AP Calculus BC

Units

1. 1

   Limits and continuity

   10 lessons

   1. Can change occur at an instant? Limits and limit notationUse limit notation to describe instantaneous rates and distinguish a limit from a function value.
   2. Estimating limit values from graphs and tablesEstimate limits from graphs and two-sided tables, and explain when an approaching height does not exist.
   3. Determining limits using algebraic properties of limitsCalculate combined, quotient, and composite limits, and state when each method applies.
   4. Determining limits using algebraic manipulationEvaluate 0/0 limits by rewriting expressions and explain why the rewrite preserves the limit.
   5. Determining limits using the squeeze theoremJustify a limit by trapping a function between two functions approaching the same number.
   6. Selecting procedures for limits and connecting multiple representationsChoose a limit procedure, calculate the limit, and distinguish exact results from graphical or numerical evidence.
   7. Types of discontinuities and continuity at a pointClassify breaks and justify continuity at a point using the function’s value and limits.
   8. Continuity over an interval and removing discontinuitiesIdentify intervals of continuity and choose values that repair joins and holes.
   9. Infinite limits, limits at infinity and asymptotesFind and justify vertical and horizontal asymptotes using one-sided limits and limits at infinity.
   10. Working with the intermediate value theoremJustify that a continuous function reaches a specified value inside an interval.
2. 2

   Differentiation: definition and basic rules

   8 lessons

   1. Average and instantaneous rates of change at a pointCalculate an average rate over an interval and use a limit to find an instantaneous rate at a point.
   2. Defining the derivative and using derivative notationFind derivatives from limits, interpret derivative notation, and write tangent-line equations.
   3. Estimating derivatives of a function at a pointEstimate a derivative from a table, a drawn tangent, or a calculator model; justify the method and include units.
   4. Connecting differentiability and continuityDetermine where a derivative exists by checking continuity and finite one-sided slopes, and justify the conclusion.
   5. The power rule and the constant, sum, difference and constant multiple rulesDifferentiate sums of powers, including rewritten roots and reciprocals, and find tangent lines.
   6. Derivatives of cos x, sin x, e^x and ln xDifferentiate combinations of these functions and use derivatives to find slopes and tangent lines.
   7. The product rule and the quotient ruleDifferentiate products and quotients, then check whether a simpler rule applies.
   8. Derivatives of tangent, cotangent, secant and cosecantDifferentiate these four functions and use their derivatives in products, quotients and tangent lines.
3. 3

   Differentiation: composite, implicit and inverse functions

   5 lessons

   1. The chain ruleDifferentiate compositions, including nested and tabulated functions, and use their derivatives to find rates and tangent lines.
   2. Implicit differentiationDifferentiate equations in x and y, then use the derivative to find tangent lines.
   3. Differentiating inverse functionsFind an inverse function’s derivative at a specified input from a formula or table.
   4. Differentiating inverse trigonometric functionsDifferentiate inverse trigonometric functions, including compositions, and use a derivative to find a tangent line.
   5. Selecting procedures for derivatives and higher-order derivativesChoose a differentiation rule from structure, then calculate explicit and implicit second derivatives.
4. 4

   Contextual applications of differentiation

   6 lessons

   1. Interpreting the derivative in context and rates in applied settingsInterpret first and second derivatives with direction and units, and estimate a nearby change.
   2. Straight-line motion: position, velocity and accelerationFind velocity and acceleration from position; justify direction, turns, and speed changes using signs.
   3. Related rates: setting up the relation and differentiating in timeFind a linked rate by differentiating an equation with respect to time, then interpret its sign and units.
   4. Related rates: solving and interpreting (cones, shadows, ladders)Solve complete related-rates problems: remove extra variables with similar triangles, differentiate in time, find the missing value at the instant, substitute last, and state the rate as a sentence with sign and units.
   5. Approximating values using local linearity and linearizationUse a tangent line to estimate a nearby value and justify whether the estimate is high or low.
   6. L'Hospital's rule for indeterminate formsCheck the form of a limit, apply L'Hospital's rule when permitted, and repeat it when needed.
5. 5

