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AP Calculus BC

One lesson for each of the course’s 111 topics, in the order the course teaches them. The first 3 units are free for everyone; the rest come with the AP Calculus BC plan. Read a lesson, try its quick checks, then practice the topic with exam-style questions.

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Limits and Continuity

Unit 1Free16 lessons
  1. 1.1Introducing Calculus: Can Change Occur at an Instant?Explain why an average rate of change can't be computed at a single instant, and find the instantaneous rate as the limit of average rates over shrinking intervals.
  2. 1.2Defining Limits and Using Limit NotationRead and write limit notation correctly, including one-sided limits, and say exactly what a limit statement does and does not tell you about a function.
  3. 1.3Estimating Limit Values from GraphsRead one-sided and two-sided limits off a graph, recognize the three ways a limit fails to exist, and know when a graph's scale can hide what a function does.
  4. 1.4Estimating Limit Values from TablesEstimate a limit from a table of values on both sides of the point, decide how many digits to trust, and spot tables that point to the wrong answer.
  5. 1.5Determining Limits Using Algebraic Properties of LimitsFind limits of sums, products, quotients, powers and compositions from formulas, given values or graphs, and know exactly when each limit law applies.
  6. 1.6Determining Limits Using Algebraic ManipulationTurn a 0/0 limit into one you can evaluate by factoring, using a conjugate, combining fractions, or applying a trig identity, and know why the rewrite is allowed.
  7. 1.7Selecting Procedures for Determining LimitsDecide how to evaluate any limit from what substitution gives you and from the expression's structure, then carry out the method that fits.
  8. 1.8Determining Limits Using the Squeeze TheoremFind a limit by trapping a function between two simpler ones, check the theorem's conditions, and use the limits of sin(x)/x and (1 - cos x)/x that it proves.
  9. 1.9Connecting Multiple Representations of LimitsRead the same limit from a graph, a table, a formula or a sentence, know what each one can prove, and combine limits given in different forms.
  10. 1.10Exploring Types of DiscontinuitiesTell removable, jump and infinite discontinuities apart by their one-sided limits, and justify each classification from a formula, a piecewise rule or a graph.
  11. 1.11Defining Continuity at a PointState the three conditions for continuity at a point, check each one for formulas and piecewise functions, and write the justification the exam awards points for.
  12. 1.12Confirming Continuity over an IntervalFind every interval on which a function is continuous, using the families that are continuous on their domains, one-sided continuity at endpoints, and boundary checks.
  13. 1.13Removing DiscontinuitiesDecide whether a discontinuity can be removed, then find the function value or the constant in a piecewise formula that makes the function continuous.
  14. 1.14Connecting Infinite Limits and Vertical AsymptotesFind and write one-sided infinite limits by checking signs near a zero denominator, and use them to locate and justify vertical asymptotes.
  15. 1.15Connecting Limits at Infinity and Horizontal AsymptotesFind limits as x grows without bound, read them as end behavior and horizontal asymptotes, and compare how fast functions grow using a limit of their ratio.
  16. 1.16Working with the Intermediate Value TheoremUse the Intermediate Value Theorem to prove a continuous function takes a value or has a zero, check its conditions, and write the justification graders expect.

Differentiation: Definition and Fundamental Properties

Unit 2Free10 lessons
  1. 2.1Defining Average and Instantaneous Rates of Change at a PointCompute an average rate of change as a difference quotient, then take its limit to get the instantaneous rate of change, the derivative at a point.
  2. 2.2Defining the Derivative of a Function and Using Derivative NotationFind the derivative function from the limit definition, read and write every derivative notation, and use the derivative to write tangent lines.
  3. 2.3Estimating Derivatives of a Function at a PointEstimate a derivative from a table with the right difference quotient, from a graph with a tangent line, and with a calculator, and know each method's limits.
  4. 2.4Connecting Differentiability and ContinuityExplain why a differentiable function must be continuous, find the corners, cusps and vertical tangents where a continuous function has no derivative, and justify it.
  5. 2.5Applying the Power RuleDifferentiate any power of x, including roots and reciprocals rewritten as powers, using a rule you can derive from the limit definition.
  6. 2.6Derivative Rules: Constant, Sum, Difference, and Constant MultipleDifferentiate any polynomial, and any sum of powers, term by term, and see why constants vanish while constant multipliers stay.
  7. 2.7Derivatives of cos x, sin x, eˣ, and ln xKnow and use the derivatives of sin x, cos x, eˣ and ln x, see where each comes from, and spot a limit that is secretly one of these derivatives.
  8. 2.8The Product RuleDifferentiate a product of two or more functions, given by formulas or by tables of values, and see from a growing rectangle why the rule has two terms.
  9. 2.9The Quotient RuleDifferentiate a quotient of two functions with the quotient rule, know where the rule comes from, and recognize when rewriting is faster.
  10. 2.10Derivatives of Tangent, Cotangent, Secant, and CosecantDerive and use the derivatives of tan x, cot x, sec x and csc x by rewriting them in sine and cosine, and simplify with identities before differentiating.

