AP Calculus BC

331 core topics + 85 prerequisite topics taught as needed · approximately 127 hours of instruction including spaced review

How the course runs

An adaptive diagnostic (up to 40 questions) places the student on the course's knowledge graph - topics already known are credited, and instruction begins exactly at the learning frontier. Every topic is taught with a worked-example lesson and auto-graded practice; a topic is mastered at 75%+ and then maintained through spaced reviews on an expanding schedule. Mixed checks follow every 6 lessons; each unit ends with a 12-item quiz, and course-wide assessments appear at 25%, 50%, 75%, and 100% mastery. Prerequisite gaps below the course are detected and taught rather than skipped, so completion certifies the whole tower, not just the top.

Core curriculum

Limits & Continuity · 10 topics

Limits: Graphical & Numerical [E]	What value a function approaches - which need not be the value it takes.
Evaluating Limits Algebraically [M]	Direct substitution - and the factor-and-cancel fix for 0/0.
One-Sided Limits [M]	Approaching from the left or right - and when the two disagree.
Limits by Rationalization [H]	Clearing a 0/0 form by multiplying by the conjugate.
Infinite Limits & Vertical Asymptotes [M]	A limit of +∞ or −∞ at a vertical asymptote; the sign of the shrinking denominator decides which.
Limits at Infinity [M]	The value a function approaches as x goes to +∞ or −∞; for a rational function, compare the degrees of the top and bottom.
Trig Limits [M]	Limits of trig expressions, evaluated with the standard results sin(x)/x → 1 and (1 − cos x)/x → 0 as x → 0.
The Squeeze Theorem [M]	If g(x) ≤ f(x) ≤ h(x) near a point and g and h share the limit L there, then f has limit L too.
Continuity [H]	No breaks in the graph: at the point, the limit equals the function's value.
Intermediate Value Theorem [M]	A continuous function can't skip values: sign changes force roots.

Differentiation · 16 topics

The Limit Definition of the Derivative [M]	The derivative is the limit of average rates of change.
Derivatives Graphically [M]	Reading slopes off a graph.
The Power Rule [M]	d/dx xⁿ = n·xⁿ⁻¹ for any real n.
Sum & Constant-Multiple Rules [M]	Differentiate term by term.
The Product Rule [M]	(fg)′ = f′g + fg′.
The Quotient Rule [M]	(f/g)′ = (f′g − fg′)/g².
Derivatives of Trig Functions [M]	d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec²x.
Derivatives of Exponentials & Logs [M]	eˣ is its own derivative; (ln x)′ = 1/x.
The Chain Rule [H]	d/dx f(g(x)) = f′(g(x)) · g′(x).
Combining Differentiation Rules [H]	Product, quotient and chain rules together.
Implicit Differentiation [H]	Differentiating equations that mix x and y.
Derivatives of Inverse Trig [M]	(arcsin x)′ = 1/√(1−x²), (arctan x)′ = 1/(1+x²).
Derivatives of Inverse Functions [M]	(f⁻¹)′(b) = 1 / f′(f⁻¹(b)).
Higher-Order Derivatives [M]	Differentiating again: f″, f‴, …
Differentiability & Continuity [M]	Differentiable ⇒ continuous, but not conversely.
Tangent Lines & Linear Approximation [M]	The tangent line is the best local linear stand-in for f.

Applications of Differentiation · 10 topics

Motion: Position, Velocity, Acceleration [M]	v = s′, a = v′; at rest when v = 0.
Related Rates [H]	Differentiating a geometric relationship with respect to time.
Critical Points & Extrema [M]	Where f′ = 0 or is undefined - the candidates for extrema.
The Mean Value Theorem [M]	Somewhere, instantaneous rate equals average rate.
Increasing & Decreasing Intervals [M]	Sign of f′ decides the direction of f.
Concavity & Inflection Points [M]	f″ > 0 bends up, f″ < 0 bends down.
Curve Sketching & f, f', f'' [M]	Reading the shape of f from its derivatives.
Optimization [H]	Maximizing or minimizing with calculus.
L'Hôpital's Rule [M]	For 0/0 or ∞/∞, differentiate top and bottom.
Indeterminate Forms [M]	0·∞ and repeated applications.

Integration · 12 topics

Antiderivatives [M]	Reversing differentiation: the power rule backwards.
Antiderivatives: Trig & Exponential [M]	∫cos = sin, ∫sin = −cos, ∫eˣ = eˣ, ∫1/x = ln|x|.
Riemann Sums [M]	Approximating area with rectangles.
The Trapezoidal Rule [M]	Averaging left and right sums.
Properties of Definite Integrals [M]	Linearity, additivity, and orientation.
The Fundamental Theorem: Evaluating Integrals [M]	∫ₐᵇ f = F(b) − F(a).
Accumulation Functions & FTC Part 1 [H]	d/dx ∫ₐˣ f(t) dt = f(x).
u-Substitution [H]	The chain rule in reverse.
Integration by Parts (BC) [H]	∫u dv = uv − ∫v du.
Partial Fractions (BC) [H]	Splitting rational functions to integrate them.
Improper Integrals (BC) [H]	Integrals to infinity, defined by limits.
Integrals Yielding Inverse Trig [H]	1/(1+x²) → arctan, 1/√(1−x²) → arcsin.