   Analytical applications of differentiation

   9 lessons

   1. The mean value theoremCheck the conditions, justify a matching instantaneous rate, and locate it when given a formula.
   2. Extreme value theorem, global and local extrema, and critical pointsFind critical points and justify absolute and interior local extrema on a closed interval.
   3. Increasing and decreasing intervals and the first derivative testUse a graph or formula for f′ to find intervals of increase and decrease and justify relative extrema.
   4. The candidates test for absolute extremaFind and justify absolute extrema on a closed interval, stating each value and where it occurs.
   5. Concavity and the second derivative testJustify concavity, inflection points, and relative extrema using derivatives.
   6. Sketching f' from f and f from f'Sketch f from f′ and justify extrema, concavity, and inflection points using aligned graphs.
   7. Reading f, f' and f'' together: justify extrema, concavity and inflection from a graphRead a graph of f′ and write justified claims about f: relative extrema, concavity, inflection points and absolute extrema, naming the function and the point or interval each time.
   8. Optimization problemsModel a quantity with one variable, justify its optimum, and answer with the requested units.
   9. Exploring behaviours of implicit relationsLocate horizontal and vertical tangents and justify extrema on an implicit curve.
6. 6

   Integration and accumulation of change

   14 lessons

   1. Exploring accumulations of changeUse rate-graph areas to find accumulated change and final amounts, with correct signs and units.
   2. Approximating areas with Riemann sumsCalculate left, right, midpoint, and trapezoidal estimates; justify error direction when the graph’s behavior is known.
   3. Riemann sums, summation notation and definite integral notationConvert between right Riemann-sum limits and definite integrals, identifying the interval, function and variable.
   4. The FTC and accumulation functions: d/dx of an integral, moving upper limitFind accumulated values, derivatives, and behavior from an integrand’s graph, including when the upper limit moves.
   5. Behaviour of an accumulation function from the graph of its rate (FRQ)From the graph of f, find values, extrema, concavity and inflection points of g(x) = ∫ f(t) dt, and justify each claim by naming g′ = f or g″ = f′.
   6. Applying properties of definite integralsCalculate definite integrals from known values and signed geometric areas.
   7. Finding antiderivatives and indefinite integrals: basic rules and notationFind basic indefinite integrals, choose an antiderivative using a given value, and check by differentiation.
   8. The fundamental theorem of calculus and definite integralsEvaluate definite integrals with antiderivatives and use accumulated change to find a final value.
   9. Integrating using substitutionChoose a substitution and evaluate indefinite and definite integrals, adjusting constants and limits correctly.
   10. Integrating using long division and completing the squareRewrite rational integrands using division or a completed square, then find their antiderivatives.
   11. Integrating using integration by partsChoose and apply integration by parts to products, including repeated and cyclic cases.
   12. Integrating using linear partial fractionsDecompose rational functions with distinct linear factors, then find indefinite and definite integrals.
   13. Evaluating improper integralsEvaluate an improper integral by writing its defining limit and deciding whether that limit is finite.
   14. Selecting techniques for antidifferentiation (BC)Choose a method, find an antiderivative, and check it by differentiating.
7. 7

   Differential equations

   6 lessons

   1. Modelling situations with differential equations and verifying solutionsTranslate a rate description into a differential equation, verify a function, and use a condition to choose a solution.
   2. Sketching and reasoning with slope fieldsSketch slope segments, trace a solution through a given point, and recognize patterns and equilibria.
   3. Approximating solutions using Euler's methodEstimate a solution with Euler’s method and justify whether the estimate is above or below the actual value.
   4. General and particular solutions by separation of variablesFind a general solution, use an initial condition to select a particular solution, and state its largest domain.
   5. Exponential models with differential equationsSolve proportional-rate equations, use half-life or doubling time, and interpret a nonzero baseline.
   6. Logistic models with differential equationsFind a logistic model’s limit, fastest-growth population, and solution from an initial value.
8. 8