Differentiation: Composite, Implicit, and Inverse Functions

Unit 3Free6 lessons
  1. 3.1The Chain RuleDifferentiate any composition, given as a formula, a table, or a graph, by multiplying the outside's derivative at the inside by the inside's derivative.
  2. 3.2Implicit DifferentiationFind dy/dx for a curve given by an equation in x and y, without solving for y, and use it for slopes, tangent lines, and horizontal or vertical tangents.
  3. 3.3Differentiating Inverse FunctionsFind the derivative of an inverse function at a point from the original function alone, using the matching point and the reciprocal of its slope.
  4. 3.4Differentiating Inverse Trigonometric FunctionsDifferentiate arcsine, arccosine and arctangent of any inside function, and know where each formula comes from and where it holds.
  5. 3.5Selecting Procedures for Calculating DerivativesLook at any function, rewrite it if that helps, pick the rule that applies first, and recognize derivatives hidden in tables, equations and limits.
  6. 3.6Calculating Higher-Order DerivativesFind second and higher derivatives from formulas, tables and implicit equations, read every notation for them, and know when they fail to exist.

Contextual Applications of Differentiation

Unit 4With the plan7 lessons
  1. 4.1Interpreting the Meaning of the Derivative in ContextTurn a derivative such as f'(a) = -3 into a correct sentence about its context, naming the input value, the direction of change, and the units.
  2. 4.2Straight-Line Motion: Position, Velocity, and AccelerationUse derivatives of a position function to find velocity and acceleration, and decide which way a particle moves and whether it is speeding up or slowing down.
  3. 4.3Rates of Change in Applied Contexts Other Than MotionCompute and interpret rates of change for any quantity, including net rates from inflow and outflow, rates of rates, and marginal cost and revenue.
  4. 4.4Introduction to Related RatesFind how fast one quantity changes from how fast a related one changes, by differentiating the equation that links them with respect to time.
  5. 4.5Solving Related Rates ProblemsCarry a related rates problem from a word description to a rate with the right sign and units, and state what that rate means at that instant.
  6. 4.6Approximating Values Using Local Linearity and LinearizationEstimate a function's value with its tangent line, and use concavity to say whether the estimate is too high or too low, with the justification the exam wants.
  7. 4.7Using L'Hospital's Rule for Indeterminate FormsSpot the indeterminate forms 0/0 and infinity over infinity, check the conditions of L'Hospital's rule, and use it to evaluate limits with the work the exam requires.