Applications of Integration · 8 topics

Average Value of a Function [M]	f_avg = (1/(b−a)) ∫ₐᵇ f.
Motion: Displacement & Distance [H]	Displacement is ∫v; distance is ∫|v|.
Accumulation & Net Change [H]	Final amount = initial + ∫(rate).
Analyzing Accumulation Functions [H]	Reading g(x) = ∫f from the graph of f.
Area Between Curves [H]	∫(top − bottom) between the intersections.
Volumes: Disc & Washer [H]	V = π∫R² dx (discs), π∫(R² − r²) dx (washers).
Volumes by Cross-Section [H]	V = ∫A(x) dx for known cross-sectional areas.
Arc Length (BC) [H]	L = ∫√(1 + (y′)²) dx.

Differential Equations · 6 topics

Differential Equations: Verifying Solutions [M]	A solution is a function that satisfies the equation.
Slope Fields [M]	A picture of dy/dx at every point.
Euler's Method (BC) [H]	Stepping along tangent lines.
Separation of Variables [H]	Move all y's left, all x's right, integrate both sides.
Exponential Growth & Decay Models [M]	dy/dt = ky means y = y₀e^{kt}.
Logistic Growth (BC) [H]	Growth limited by a carrying capacity L.

Parametric, Polar & Vector Calculus · 8 topics

Parametric Derivatives (BC) [H]	dy/dx = (dy/dt)/(dx/dt).
Parametric Second Derivatives (BC) [H]	Differentiate dy/dx with respect to t, divide by dx/dt again.
Parametric Arc Length (BC) [H]	L = ∫√((dx/dt)² + (dy/dt)²) dt.
Vector-Valued Functions (BC) [H]	Differentiate component by component.
Motion in the Plane (BC) [H]	Speed is the magnitude of velocity.
Slopes of Polar Curves (BC) [H]	Convert to parametric: x = r cos θ, y = r sin θ.
Area in Polar Coordinates (BC) [H]	A = ½∫r² dθ.
Area Between Polar Curves (BC) [H]	Subtract the inner sweep from the outer sweep.

Infinite Series · 15 topics

Convergence of Sequences (BC) [M]	A sequence converges if aₙ approaches a limit.
Geometric Series (BC) [M]	Σarⁿ = a/(1−r) when |r| < 1.
The nth-Term Test (BC) [M]	If terms don't go to 0, the series diverges - but 0 proves nothing.
Integral Test & p-Series (BC) [M]	Σ1/nᵖ converges iff p > 1.
Comparison Tests (BC) [M]	Compare with a series you already understand.
Alternating Series (BC) [M]	Alternating + decreasing to 0 ⇒ converges.
Alternating Series Error Bound (BC) [H]	|error| ≤ first omitted term.
The Ratio Test (BC) [H]	L = lim|aₙ₊₁/aₙ|: L<1 converges, L>1 diverges, L=1 says nothing.
Absolute vs Conditional Convergence (BC) [M]	Does it still converge with all terms made positive?
Radius of Convergence (BC) [H]	The ratio test gives |x − c| < R.
Interval of Convergence (BC) [H]	Check both endpoints separately.
Taylor Polynomials (BC) [H]	Matching derivatives at a point: Pₙ(x) = Σ f⁽ᵏ⁾(a)(x−a)ᵏ/k!.
Taylor & Maclaurin Series (BC) [H]	The big four: eˣ, sin x, cos x, 1/(1−x).
Manipulating Known Series (BC) [H]	Substitute, multiply, differentiate, integrate known series.
Lagrange Error Bound (BC) [H]	|Rₙ| ≤ M|x−a|ⁿ⁺¹/(n+1)!.

AP Calculus: Limits, Continuity & Differentiation - Depth · 12 topics

Classifying Discontinuities [M]	Removable, jump, or infinite - three ways continuity can fail.
Limits of Piecewise Functions [M]	One-sided limits pick the piece the input comes from.
Continuity on an Interval & Existence [M]	Match the pieces; then let the IVT do the existence work.
The Extreme Value Theorem [M]	Continuous on a closed interval ⇒ an absolute max and min exist.
The First-Derivative Test [M]	Where f′ switches + → − is a max; − → + is a min.
The Second-Derivative Test [M]	At a critical point, concavity settles max vs. min.
The Candidates Test for Absolute Extrema [H]	On [a, b], the absolute extremes hide among critical points and endpoints.
Reasoning Across f, f′ and f″ [M]	Given one graph, deduce the sign story of the others.
Linearization & Differentials [M]	L(x) = f(a) + f′(a)(x − a); concavity says over or under.
Indeterminate Forms in Depth [H]	Turn 0·∞, ∞−∞ and 1^∞ into a quotient, then apply L'Hôpital.
Interpreting the Derivative in Context [M]	f′(a) is a rate: its units are (units of f) per (unit of x).
Rates of Change in Applied Contexts [M]	Instantaneous rate is a derivative; average rate is a difference quotient.

