AP Calculus BC
83 of 83 topics written so far. Greyed-out topics are coming soon.
Unit 1: Limits and Continuity
- 1Introduction to Limits
The idea of a limit — what a function approaches near a point, limit notation, one-sided limits, and when a limit does not exist.
Draft - 2Estimating Limits from Graphs and Tables
Reading limits and one-sided limits from a graph, estimating limits from tables of values, and checking that graphs, tables, and equations agree.
Draft - 3Limit Laws
The algebraic properties of limits — sums, differences, constant multiples, products, quotients, powers, and roots — plus direct substitution and limits of composite functions.
Draft - 4Algebraic Techniques for Limits
Evaluating 0/0 limits by factoring, rationalizing with conjugates, and combining fractions, and choosing the right procedure for a limit.
Draft - 5The Squeeze Theorem
Finding a limit by trapping a function between two others, and the special trig limits sin x / x → 1 and (1 − cos x)/x → 0 (radians).
Draft - 6Types of Discontinuities
Removable, jump, and infinite discontinuities, how to remove a removable discontinuity, and how to choose constants that make a piecewise function continuous.
Draft - 7Continuity
The three conditions for continuity at a point, continuity on an interval, and where polynomial, rational, root, exponential, logarithmic, and trig functions are continuous.
Draft - 8Infinite Limits and Vertical Asymptotes
Limits that grow without bound, one-sided infinite limits, the sign analysis for nonzero/0, and finding vertical asymptotes.
Draft - 9Limits at Infinity
End behaviour and horizontal asymptotes — limits of rational functions by comparing degrees, exponential and logarithmic functions, square roots, and relative growth rates.
Draft - 10The Intermediate Value Theorem
What the Intermediate Value Theorem says, why continuity on a closed interval is essential, and how to write an AP-style IVT justification.
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Unit 2: Differentiation: Definition and Fundamental Properties
- 1Average and Instantaneous Rates of Change
Average rate of change as the slope of a secant line, the difference quotient, and the instantaneous rate of change as a limit.
Draft - 2The Definition of the Derivative
The derivative as a limit, derivative notation, tangent lines, and estimating derivatives from tables and graphs.
Draft - 3Differentiability and Continuity
Why differentiable functions are continuous, where derivatives fail to exist (corners, cusps, vertical tangents, discontinuities), and checking piecewise functions.
Draft - 4The Power Rule and Basic Derivative Rules
Differentiating powers of x (including negative and fractional exponents), constants, sums, differences, and constant multiples.
Draft - 5Derivatives of Sine, Cosine, eˣ, and ln x
The derivatives of sin x, cos x, e^x, and ln x, why radians matter, and using them with the basic derivative rules.
Draft - 6The Product Rule
Differentiating a product of two (or three) functions, including from tables of values and in context.
Draft - 7The Quotient Rule and Trig Derivatives
Differentiating quotients, and using the quotient rule to find the derivatives of tan x, cot x, sec x, and csc x.
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Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
- 1The Chain Rule
Differentiating composite functions with f'(g(x)) · g'(x) and dy/du · du/dx — powers, trig, exponential, and log compositions, tables of values, and choosing which rules to combine.
Draft - 2Implicit Differentiation
Finding dy/dx for curves like circles where y isn't written as a function of x — differentiating both sides, solving for dy/dx, and finding tangent lines.
Draft - 3Derivatives of Inverse Functions
Finding the slope of an inverse function with (f⁻¹)'(a) = 1 / f'(f⁻¹(a)) — from equations, tables, and graphs — and why the slopes are reciprocals.
Draft - 4Derivatives of Inverse Trig Functions
Derivatives of arcsin x, arccos x, and arctan x (also written sin⁻¹ x, cos⁻¹ x, tan⁻¹ x), where they come from, and using them with the chain rule.
Draft - 5Higher-Order Derivatives
Second, third, and higher derivatives — notation like f''(x), d²y/dx², and y'', what the second derivative means, and second derivatives of implicit relations.
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Unit 4: Contextual Applications of Differentiation
- 1Derivatives in Context
Interpreting a derivative as a rate of change with units, and writing AP-style sentences about what f'(a) means in a real situation.
Draft - 2Straight-Line Motion with Derivatives
Position, velocity, and acceleration of a particle on a line — speed, direction, changing direction, and speeding up versus slowing down.
Draft - 3Introduction to Related Rates
Differentiating an equation with respect to time — using the chain rule to connect rates like dr/dt and dA/dt for growing circles, squares, and spheres.
Draft - 4Related Rates Problems
A step-by-step strategy for multi-step related rates — sliding ladders, filling cones, moving shadows, and approaching cars.
Draft - 5Linearization and Tangent Line Approximation
Local linearity, the tangent line approximation L(x) = f(a) + f'(a)(x − a), estimating values, and using concavity to tell an overestimate from an underestimate.
