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Family Table Math

AP Calculus AB

57 of 57 topics written so far. Greyed-out topics are coming soon.

Unit 1: Limits and Continuity

  1. Introduction to LimitsDraft — The idea of a limit — what a function approaches near a point, limit notation, one-sided limits, and when a limit does not exist.
  2. Estimating Limits from Graphs and TablesDraft — Reading limits and one-sided limits from a graph, estimating limits from tables of values, and checking that graphs, tables, and equations agree.
  3. Limit LawsDraft — The algebraic properties of limits — sums, differences, constant multiples, products, quotients, powers, and roots — plus direct substitution and limits of composite functions.
  4. Algebraic Techniques for LimitsDraft — Evaluating 0/0 limits by factoring, rationalizing with conjugates, and combining fractions, and choosing the right procedure for a limit.
  5. The Squeeze TheoremDraft — 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).
  6. Types of DiscontinuitiesDraft — Removable, jump, and infinite discontinuities, how to remove a removable discontinuity, and how to choose constants that make a piecewise function continuous.
  7. ContinuityDraft — The three conditions for continuity at a point, continuity on an interval, and where polynomial, rational, root, exponential, logarithmic, and trig functions are continuous.
  8. Infinite Limits and Vertical AsymptotesDraft — Limits that grow without bound, one-sided infinite limits, the sign analysis for nonzero/0, and finding vertical asymptotes.
  9. Limits at InfinityDraft — End behaviour and horizontal asymptotes — limits of rational functions by comparing degrees, exponential and logarithmic functions, square roots, and relative growth rates.
  10. The Intermediate Value TheoremDraft — What the Intermediate Value Theorem says, why continuity on a closed interval is essential, and how to write an AP-style IVT justification.

Unit 2: Differentiation: Definition and Fundamental Properties

  1. Average and Instantaneous Rates of ChangeDraft — Average rate of change as the slope of a secant line, the difference quotient, and the instantaneous rate of change as a limit.
  2. The Definition of the DerivativeDraft — The derivative as a limit, derivative notation, tangent lines, and estimating derivatives from tables and graphs.
  3. Differentiability and ContinuityDraft — Why differentiable functions are continuous, where derivatives fail to exist (corners, cusps, vertical tangents, discontinuities), and checking piecewise functions.
  4. The Power Rule and Basic Derivative RulesDraft — Differentiating powers of x (including negative and fractional exponents), constants, sums, differences, and constant multiples.
  5. Derivatives of Sine, Cosine, eˣ, and ln xDraft — The derivatives of sin x, cos x, e^x, and ln x, why radians matter, and using them with the basic derivative rules.
  6. The Product RuleDraft — Differentiating a product of two (or three) functions, including from tables of values and in context.
  7. The Quotient Rule and Trig DerivativesDraft — Differentiating quotients, and using the quotient rule to find the derivatives of tan x, cot x, sec x, and csc x.

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

  1. The Chain RuleDraft — 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.
  2. Implicit DifferentiationDraft — 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.
  3. Derivatives of Inverse FunctionsDraft — 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.
  4. Derivatives of Inverse Trig FunctionsDraft — 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.
  5. Higher-Order DerivativesDraft — Second, third, and higher derivatives — notation like f''(x), d²y/dx², and y'', what the second derivative means, and second derivatives of implicit relations.

Unit 4: Contextual Applications of Differentiation

  1. Derivatives in ContextDraft — Interpreting a derivative as a rate of change with units, and writing AP-style sentences about what f'(a) means in a real situation.
  2. Straight-Line Motion with DerivativesDraft — Position, velocity, and acceleration of a particle on a line — speed, direction, changing direction, and speeding up versus slowing down.
  3. Introduction to Related RatesDraft — 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.
  4. Related Rates ProblemsDraft — A step-by-step strategy for multi-step related rates — sliding ladders, filling cones, moving shadows, and approaching cars.
  5. Linearization and Tangent Line ApproximationDraft — 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.
  6. L'Hospital's RuleDraft — Using derivatives to evaluate limits of the indeterminate forms 0/0 and ∞/∞ — checking the form first, and knowing when not to use the rule.

Unit 5: Analytical Applications of Differentiation

  1. Mean Value TheoremDraft — The Mean Value Theorem and Rolle's theorem — checking the hypotheses, finding c, and writing the justification AP graders expect.
  2. Critical Points and ExtremaDraft — Absolute and relative maximums and minimums, the Extreme Value Theorem, and finding critical points where f'(x) = 0 or f'(x) does not exist.
  3. First Derivative TestDraft — 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.
  4. Candidates Test for Absolute ExtremaDraft — Finding the absolute maximum and minimum of a continuous function on a closed interval by checking every critical point and both endpoints.
  5. Concavity and the Second Derivative TestDraft — 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.
  6. Connecting f, f′, and f″Draft — 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.
  7. OptimizationDraft — Setting up and solving optimization problems with derivatives — variables, constraint, objective function, domain — and justifying the answer for fencing, boxes, cans, cost, and distance.
  8. Analyzing Implicit RelationsDraft — 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.

Unit 6: Integration and Accumulation of Change

  1. Accumulation of ChangeDraft — Why the area under a rate graph tells you the total change, with units, and how positive and negative contributions add up.
  2. Riemann SumsDraft — Approximating the area under a curve with left, right, midpoint, and trapezoidal sums, from graphs, tables, and formulas.
  3. The Definite IntegralDraft — The definite integral as the limit of Riemann sums, sigma notation, and translating between a limit of a sum and an integral.
  4. Accumulation Functions and the FTCDraft — 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.
  5. Properties of Definite IntegralsDraft — Reversing limits, zero-width intervals, adding intervals, constant multiples and sums, and evaluating integrals with geometry.
  6. Evaluating Definite Integrals with the FTCDraft — The Fundamental Theorem of Calculus, part 2 — evaluating a definite integral as F(b) − F(a) using an antiderivative, and the net change theorem.
  7. Antiderivatives and Indefinite IntegralsDraft — Finding antiderivatives with the basic rules (power, 1/x, exponential, trig, inverse trig), the + C, and solving initial-value problems.
  8. Integration by SubstitutionDraft — Undoing the chain rule with u-substitution, for indefinite and definite integrals (including changing the limits), and choosing an integration technique.
  9. Integrating with Long Division and Completing the SquareDraft — Integrating rational functions by first using polynomial long division, or by completing the square to get an arctangent.

Unit 7: Differential Equations

  1. Introduction to Differential EquationsDraft — Writing differential equations from verbal descriptions, verifying solutions by substitution, and telling general solutions from particular ones.
  2. Slope FieldsDraft — Sketching slope fields from a differential equation, matching fields to equations, and drawing and reading solution curves.
  3. Separation of VariablesDraft — Solving separable differential equations, finding particular solutions from initial conditions, and stating where a solution is defined.
  4. Exponential Models with Differential EquationsDraft — 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.

Unit 8: Applications of Integration

  1. Average Value of a FunctionDraft — 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.
  2. Motion Along a Line with IntegralsDraft — Going backwards from velocity to position — initial conditions, displacement versus total distance, and velocity from acceleration.
  3. Definite Integrals in ContextDraft — 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.
  4. Area Between CurvesDraft — 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.
  5. Area Between Curves Using Horizontal SlicesDraft — 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.
  6. Volumes with Known Cross SectionsDraft — Finding volumes by slicing — squares, rectangles, triangles, and semicircles standing on a base region, perpendicular to the x-axis or the y-axis.
  7. Volumes of Revolution — the Disc MethodDraft — 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.
  8. Volumes of Revolution — the Washer MethodDraft — 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.