AP Calculus: Integration & Applications - Depth · 11 topics

Riemann Sums in Sigma Notation [M]	Package a Riemann sum as Σ f(x_i)·Δx and use summation formulas.
The Definite Integral as a Limit [M]	Send n → ∞ and the Riemann sum becomes the definite integral.
Over- and Under-Estimates of Sums [M]	Monotonicity fixes left/right; concavity fixes the trapezoid.
Area Between Curves (with respect to y) [M]	When curves are functions of y, integrate (right − left) dy.
Volumes About a Non-Axis Line [H]	Shift the radii: distance to y = k, not to the axis.
Accumulation Functions & FTC Analysis [H]	g(x) = ∫ₐˣ f: its slope is f, so f's sign shapes g.
Reasoning with Slope Fields [M]	Read where slopes are zero, and whether they depend on x or y.
Interpreting a Definite Integral in Context [M]	∫ of a rate is a net change - with the rate's units × x's units.
Average Value vs. Average Rate of Change [M]	One is an integral ÷ length; the other is a difference ÷ length.
Total Distance vs. Displacement [H]	Displacement integrates v; distance integrates |v|.
Setting Up a Differential Equation [M]	Translate 'rate proportional to …' into dy/dt = k·(…).

Calculus - Limits & Continuity (Deep) · 37 topics

Combining Limits with the Limit Laws [H]	The limit of a sum, product, or quotient is that operation on the limits.
Limits by Direct Substitution [H]	Where a function is continuous, the limit is just the value there.
0/0 Limits by Non-Monic Factoring [H]	Factor the shared (x - c) from top and bottom, cancel, then substitute.
0/0 Limits with Cubes and Higher Powers [H]	Difference-of-powers factoring cancels the zero: xⁿ - aⁿ has factor x - a.
0/0 Limits with Compound Fractions [H]	Combine the little fractions over a common denominator, then cancel.
Conjugate Limits (Radical in the Denominator) [H]	Multiply top and bottom by the denominator's conjugate to clear the 0/0.
Conjugate Limits with a Rational Value [H]	Conjugate the radical numerator; the leftover root evaluates to a number.
One-Sided Limits of Absolute-Value Functions [H]	|x - c|/(x - c) is +1 on the right of c and -1 on the left.
One-Sided Limits of Step Functions [H]	The floor function drops by 1 across each integer: a jump discontinuity.
Limits at Infinity with Radicals [H]	For large |x|, √(ax² + ...) behaves like √a·|x|; watch the sign at -∞.
Growth Rates: Exponential vs Polynomial vs Log [H]	As x → ∞, exponentials beat any power, and any power beats logarithms.
Counting Horizontal Asymptotes [H]	Each finite limit as x → +∞ and as x → -∞ gives one horizontal asymptote.
Bounded-Over-Growing Limits at Infinity [H]	A bounded numerator divided by something growing to ∞ tends to 0.
Identifying Vertical Asymptotes [H]	A denominator zero that does NOT cancel is a vertical asymptote; one that cancels is a hole.
Sign of a One-Sided Infinite Limit [H]	Near an asymptote, count the signs of every factor to choose +∞ or -∞.
Infinite Limits of ln, tan, and Reciprocal Powers [H]	Infinite limits of ln x, tan x, and 1/xⁿ, where each grows without bound at the given point.
The Squeeze Theorem with Computed Bounds [H]	When the two bounding functions meet at x = c, evaluate them to get L.
The Squeeze Theorem on Bounded Oscillation [H]	A factor going to 0 times a bounded oscillation is squeezed to 0.
The (1 - cos)/x² Limit [H]	As x → 0, (1 - cos x)/x² → 1/2; scaling the angle scales by the square.
Products and Quotients of sin, tan at 0 [H]	Rewrite each factor as sin(kx)/(kx) or tan(kx)/(kx), each → 1, then read off the constants.
Trig Limits Needing a Substitution [H]	A shift like u = x - π turns an unfamiliar trig limit into a standard one.
Classifying Discontinuities of Rational Functions [H]	A canceling factor gives a removable hole; a surviving one gives an infinite discontinuity.
Counting Discontinuities [H]	Every zero of the original denominator is a discontinuity; canceling ones are the holes.
Which Continuity Condition Fails [H]	Continuity needs f(c) defined, the limit to exist, and the two to agree.
Gluing Pieces with a Coefficient Unknown [H]	Set the two pieces equal at the boundary and solve for the unknown coefficient.
Filling a Removable Hole [H]	Cancel the common factor, then evaluate the simplified form to get k.
Where Is a Function Continuous [H]	Rational functions break at denominator zeros; roots need the inside ≥ 0.
Continuity on an Interval [H]	The interval of continuity is the domain: root needs ≥ 0, log needs > 0.
Counting Guaranteed Roots with the IVT [H]	Each sign change between consecutive points guarantees at least one root.
Guaranteeing a Value with the IVT [H]	The IVT only guarantees values that lie between f(a) and f(b).
The IVT in Applied Contexts [H]	A continuously changing real quantity attains every value between its endpoints.
Limit Definition on a General Quadratic [H]	Expand f(a + h), subtract f(a), divide by h, cancel, then let h → 0.
Recognizing a Difference Quotient [H]	Match lim (f(a + h) - f(a))/h to its f and a by reading the base and the power.
Limit Definition for Roots and Reciprocals [H]	Rationalize (for √x) or combine over a common denominator (for 1/x), then let h → 0.
The x → a Form of the Derivative Limit [H]	f′(a) = lim (f(x) - f(a))/(x - a); factor out (x - a) and substitute.
Two-Sided Limit vs the Function Value [H]	When both one-sided limits agree, the limit is their common value - not f(c).
Limits of Composite Functions [H]	If the outer function is continuous, take the limit inside first.