Analytical Applications of Differentiation

Unit 5With the plan12 lessons
  1. 5.1Using the Mean Value TheoremState the Mean Value Theorem with its two conditions, see why it is true, and use it to justify that a derivative must take a given value.
  2. 5.2Extreme Value Theorem, Global Versus Local Extrema, and Critical PointsTell absolute from relative extrema, use the Extreme Value Theorem to justify that they exist, and find the critical points where relative extrema can occur.
  3. 5.3Determining Intervals on Which a Function Is Increasing or DecreasingUse the sign of f' to find where f is increasing or decreasing, from a formula or from the graph of f', and justify the answer the way the exam expects.
  4. 5.4Using the First Derivative Test to Determine Relative ExtremaDecide whether each critical point of a function is a relative maximum, a relative minimum, or neither, from the sign of f' on each side, and justify it.
  5. 5.5Using the Candidates Test to Determine Absolute ExtremaFind the absolute maximum and minimum of a continuous function on a closed interval by testing critical points and endpoints, and justify it as the exam expects.
  6. 5.6Determining Concavity of Functions over Their DomainsDecide where a graph is concave up or concave down from its first or second derivative, and find and justify its points of inflection.
  7. 5.7Using the Second Derivative Test to Determine ExtremaClassify a critical point as a relative maximum or minimum from the sign of the second derivative, and know when a single relative extremum is the absolute one.
  8. 5.8Sketching Graphs of Functions and Their DerivativesSketch the graph of a function from its formula, from the graph of its derivative, or from a table, and sketch a derivative from the graph of the function.
  9. 5.9Connecting a Function, Its First Derivative, and Its Second DerivativeRead the graphs or formulas of f, f' and f'' against each other, and justify where f increases, peaks, bends, and has points of inflection.
  10. 5.10Introduction to Optimization ProblemsTurn a "largest" or "smallest" question into a function of one variable on a domain, then find its absolute maximum or minimum with derivatives.
  11. 5.11Solving Optimization ProblemsSolve applied optimization problems from setup to a justified absolute extremum, and state what the maximum or minimum means in context, with units.
  12. 5.12Exploring Behaviors of Implicit RelationsFind the critical points of a curve defined implicitly, and use dy/dx and the second derivative in x, y and dy/dx to justify relative extrema and concavity.

Integration and Accumulation of Change

Unit 6With the plan14 lessons
  1. 6.1Exploring Accumulations of ChangeRead the area between a rate-of-change graph and the horizontal axis as the total change in a quantity, with its sign and its units, using geometry.
  2. 6.2Approximating Areas with Riemann SumsApproximate a definite integral from a graph, table or formula with left, right, midpoint and trapezoidal sums, and decide whether each is too big or too small.
  3. 6.3Riemann Sums, Summation Notation, and Definite Integral NotationWrite Riemann sums in sigma notation, define the definite integral as their limit, and convert between a limit of Riemann sums and a definite integral.
  4. 6.4The Fundamental Theorem of Calculus and Accumulation FunctionsDefine functions with integrals such as g(x) = integral from a to x of f(t) dt, and differentiate them with the Fundamental Theorem of Calculus and the chain rule.
  5. 6.5Interpreting the Behavior of Accumulation Functions Involving AreaGiven f by a graph, table or formula, find the values, increasing intervals, extrema and concavity of g(x), the integral of f from a to x, and justify them.
  6. 6.6Applying Properties of Definite IntegralsEvaluate definite integrals from areas of familiar shapes and from given integral values, using linearity, reversed limits and adjacent intervals.
  7. 6.7The Fundamental Theorem of Calculus and Definite IntegralsEvaluate definite integrals exactly as F(b) minus F(a) from an antiderivative, check the continuity condition, and use the theorem with tables of f and f'.
  8. 6.8Finding Antiderivatives and Indefinite IntegralsWrite indefinite integrals with correct notation and the constant C, find antiderivatives by reversing derivative rules, and recognize integrands with no formula.
  9. 6.9Integrating Using SubstitutionRecognize an inside function and its derivative in an integrand, substitute u to find indefinite integrals, and change the limits when evaluating definite ones.
  10. 6.10Integrating Using Long Division and Completing the SquareRewrite fractions by polynomial long division and quadratics by completing the square, so the integral becomes a power, a logarithm, an arctangent or an arcsine.
  11. 6.11Integrating Using Integration by PartsAntidifferentiate products like x e^(2x), x^2 sin x and ln x by running the product rule backward, and evaluate definite integrals the same way.
  12. 6.12Integrating Using Linear Partial FractionsSplit a rational function with distinct linear factors in its denominator into simple fractions, then integrate each one to a natural log.
  13. 6.13Evaluating Improper IntegralsEvaluate integrals over infinite intervals or of unbounded functions as limits of ordinary definite integrals, and decide when they diverge.
  14. 6.14Selecting Techniques for AntidifferentiationRead an integrand's structure and pick the technique that fits it, whether a basic rule, a rewrite, substitution, division or completing the square.