Draft - 6L'Hospital's Rule
Using derivatives to evaluate limits of the indeterminate forms 0/0 and ∞/∞ — checking the form first, and knowing when not to use the rule.
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Unit 5: Analytical Applications of Differentiation
- 1Mean Value Theorem
The Mean Value Theorem and Rolle's theorem — checking the hypotheses, finding c, and writing the justification AP graders expect.
Draft - 2Critical Points and Extrema
Absolute and relative maximums and minimums, the Extreme Value Theorem, and finding critical points where f'(x) = 0 or f'(x) does not exist.
Draft - 3First Derivative Test
Finding where a function increases and decreases from the sign of f', building sign charts, and using the first derivative test to classify relative extrema.
Draft - 4Candidates Test for Absolute Extrema
Finding the absolute maximum and minimum of a continuous function on a closed interval by checking every critical point and both endpoints.
Draft - 5Concavity and the Second Derivative Test
Concave up and concave down from the sign of f'', points of inflection, and the second derivative test for relative extrema — including when it's inconclusive.
Draft - 6Connecting f, f′, and f″
Reading the features of f from graphs of f' and f'', and sketching f and f' from each other — increasing, extrema, concavity, and inflection points.
Draft - 7Optimization
Setting up and solving optimization problems with derivatives — variables, constraint, objective function, domain — and justifying the answer for fencing, boxes, cans, cost, and distance.
Draft - 8Analyzing Implicit Relations
Using dy/dx and d²y/dx² from implicit differentiation to find horizontal and vertical tangents, highest and lowest points, and concavity on curves that aren't functions.
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Unit 6: Integration and Accumulation of Change
- 1Accumulation of Change
Why the area under a rate graph tells you the total change, with units, and how positive and negative contributions add up.
Draft - 2Riemann Sums
Approximating the area under a curve with left, right, midpoint, and trapezoidal sums, from graphs, tables, and formulas.
Draft - 3The Definite Integral
The definite integral as the limit of Riemann sums, sigma notation, and translating between a limit of a sum and an integral.
Draft - 4Accumulation Functions and the FTC
Accumulation functions defined by integrals, the Fundamental Theorem of Calculus (g'(x) = f(x)), the chain rule with a variable limit, and reading g's behaviour from the graph of f.
Draft - 5Properties of Definite Integrals
Reversing limits, zero-width intervals, adding intervals, constant multiples and sums, and evaluating integrals with geometry.
Draft - 6Evaluating Definite Integrals with the FTC
The Fundamental Theorem of Calculus, part 2 — evaluating a definite integral as F(b) − F(a) using an antiderivative, and the net change theorem.
Draft - 7Antiderivatives and Indefinite Integrals
Finding antiderivatives with the basic rules (power, 1/x, exponential, trig, inverse trig), the + C, and solving initial-value problems.
Draft - 8Integration by Substitution
Undoing the chain rule with u-substitution, for indefinite and definite integrals (including changing the limits), and choosing an integration technique.
Draft - 9Integrating with Long Division and Completing the Square
Integrating rational functions by first using polynomial long division, or by completing the square to get an arctangent.
Draft - 10Integration by Parts
Undoing the product rule — choosing u and dv (LIATE), repeated parts, the tabular method, definite integrals by parts, and integrals that return to themselves.
Draft - 11Integration Using Partial Fractions
Splitting a rational function with distinct linear factors into simple fractions that integrate to logarithms, including when to use long division first.
Draft - 12Improper Integrals
Integrals over infinite intervals or across vertical asymptotes, written as limits — deciding whether they converge or diverge, including the p-integral rule.
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Unit 7: Differential Equations
- 1Introduction to Differential Equations
Writing differential equations from verbal descriptions, verifying solutions by substitution, and telling general solutions from particular ones.
Draft - 2Slope Fields
Sketching slope fields from a differential equation, matching fields to equations, and drawing and reading solution curves.
Draft - 3Euler's Method
Approximating a solution of a differential equation step by step with tangent lines, organizing the work in a table, and deciding whether the estimate is too high or too low.
Draft - 4Separation of Variables
Solving separable differential equations, finding particular solutions from initial conditions, and stating where a solution is defined.
Draft - 5Exponential Models with Differential Equations
Why dy/dt = ky gives y = y₀e^(kt), and how to use it for growth, decay, half-life, doubling time, and Newton's law of cooling.
Draft - 6Logistic Models
The logistic differential equation dP/dt = kP(1 − P/L) — carrying capacity, long-run behaviour, where growth is fastest, and the solution formula.
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Unit 8: Applications of Integration
- 1Average Value of a Function
The average height of a function on an interval — the formula, what it means as a picture, units, and where the function actually reaches its average.
Draft - 2Motion Along a Line with Integrals
Going backwards from velocity to position — initial conditions, displacement versus total distance, and velocity from acceleration.