Calculus - Differentiation Rules (Deep) · 38 topics

Power Rule for Negative Exponents [H]	Rewrite 1/x^n as x^(-n), then bring the negative exponent down.
Power Rule for Rational Exponents [H]	d/dx x^(p/q) = (p/q) x^((p-q)/q) - subtract 1 from the fraction.
Rewrite Radicals and Reciprocals, then Differentiate [H]	Turn every root and reciprocal into a power of x before applying the rule.
Differentiating a Polynomial (the Derivative Function) [H]	Differentiate term by term to get f' as a function, then simplify.
Product Rule: Polynomial Times Polynomial [H]	(fg)' = f'g + fg'; differentiate each factor, keep the other whole.
Product Rule with a Radical Factor [H]	One factor is a root: differentiate it as x^(1/2) inside the product rule.
Product Rule for Three Factors [H]	(fgh)' = f'gh + fg'h + fgh' - one term per factor differentiated.
Quotient Rule on Rational Functions [H]	(f/g)' = (f'g - fg')/g^2; the numerator collapses to a constant here.
Quotient Rule with a Trig Function [H]	Numerator or denominator is trig: apply (f'g - fg')/g^2 with cos/-sin.
Evaluating a Quotient-Rule Derivative [H]	Differentiate with the quotient rule, then substitute the x-value.
Chain Rule: a Power of a Polynomial [H]	d/dx u^n = n u^(n-1) u'; the inner derivative is the outside factor.
Chain Rule: Trig of a Polynomial Argument [H]	d/dx sin(u) = cos(u) u'; the argument's derivative multiplies on.
Chain Rule: Exponential of a Function [H]	d/dx e^u = e^u u'; the exponential is reproduced, times u'.
Chain Rule: Logarithm of a Function [H]	d/dx ln(u) = u'/u - the inside derivative over the inside.
The Chain Rule Twice (Nested Compositions) [H]	Peel one layer at a time: outer, then middle, then inner.
Implicit Differentiation: Solving for dy/dx [H]	Differentiate every term (each y gives a y' factor), then solve for y'.
Implicit Differentiation at a Point [H]	Find dy/dx implicitly, then substitute both coordinates of the point.
Implicit Differentiation of a Product of Powers [H]	Differentiate x^m y^n as a product; the a on the right has derivative 0.
Derivative of a^x [H]	d/dx a^x = a^x ln a - the base exponential picks up a factor of ln a.
Derivative of log_a(x) [H]	d/dx log_a x = 1/(x ln a) - divide the natural-log rule by ln a.
Derivative of a^(g(x)) [H]	d/dx a^u = a^u (ln a) u' - base rule times the argument's derivative.
Logarithmic Differentiation of x^(ax) [H]	Take ln of both sides, differentiate implicitly, then multiply back by f.
Derivatives of tan, sec, csc, and cot [H]	The four reciprocal-family rules, with the two minus signs on csc and cot.
Trig Derivatives with a Coefficient Argument [H]	d/dx trig(kx) multiplies the basic rule by the inside derivative k.
Products with Trig Functions [H]	Combine the product rule with a trig derivative.
Exact Trig Derivative Values at Special Angles [H]	Differentiate, then evaluate with exact unit-circle values.
Derivatives of arccos and arccot [H]	The co-inverse rules are the negatives of arcsin's and arctan's.
Inverse Trig with the Chain Rule [H]	d/dx arctan(u) = u'/(1+u^2); the inside derivative multiplies on top.
Evaluating Inverse-Trig Derivatives [H]	Substitute into 1/(1+x^2) (arctan) or -1/(1+x^2) (arccot).
The Second Derivative as a Function [H]	Differentiate twice; report f'' as an expression in x.
n-th Derivative Patterns [H]	Repeated differentiation of e^(kx), ln x and 1/x follows a clean pattern.
Second Derivatives of Sine and Cosine [H]	Two derivatives of sin(kx) return -k^2 times the original sinusoid.
Making a Piecewise Function Differentiable [H]	Differentiable needs BOTH matching slopes and matching values at the join.
Corners, Cusps, and Vertical Tangents [H]	Continuous but not differentiable: corner, cusp, or vertical tangent.
Differentiability Implies Continuity (Not Conversely) [H]	Differentiable forces continuous; continuous does not force differentiable.
Estimating a Derivative from a Table [H]	Approximate f'(x1) by the symmetric difference (f(x2)-f(x0))/(x2-x0).
Reading f'(x) off a Piecewise-Linear Graph [H]	On a straight segment, f'(x) is that segment's constant slope.
The Chain Rule from a Table of Values [H]	h'(a) = f'(g(a)) g'(a): look up f' at the INNER output g(a), not at a.

