria →
• Weight an unequal partition correctlyCheck understanding · Accumulation and definite integrals

  Compute left, right, and trapezoidal sums with each width and explain why an unweighted sample mean gives a different result.

  This table-based task does not assess the full graphical, analytic, and verbal range of integral approximations.

  Separate worked reasoning and criteria →
• Differentiate a moving-endpoint accumulationCheck understanding · Accumulation and definite integrals

  Apply continuity, the Fundamental Theorem, and the endpoint’s chain factor without evaluating the original integral.

  The integral is supplied; this checks differentiation rather than constructing an integral representation.

  Separate worked reasoning and criteria →
• Construct and sketch accumulation from a rate graphCheck understanding · Accumulation and definite integrals

  Build net-change and amount integrals, use signed areas and endpoint comparisons, then justify and draw the accumulation curve.

  Uses the supplied graph; numerical, analytic, and verbal construction contexts require separate checks.

  Separate worked reasoning and criteria →
• Check the interval and integrate after divisionCheck understanding · Antiderivatives and integration tools

  Verify the denominator on the bounds, use long division, and evaluate the endpoint difference.

  Separate worked reasoning and criteria →
• Select a method and justify improper convergenceCheck understanding · Antiderivatives and integration tools

  Decompose the integrand on finite intervals, then take an endpoint limit to establish a finite value.

  Separate worked reasoning and criteria →
• Solve an initial-value problem with a poleCheck understanding · Differential equations and growth

  Separate variables, use the initial condition, and state the largest connected solution interval containing it.

  Separate worked reasoning and criteria →
• Fit and interpret an exponential modelCheck understanding · Differential equations and growth

  Determine the proportional rate from two observations, retain inverse-time units, and predict the later amount.

  Separate worked reasoning and criteria →
• Track two Euler stepsCheck understanding · Differential equations and growth

  Evaluate the slope at each current estimated point and report the result as an approximation.

  Separate worked reasoning and criteria →
• Compare three logistic starting statesCheck understanding · Differential equations and growth

  Distinguish a trajectory that passes through fastest growth, one whose growth rate falls from the start, and the zero equilibrium.

  Separate worked reasoning and criteria →
• Separate final position from total distanceCheck understanding · Motion, area, and volume

  Add signed velocity accumulation to the initial position and split absolute-speed accumulation at a reversal.

  Separate worked reasoning and criteria →
• Keep two area lobes from cancellingCheck understanding · Motion, area, and volume

  Find all crossings, justify the upper curve on each piece, and accumulate positive geometric area.

  Separate worked reasoning and criteria →
• Measure washer radii from a shifted axisCheck understanding · Motion, area, and volume

  Choose the farther and nearer boundaries, square both distances separately, and integrate positive volume.

  Separate worked reasoning and criteria →
• Find length across a cornerCheck understanding · Motion, area, and volume

  Apply the smooth arc-length formula to each straight piece and explain why the corner requires a split.

  Separate worked reasoning and criteria →
• Distinguish velocity, speed, slope, and positionCheck understanding · Parametric curves and vector motion

  Use exact coordinate rates for a velocity vector, magnitude, and tangent slope, and explain why position remains unknown.

  Separate worked reasoning and criteria →
• Recover a planar position from accelerationCheck understanding · Parametric curves and vector motion

  Integrate both components twice, using separate initial velocity and position conditions.

  Separate worked reasoning and criteria →
• Account for repeated tracingCheck understanding · Parametric curves and vector motion

  Compare distance over two parameter laps with the length of the distinct geometric circle.

  Separate worked reasoning and criteria →
• Interpret a negative polar radius and its tangentCheck understanding · Polar coordinates and regions

  Convert the signed-radius point, find an equivalent positive-radius pair, and check both coordinate tangent rates.

  Separate worked reasoning and criteria →
• Explain changing polar overlap boundariesCheck understanding · Polar coordinates and regions

  Find and restrict intersection angles, choose the smaller radial upper boundary on each piece, and justify the area integral.

  Separate worked reasoning and criteria →
• Repair a zero-term convergence claimCheck understanding · Sequences, series, and convergence

  Explain why a zero term limit is only necessary and use the harmonic benchmark to classify the series.

  Separate worked reasoning and criteria →
• Reject an inapplicable alternating testCheck understanding · Sequences, series, and convergence

  Check the magnitude behavior, then use finite partial-sum decomposition to make a valid divergence argument.

  Separate worked reasoning and criteria →
• Trap an alternating sumCheck understanding · Sequences, series, and convergence

  Verify decreasing-to-zero magnitudes, use the first omitted term, and state both an error bound and trapping interval.

  Separate worked reasoning and criteria →
• Justify a Taylor approximation’s error boundCheck understanding · Power series and Taylor approximation

  Build the degree-two polynomial and apply the supplied derivative bound over the entire center-to-target interval.

  Separate worked reasoning and criteria →
• Test both convergence endpointsCheck understanding · Power series and Taylor approximation

  Determine the radius from a ratio inequality, then classify each endpoint in the original series.

  Uses one given power series with a finite positive radius. It does not ask students to state the general definition or decide center-only or infinite-radius cases.

  Separate worked reasoning and criteria →
• Derive and check an integrated seriesCheck understanding · Power series and Taylor approximation

  Integrate a geometric series, fix its constant, justify endpoint representation, and compare a four-term approximation with its alternating error bound.

  The task derives an integrated series and interval; it does not explicitly ask students to compare its radius with the starting series.

  Separate worked reasoning and criteria →
• Compare center-only, finite, and infinite radiiCheck understanding · Power series and Taylor approximation

  Evaluate the center separately, decide each convergence set, and draw number lines with explicit boundary inclusion.

  Separate worked reasoning and criteria →