Draft - 3Definite Integrals in Context
Using integrals of rates to find amounts — water in a tank, people at an event, rates in and out, units, and finding when an amount is largest.
Draft - 4Area Between Curves
Finding the area between two curves with vertical slices — top minus bottom, finding intersection points, and splitting the integral when the curves cross more than twice.
Draft - 5Area Between Curves Using Horizontal Slices
Integrating with respect to y — right minus left, rewriting curves as x in terms of y, and choosing the slice direction that makes the problem easier.
Draft - 6Volumes with Known Cross Sections
Finding volumes by slicing — squares, rectangles, triangles, and semicircles standing on a base region, perpendicular to the x-axis or the y-axis.
Draft - 7Volumes of Revolution — the Disc Method
Spinning a region around a line to make a solid — discs with radius R, revolving about the x-axis, the y-axis, and other horizontal or vertical lines.
Draft - 8Volumes of Revolution — the Washer Method
Solids with a hole — washers with outer radius R and inner radius r, revolving about the x-axis, the y-axis, and other horizontal or vertical lines.
Draft - 9Arc Length
Finding the length of a curve y = f(x) (or x = g(y)) with the integral of √(1 + (dy/dx)²) — setting it up, exact answers, and calculator evaluation.
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Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
- 1Derivatives of Parametric Equations
Finding dy/dx and d²y/dx² for curves given by x(t) and y(t) — tangent lines, horizontal and vertical tangents, and concavity.
Draft - 2Arc Length of Parametric Curves
Finding the length of a curve given by x(t) and y(t), and the total distance travelled by a moving particle compared with its displacement.
Draft - 3Vector-Valued Functions
Writing curves as r(t) = ⟨x(t), y(t)⟩, and differentiating and integrating them one component at a time, including initial conditions.
Draft - 4Motion in the Plane with Vectors
Position, velocity, and acceleration vectors for a particle moving in the plane — speed, direction of motion, position from velocity, and total distance travelled.
Draft - 5Polar Coordinates and Derivatives
Converting between polar and rectangular coordinates, recognizing circles, cardioids, limaçons and roses, finding dy/dx for r = f(θ), and what dr/dθ says about distance from the pole.
Draft - 6Area in Polar Coordinates
Finding the area enclosed by a polar curve with A = ½∫r² dθ — one petal, a whole curve, and the region between two polar curves, including intersections at the pole.
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Unit 10: Infinite Sequences and Series
- 1Introduction to Infinite Series
Sequences versus series, partial sums, what it means for an infinite series to converge, telescoping series, and the nth term test for divergence.
Draft - 2Infinite Geometric Series
When an infinite geometric series converges, its sum a/(1 − r), and how to use it for repeating decimals, shifted indices, and bouncing-ball problems.
Draft - 3The Integral Test
Deciding whether a series converges by comparing it with an improper integral — the hypotheses (positive, continuous, decreasing) and why the series and the integral don't have the same value.
Draft - 4Harmonic Series and p-Series
Why the harmonic series diverges, and the p-series rule — the sum of 1/n^p converges exactly when p > 1.
Draft - 5Comparison Tests for Series
The direct comparison test and the limit comparison test — deciding whether a series of positive terms converges by comparing it with a p-series or geometric series you already know.
Draft - 6The Alternating Series Test
Series whose terms switch sign — the two conditions of the alternating series test, why the partial sums zig-zag toward the sum, and the alternating harmonic series.
Draft - 7The Ratio Test
Deciding convergence from the limit of |a(n+1)/a(n)| — especially useful for factorials and exponentials — and why L = 1 tells you nothing.
Draft - 8Absolute and Conditional Convergence
Absolute versus conditional convergence, why absolute convergence implies convergence, how to classify a series, and a strategy for choosing a convergence test.
Draft - 9Alternating Series Error Bound
How far a partial sum of an alternating series can be from the true sum — the error is at most the first omitted term — plus the sign of the error and how many terms you need.
Draft - 10Taylor Polynomials
Building the nth-degree Taylor polynomial about x = a from derivative values (including values from a table), reading derivatives back from coefficients, and approximating function values.
Draft - 11Lagrange Error Bound
Bounding the error of a Taylor polynomial approximation with |R_n(x)| ≤ M|x − a|^(n+1)/(n+1)!, choosing the bound M, and finding the degree needed for a given accuracy.
Draft - 12Radius and Interval of Convergence
Power series centred at x = a, finding the radius of convergence with the ratio test, and checking each endpoint separately to get the interval of convergence.
Draft - 13Taylor and Maclaurin Series
Taylor series as never-ending Taylor polynomials, the four series AP expects you to know (e^x, sin x, cos x, 1/(1 − x)), writing the general term, and building a series from derivatives.
Draft - 14Representing Functions as Power Series
Building new power series from known ones — substitution, multiplying by x, and differentiating or integrating term by term (radius unchanged, endpoints may change) — including the series for ln(1 + x) and arctan x.
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