Calculus - Applications of Differentiation (Deep) · 36 topics

Related Rates: Expanding Ripple [H]	Differentiate the circle relation, then solve for the wanted rate.
Related Rates: Inflating Balloon [H]	Differentiate the sphere formula; a radius rate and a volume rate trade.
Related Rates: Draining/Filling Cone [H]	Use similar triangles to reduce V to one variable, then differentiate.
Related Rates: Walking Shadow [H]	Similar triangles relate shadow length to distance; differentiate.
Related Rates: Sliding-Ladder Triangle Area [H]	Combine the ladder's dy/dt with the product rule on A = (1/2)xy.
Related Rates: Differentiating the Relation [H]	Every variable that changes with time carries a chain-rule factor.
Differentials: Estimating a Change [H]	dy = f'(x) dx estimates a small change from the derivative.
Linear Approximation of Roots & Reciprocals [H]	L(x) = f(a) + f'(a)(x - a) with a chosen so f(a) is exact.
Differentials: Relative Error Propagation [H]	For a power law y = x^n, the relative error multiplies by n.
L'Hopital: 0/0 Trigonometric Limits [H]	Confirm 0/0, then differentiate top and bottom (sometimes twice).
L'Hopital: Infinity Over Infinity [H]	For a rational or radical quotient, the leading terms decide.
Recognizing Indeterminate Forms [H]	L'Hopital needs 0/0 or infinity/infinity - check before applying.
Mean Value Theorem: Solving for c [H]	Set f'(c) equal to the average rate, then solve for c in (a, b).
Rolle's Theorem: Solving for c [H]	Equal endpoint values force a flat spot; solve f'(c) = 0 inside.
Bounding a Function with the MVT [H]	f(b) - f(a) = f'(c)(b - a), so derivative bounds bound the change.
Rolle & MVT: Checking the Hypotheses [H]	Continuity on [a,b], differentiability on (a,b), and (Rolle) equal ends.
First-Derivative Test with a Repeated Factor [H]	A squared factor gives a critical point with no sign change - no extremum.
Counting Critical Points [H]	Count zeros of f' AND points where f' is undefined; a double root is one.
Critical Points via the Product/Quotient Rule [H]	Differentiate the product or quotient, factor, and set equal to zero.
Second Derivative at a Critical Point [H]	Find the critical points from f', then evaluate f'' there.
Second-Derivative Test: Classifying Extrema [H]	f'' > 0 means min, f'' < 0 means max, f'' = 0 means inconclusive.
Absolute Extrema: The Candidates Test [H]	Evaluate f at every interior critical point and both endpoints.
Absolute Extrema: Where They Occur [H]	The winner can be an endpoint - compare all candidates, do not assume.
Absolute Extrema of a Rational Function [H]	Same Candidates Test - just differentiate the quotient to find the crit.
Optimization: Fence Against a River [H]	One free side changes the constraint: fence = 2x + y, area = x(L - 2x).
Optimization: Open Box from a Sheet [H]	V(x) = x(a - 2x)^2; the maximizing cut is x = a/6.
Optimization: Minimum-Cost Enclosure [H]	Fix the area constraint, write cost as C(x), minimize with calculus.
Optimization: Constrained Two-Number Problems [H]	Use the constraint to reduce to one variable, then optimize.
Optimization: Building the Objective Function [H]	Translate the constraint into a one-variable formula for what to optimize.
Motion: Speeding Up or Slowing Down [H]	Speeding up when velocity and acceleration share a sign.
Motion: Total Distance via Turning Points [H]	Split at the times v = 0, then add the absolute position changes.
Motion: Maximum Velocity [H]	Velocity is extreme where acceleration is zero (or at an endpoint).
Inflection Points of a Quartic [H]	Solve f'' = 0 and confirm the sign of f'' actually flips.
Concavity Intervals of a Cubic [H]	Concave up where f'' > 0; a cubic switches once at its inflection.
Verifying an Inflection Point [H]	f'' = 0 is necessary but not sufficient - the sign must actually flip.
Counting Inflection Points from f'' [H]	Count the sign changes of f'' - simple roots flip, even-power roots do not.