Differential Equations

Unit 7With the plan9 lessons
  1. 7.1Modeling Situations with Differential EquationsTurn a description like "decreases at a rate proportional to the square root of the depth" into a differential equation, and read what a given one says.
  2. 7.2Verifying Solutions for Differential EquationsCheck whether a function solves a differential equation by differentiating and substituting, and see why one equation has a whole family of solutions.
  3. 7.3Sketching Slope FieldsDraw the slope field of a differential equation by hand, match a slope field to its equation, and read off where solutions rise, fall, or stay constant.
  4. 7.4Reasoning Using Slope FieldsSketch solution curves through a slope field, predict where solutions rise, fall, bend and level off, and use the second derivative found from the equation.
  5. 7.5Approximating Solutions Using Euler's MethodApproximate a value of a solution to a differential equation with Euler's method, step by step, and decide whether the estimate is too high or too low.
  6. 7.6Finding General Solutions Using Separation of VariablesSolve a differential equation by antidifferentiating, or by separating x and y and integrating both sides, and handle the constant correctly.
  7. 7.7Finding Particular Solutions Using Initial ConditionsUse an initial condition to pick the one solution of a differential equation through a given point, write it as an accumulation function, and state its domain.
  8. 7.8Exponential Models with Differential EquationsTurn "rate proportional to amount" into dy/dt = ky, solve it to y = y0 e^(kt), find k from data, and interpret rate equations, including motion, in context.
  9. 7.9Logistic Models with Differential EquationsRead a logistic differential equation to find its carrying capacity, its long-run limit, and the value where the quantity changes fastest, all without solving it.

Applications of Integration

Unit 8With the plan13 lessons
  1. 8.1Finding the Average Value of a Function on an IntervalCompute the average value of a function on an interval with a definite integral, read it from a graph, and find where the function reaches it.
  2. 8.2Position, Velocity, and Acceleration Using IntegralsUse definite integrals of velocity and speed to find a moving particle's position, displacement and total distance, and integrate acceleration to get velocity.
  3. 8.3Using Accumulation Functions and Definite Integrals in Applied ContextsInterpret a definite integral of a rate as a net change with correct units, and build and analyze amount functions from rates in and rates out.
  4. 8.4Area Between Curves Expressed as Functions of xFind the area of a region between two graphs with a definite integral of top minus bottom, choosing the limits and splitting where the curves cross.
  5. 8.5Area Between Curves Expressed as Functions of yFind the area of a region by slicing it horizontally and integrating right minus left with respect to y, and decide when that beats slicing vertically.
  6. 8.6Area Between Curves That Intersect at More Than Two PointsFind the area between curves that cross each other, by splitting the integral at every crossing or by integrating the absolute value of the difference.
  7. 8.7Volumes with Cross Sections: Squares and RectanglesFind the volume of a solid built on a flat base whose slices are squares or rectangles, by integrating the area of one slice across the base.
  8. 8.8Volumes with Cross Sections: Triangles and SemicirclesFind volumes of solids whose slices are triangles, semicircles, or other known shapes by writing each slice's area in terms of its edge in the base.
  9. 8.9Volume with Disc Method: Revolving Around the x- or y-AxisFind the volume of a solid formed by spinning a region around the x- or y-axis by adding up circular slices, each with area pi times radius squared.
  10. 8.10Volume with Disc Method: Revolving Around Other AxesFind the volume of a solid formed by revolving a region around any horizontal or vertical line, using the distance from that line to the curve as the radius.
  11. 8.11Volume with Washer Method: Revolving Around the x- or y-AxisFind the volume of a solid made by spinning the region between two curves around the x- or y-axis, using outer and inner radii.
  12. 8.12Volume with Washer Method: Revolving Around Other AxesSet up and evaluate washer-method volumes when a region spins around a line like y = -1, y = 4 or x = 3, measuring every radius from that line.
  13. 8.13Arc Length of a Smooth, Planar Curve and Distance TraveledFind the length of a curve y = f(x) or x = g(y) with an integral, know when it needs a calculator, and use it for the distance an object travels along a path.