   Applications of integration

   9 lessons

   1. Average value of a function on an intervalCalculate and interpret average value using a definite integral, and distinguish it from average rate of change.
   2. Position, velocity and acceleration using integralsFind final position, total distance, and final velocity from motion integrals.
   3. Accumulation functions and definite integrals in applied contextsFind an amount from entering and leaving rates, and justify when it is greatest or least.
   4. Area between curves expressed as functions of x and of ySketch a bounded region, choose a slice direction, and calculate its area with correct limits.
   5. Area between curves that intersect at more than two pointsFind crossings, calculate the area of every region, and justify a split-integral or absolute-value setup.
   6. Volumes with cross sections: squares, rectangles, triangles and semicirclesFind volume from a base region and the shape of its perpendicular cross sections.
   7. Volume with the disc method about axes and other linesFind a disc radius, set up its integral, and calculate volume about a horizontal or vertical line.
   8. Volume with the washer method about axes and other linesDraw both radii, set up washer integrals, and calculate volumes about horizontal or vertical lines.
   9. The arc length of a smooth planar curve and distance travelledFind distance along a curve using an arc-length integral, and count every pass along a path.
9. 9

   Parametric equations, polar coordinates and vector-valued functions

   7 lessons

   1. Defining and differentiating parametric equationsLocate points, find slopes and tangent lines, and identify horizontal and vertical tangencies.
   2. Second derivatives of parametric equationsCalculate d²y/dx² and use its sign to determine concavity at a given parameter value.
   3. Arc lengths of curves given by parametric equationsFind distance traveled along a parametric path, including when it retraces itself.
   4. Differentiating and integrating vector-valued functionsFind velocity and acceleration from a position vector; recover position from velocity and an initial position.
   5. Motion problems using parametric and vector-valued functionsCalculate speed, position, and total distance from a particle’s velocity and starting position.
   6. Polar coordinates and differentiating in polar formFind a polar curve’s tangent slope and tell whether its point moves toward or away from the origin.
   7. Area of a polar region and between two polar curvesCalculate polar area by choosing angles that trace a region once and, when needed, subtracting inner sectors.
10. 10

    Infinite sequences and series

    14 lessons

    1. Defining convergent and divergent infinite seriesDetermine convergence from partial sums and find a series’ sum when it exists.
    2. Working with geometric seriesDetermine whether a geometric series converges and find its sum when it does.
    3. The nth term test for divergenceUse a term limit to prove divergence or recognize when the test gives no conclusion.
    4. The integral test, the harmonic series and p-seriesCheck integral-test conditions and use an improper integral or the p-series rule to classify a series.
    5. Comparison tests for convergenceChoose a known series and justify convergence or divergence by direct or limit comparison.
    6. Alternating series test for convergenceJustify convergence with both alternating series test conditions, and recognize when the nth-term test proves divergence.
    7. Ratio test for convergenceCompute consecutive-term ratio limits, apply the ratio test, and recognize when it is inconclusive.
    8. Absolute or conditional convergenceClassify a series as absolutely convergent, conditionally convergent, or divergent, justifying both relevant tests.
    9. Alternating series error boundBound the error of an alternating-series approximation and determine whether it is high or low.
    10. Finding Taylor polynomial approximations of functionsConstruct a Taylor polynomial from derivatives at its center and use it to approximate a nearby value.
    11. Lagrange error boundCalculate a Taylor polynomial error bound and use it to certify accuracy.
    12. Radius and interval of convergence of power seriesFind a power series’ radius and justify its full interval by testing both endpoints.
    13. Finding Taylor or Maclaurin series for a functionWrite first terms and a general term, then recover derivatives from coefficients.
    14. Representing functions as power seriesBuild power series from a geometric series and determine where the resulting series converge.