Calculus - Integration Techniques (Deep) · 37 topics

Antiderivative of a Coefficient Times a Power [H]	A constant multiple rides along: raise the power by one and divide.
Antiderivative of a Polynomial (as a Function) [H]	Antidifferentiate term by term; the sum rule lets each power act alone.
Antiderivative of a Negative Power [H]	The power rule still works for x^(-n) (as long as n is not 1).
Antiderivative of a Radical (Rational Power) [H]	Write the root as a fractional power, then add one to the exponent.
Reciprocal-Trig Antiderivatives [H]	Read the sec/csc derivative table backward.
Antiderivative of an Exponential Base a [H]	For a base other than e, divide by ln a.
FTC: A Polynomial Over a General Interval [H]	Antidifferentiate, then subtract F(a) from F(b) - with real endpoints.
FTC: Trig Integrals at Exact Angles [H]	Antidifferentiate the sinusoid, then read an exact unit-circle value.
FTC: Exponential Integrals [H]	The antiderivative of e^(kx) is e^(kx)/k; then subtract the endpoints.
FTC: The 1/x Integral (Logarithms) [H]	The antiderivative of 1/x is ln|x|, so the integral is a difference of logs.
FTC: Definite Integral of a Negative Power [H]	Antidifferentiate x^(-p) to x^(1-p)/(1-p), then subtract the endpoints.
FTC: Integrating an Absolute Value [H]	Split at the corner where the inside changes sign; add two triangle areas.
u-Substitution: Indefinite Polynomial Form [H]	The lone x supplies half of du = 2x dx; integrate u^n, then divide.
u-Substitution: Definite Polynomial Form [H]	Change the limits with u, then evaluate in u - no need to resubstitute.
u-Substitution: Powers of Sine or Cosine [H]	Let u be the base trig function; its derivative is the leftover factor.
u-Substitution: Gaussian-Type Exponential [H]	Let u be the exponent; the outside x is (part of) du.
u-Substitution: The ln|u| Pattern [H]	When the top is the derivative of the bottom, the integral is a log.
u-Substitution With a Radical (Change of Limits) [H]	Let u be the inside of the root; the outer x completes du.
FTC Part 1: The Derivative Function [H]	d/dx of an accumulation function just hands back the integrand at x.
FTC Part 1 With the Chain Rule [H]	Upper limit u(x): evaluate f at u, then multiply by u'.
FTC Part 1: Variable Lower Limit [H]	x on the bottom flips the sign: g'(x) = -f(x).
FTC Part 1: Both Limits Are Functions [H]	Top minus bottom: f(upper)*upper' - f(lower)*lower'.
FTC Part 1: Shape of the Accumulation [H]	Since g' = f, the sign and slope of f dictate g's shape.
Midpoint Riemann Sum From a Function [H]	Sample each subinterval at its center, not an endpoint.
Left/Right Riemann Sum From a Function [H]	Build the sample heights yourself from the formula, not a table.
Trapezoidal Rule From a Function [H]	Half-weight the two ends, full-weight every interior point.
A Riemann-Sum Limit as an Integral [H]	Read off a, b, and f from the width and the sample point.
Expressing an Integral as a Riemann Sum [M]	Width is (b - a)/n; the right endpoint starts at a and steps by that width.
Odd/Even Symmetry of Definite Integrals [H]	Odd terms vanish over [-a, a]; even terms double the half-integral.
Splitting Off an Adjacent Interval [H]	Additivity rearranged: the piece from b to c is the whole minus the first.
Linearity Across Two Functions [H]	The integral of a linear combination is the same combination of integrals.
Adding a Constant to the Integrand [H]	Adding c raises the integral by c times the interval length.
A Definite Integral as a Geometric Area [H]	y = sqrt(r^2 - x^2) is a semicircle; use the circle-area formula.
By Parts: Polynomial Times Exponential [H]	Let u = x (it differentiates away) and dv = e^(ax) dx.
By Parts: Polynomial Times Trig [H]	Let u = x; dv is the sinusoid, which integrates cleanly.
By Parts: Logarithmic Integrands [H]	Let u = ln x (it differentiates to 1/x) and dv be the power.
By Parts: Exact Definite Values [H]	Find the antiderivative by parts, then evaluate at the limits.