Parametric Equations, Polar Coordinates, and Vector-Valued Functions

Unit 9With the plan9 lessons
  1. 9.1Defining and Differentiating Parametric EquationsDescribe a curve with parametric equations, find the slope dy/dx at any point from dy/dt and dx/dt, and locate horizontal and vertical tangent lines.
  2. 9.2Second Derivatives of Parametric EquationsCompute the second derivative of a parametric curve by differentiating dy/dx with respect to t and dividing by dx/dt, then use it to decide concavity.
  3. 9.3Arc Lengths of Curves Given by Parametric EquationsSet up and evaluate the definite integral that gives the length of a parametric curve, and know when that integral measures the curve's true length.
  4. 9.4Defining and Differentiating Vector-Valued FunctionsWrite a curve as a vector-valued function, differentiate it component by component, and read the derivative vector as the direction and rate of motion along the curve.
  5. 9.5Integrating Vector-Valued FunctionsIntegrate a vector-valued function one component at a time and use a velocity or acceleration vector with an initial condition to find a particle's position.
  6. 9.6Motion Problems Using Parametric and Vector-Valued FunctionsFind velocity, speed and acceleration of a particle moving in the plane, then use integrals to get its position, displacement and total distance traveled.
  7. 9.7Defining Polar Coordinates and Differentiating in Polar FormLocate points with polar coordinates, convert to x and y, and use dr/dθ, dx/dθ, dy/dθ and dy/dx to read how a polar curve moves and where it turns.
  8. 9.8Area of a Polar Region or Bounded by a Single Polar CurveFind the area swept out by a polar curve with A = ½∫r² dθ, choosing limits that trace the region exactly once, including petals and inner loops.
  9. 9.9Area of the Region Bounded by Two Polar CurvesFind areas between two polar curves: locate where they meet, subtract inner from outer with ½∫(outer² − inner²) dθ, and split where the boundary changes curves.

Infinite Sequences and Series

Unit 10With the plan15 lessons
  1. 10.1Defining Convergent and Divergent Infinite SeriesBuild the sequence of partial sums of a series and use its limit to decide whether the series converges, and to what sum.
  2. 10.2Working with Geometric SeriesRecognize a geometric series, find its first term and common ratio, decide whether it converges, and find its exact sum when it does.
  3. 10.3The nth Term Test for DivergenceUse the limit of a series' terms to prove divergence when that limit isn't 0, and know why a limit of 0 proves nothing.
  4. 10.4Integral Test for ConvergenceDecide whether a series of positive, decreasing terms converges by testing the matching improper integral, with every condition checked.
  5. 10.5Harmonic Series and p-SeriesKnow which p-series converge and why, prove the harmonic series diverges, and see why the alternating harmonic series converges anyway.
  6. 10.6Comparison Tests for ConvergenceDecide whether a series of positive terms converges by comparing it with a p-series or a geometric series, term by term or through a limit of ratios.
  7. 10.7Alternating Series Test for ConvergenceShow that a series whose terms alternate in sign converges by checking that the sizes of its terms decrease and approach zero, with a complete justification.
  8. 10.8Ratio Test for ConvergenceDecide whether a series converges by finding the limit of the ratio of consecutive terms, the test built for factorials and exponentials.
  9. 10.9Determining Absolute or Conditional ConvergenceClassify a series as absolutely convergent, conditionally convergent, or divergent, and know why only absolute convergence makes the order of the terms irrelevant.
  10. 10.10Alternating Series Error BoundBound how far a partial sum of an alternating series is from the true sum, decide whether it over- or underestimates, and find how many terms reach a set accuracy.
  11. 10.11Finding Taylor Polynomial Approximations of FunctionsBuild the Taylor polynomial of any degree for a function about a point, use it to estimate values, and read derivatives back off its coefficients.
  12. 10.12Lagrange Error BoundBound the error of a Taylor polynomial approximation with the Lagrange error bound, or the alternating series bound when it applies, and use it to draw conclusions.
  13. 10.13Radius and Interval of Convergence of Power SeriesFind the radius of convergence of a power series with the ratio test, then test both endpoints to get the exact interval where the series converges.
  14. 10.14Finding Taylor or Maclaurin Series for a FunctionWrite the Taylor or Maclaurin series for a function from its derivatives, know the series for e^x, sin x, cos x and 1/(1-x) cold, and build new series from them.
  15. 10.15Representing Functions as Power SeriesBuild a power series for a function from a known series by substitution, algebra, the geometric series, or term-by-term differentiation and integration.

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