Calculus - Applications of Integration (Deep) · 37 topics

Area Between a Line and a Parabola [H]	Integrate (top - bottom) between the two intersection points.
Area When Curves Cross (Splitting the Integral) [H]	Where the top and bottom curve swap, split and add the pieces.
Area Between Trigonometric Curves [H]	Integrate the difference of the trig curves; cos is above sin on [0, pi/4].
Area Between Curves (Integrating in y) [H]	When curves are x = g(y), integrate (right - left) dy.
Area of a Sideways-Parabola Region [H]	A leftward parabola meets x = 0 at y = +-a; integrate its width in y.
Disc Method About the x-axis [H]	Each disc has radius equal to the curve; V = pi * integral of R^2 dx.
Disc Method About the y-axis [H]	Slice in y: radius is x = sqrt(y), so V = pi * integral of y dy.
Washer Method About the x-axis [H]	Outer radius minus inner radius: V = pi * integral of (R^2 - r^2) dx.
Washer Method About the y-axis [H]	Slice in y: outer radius sqrt(y), inner radius y/a.
Washers About a Horizontal Line [H]	Measure both radii from the line y = c, not from the axis.
Cylindrical Shells About the y-axis [H]	A shell has radius x and height f(x): V = 2*pi * integral of x f(x) dx.
Cylindrical Shells About the x-axis [H]	Slice in y: shell radius y, height (k - sqrt(y)).
Cylindrical Shells About a Vertical Line [H]	Shell radius is the distance c - x to the line x = c.
Square Cross-Sections on a Two-Curve Base [H]	The side of each square is the gap between the two curves.
Semicircular Cross-Sections [H]	Semicircle on diameter w has area (pi/8) w^2.
Equilateral-Triangle Cross-Sections [H]	Equilateral triangle of side s has area (sqrt3/4) s^2.
Isosceles-Right-Triangle Cross-Sections [H]	Leg equal to base width w gives area (1/2) w^2.
Cross-Sections Perpendicular to the y-axis [H]	Slice in y: the base width is x = sqrt(y), so a square has area y.
Square Cross-Sections on a Circular Base [H]	At position x the chord has length 2 sqrt(R^2 - x^2); square it.
Average Value of a Quadratic [H]	Average value is the integral over [a, b] divided by (b - a).
Average Value of a Trigonometric Function [H]	Average value pulls a 1/(b-a) factor out front; here 1/pi.
The Mean Value Theorem for Integrals [H]	Solve f(c) = average value for c inside the interval.
Average Value of a Rate [H]	Average rate = (net change)/(elapsed time) = integral of r, over the time.
Net Change from a Polynomial Rate [H]	Net change is the definite integral of the rate.
Final Amount from a Starting Value [H]	Final amount = initial amount + integral of the rate.
Net Change with Competing Rates [H]	Net change integrates (inflow rate - outflow rate).
Interpreting and Naming a Net Change [H]	Integrating a rate gives a total, with the rate's numerator units.
Displacement from a Quadratic Velocity [H]	Displacement is the signed integral of velocity.
Total Distance from a Quadratic Velocity [H]	Total distance integrates |v|, so split where v changes sign.
Position from Velocity and an Initial Value [H]	s(T) = s(0) + integral of v from 0 to T.
Speeding Up vs Slowing Down [H]	Speeding up when velocity and acceleration share a sign.
Average Velocity vs Average Speed [H]	Average velocity uses displacement; average speed uses total distance.
Exact Arc Length of a Rational Curve [H]	For this curve 1 + (y')^2 is a perfect square, so the root simplifies.
Arc Length of a Slanted Line [H]	A line's length is sqrt(1 + m^2) times the run.
Setting Up the Arc-Length Integrand [H]	Differentiate, square, add 1, then take the square root.
Choosing the Integral for an Area [H]	Use the full interval between intersections and top minus bottom.
Choosing a Volume Method [H]	Match discs/washers vs shells to the axis and the slice direction.

Calculus - Differential Equations (Deep) · 38 topics

Verifying: Which Function Solves It [H]	Differentiate each candidate and substitute; only a genuine solution fits.
Verifying: Constant of a Shifted Solution [H]	Plug the initial condition into the general solution and solve for C.
Verifying: Constant of an Implicit Solution [H]	An implicit solution is a level curve; the initial point fixes its level.
Verifying: Evaluating a Solution Family [H]	Use the initial value to find C, then substitute the requested x.
Separable: General Antiderivative Solution [H]	When dy/dx depends on x only, y is just an antiderivative plus C.
Separable: Solving Explicitly for y [H]	Separate, integrate both sides, use the initial condition, then isolate y.
Separable: A Reciprocal Right Side [H]	Clear the reciprocal by cross-multiplying the differentials first.
Separable: Exponential of a Square [H]	Dividing by y gives dy/y = f(x)dx; the integral of dy/y is ln|y|.
Separable: A Trigonometric Rate [H]	Same recipe, with a trig antiderivative: the integral of cos x is sin x.
Separable: Implicit-Solution Constant [H]	Separate to an implicit relation, then let the point fix the constant.
Separable: Evaluating a Particular Solution [H]	dy/dx = y/x separates to ln|y| = ln|x| + C, a line through the origin.
Slope Fields: Slope at a Point [H]	The slope of a segment is just dy/dx evaluated at that point.
Slope Fields: Matching a Field to an Equation [H]	Read where segments are horizontal and whether they vary with x or y.
Slope Fields: Where Segments Are Horizontal [H]	Horizontal segments occur exactly on the curve F(x, y) = 0.
Slope Fields: Equilibrium Solutions [H]	Constant solutions are the y-values where dy/dx is zero for all x.
Slope Fields: Long-Run Behavior [H]	Track the sign of dy/dx on each side of an equilibrium to see the drift.
Slope Fields: Concavity of a Solution Curve [H]	Differentiate dy/dx again (chain rule) to read a solution's concavity.
Exponential Models: Recovering the Rate Constant [H]	From y0·e^(kt) = (factor)·y0, take a logarithm to solve for k.
Exponential Models: Evaluating the Solution [H]	Solve dP/dt = kP as P = P0·e^(kt), then substitute the time.
Exponential Models: Doubling Time [H]	Doubling means e^(kt) = 2, so t = ln 2 / k.
Exponential Models: Half-Life and k [H]	Half-life means e^(-kH) = 1/2, so k = ln 2 / H.
Exponential Models: Relative Rate as k [H]	In dB/dt = kB, k is the constant relative (per-unit) growth rate.
Exponential Models: Continuous Decay Amount [H]	Count how many reduction periods fit in the time, then divide repeatedly.
Newton's Cooling: Evaluating the Model [H]	Substitute the time into T = T_s + (T_0 - T_s)e^(kt).
Newton's Cooling: Limiting Temperature [H]	As t grows the exponential dies, leaving the surrounding temperature.
Newton's Cooling: Recovering k [H]	Subtract the ambient temperature, form the ratio, then take a log.
Newton's Cooling: Solving for a Time [H]	Set T equal to the target, isolate the exponential, then take a log.
Logistic: Carrying Capacity from aP - bP^2 [H]	Factor dP/dt = aP - bP^2 = bP(a/b - P); the capacity is a/b.
Logistic: Fastest-Growth Population [H]	The parabola aP - bP^2 peaks at P = a/(2b), which is half the capacity.
Logistic: Maximum Growth Rate [H]	The peak rate is the vertex height a^2/(4b), reached at P = L/2.
Logistic: Sign of the Growth Rate [H]	Below L the rate is positive; above L it is negative; solutions tend to L.
Logistic: Rate at a Given Population [H]	Just substitute the population into the rate expression aP - bP^2.
Logistic: Inflection of the Solution Curve [H]	The S-curve changes concavity where growth is fastest, at P = L/2.
Euler's Method: One Step [H]	One step rides the tangent: y_new = y + h·f(x, y).
Euler's Method: Two Steps [H]	Recompute the slope at the new point before taking the second hop.
Euler's Method: A Nonlinear Field [H]	The recipe is unchanged when f(x, y) is nonlinear; just evaluate carefully.
Euler's Method: Over- or Under-Estimate [H]	Euler follows tangent lines, so concavity decides the direction of error.
Euler's Method: A Decimal Step [H]	A small decimal step works the same; convert to a fraction to stay exact.

Prerequisite material - taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.
Order of Operations	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing	Sign rules for products and quotients.
Exponents	Repeated multiplication in shorthand.
Square Roots	Undoing a square.

Fractions · 6 topics

Equivalent Fractions	Different fractions can name the same amount.
Simplifying Fractions	Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators)	Same-denominator addition.
Adding Fractions (Unlike Denominators)	Rewrite over a common denominator first.
Multiplying Fractions	Multiply straight across.
Dividing Fractions	Multiply by the reciprocal.

Decimals, Percents & Ratios · 4 topics

Fractions ↔ Decimals	Converting between the two notations.
Percent of a Number	Percent means per hundred.
Percent Increase & Decrease	Applying a percent change to a quantity.
Ratios & Proportions	Two quantities that scale together.

Expressions & Equations · 7 topics

Evaluating Expressions	Substituting a value for a variable.
Combining Like Terms	Adding the coefficients of matching variable parts.
The Distributive Property	Multiplying across a sum.
One-Step Equations	Undoing a single operation.
Two-Step Equations	Undo addition/subtraction first, then multiplication.
Multi-Step Equations	Equations needing distribution or variables on both sides.
Linear Inequalities	Solving with <, >, ≤, ≥.

Linear Functions · 4 topics

The Coordinate Plane	Locating points with (x, y) pairs.
Slope of a Line	Rise over run between two points.
Slope-Intercept Form	y = mx + b describes a whole line.
Systems of Equations (Substitution)	Two equations, two unknowns.

Quadratics & Polynomials · 10 topics

Adding & Subtracting Polynomials	Combining polynomials by collecting like terms.
Multiplying Binomials (FOIL)	Expanding products of binomials.
Factoring Out the GCF	Undoing the distributive property.
Factoring Trinomials	Reversing FOIL: finding two numbers that multiply to c and add to b.
Special Factoring Patterns	Difference of squares and perfect-square trinomials.
Solving Quadratics by Factoring	Zero-product property: if a·b = 0 then a = 0 or b = 0.
Solving x² = k	Taking square roots of both sides - remembering ±.
Completing the Square	Turning any quadratic into a perfect square plus a constant.
The Quadratic Formula	x = (−b ± √(b² − 4ac)) / 2a solves any quadratic.
Quadratic Models	Projectile motion and other parabolic models.

Radicals & Exponentials · 7 topics

Product Rule for Exponents	Multiplying powers of the same base adds the exponents.
Quotient & Power Rules	Dividing powers subtracts exponents; a power of a power multiplies them.
Zero & Negative Exponents	Anything (nonzero) to the 0 power is 1; a negative exponent flips to a reciprocal.
Simplifying Radicals	Pulling perfect-square factors out of a square root.
Operations with Radicals	Adding like radicals and multiplying square roots.
Rational Exponents	Fractional exponents are roots: x^(p/q) is the q-th root of x, raised to the p.
Exponential Growth & Decay	Quantities that multiply by the same factor each time step: y = a·bᵗ.

Geometry · 10 topics

Angle Relationships	Vertical, complementary, and supplementary angle pairs.
Triangle Angle Sum	The three angles of a triangle always add to 180°.
The Pythagorean Theorem	In a right triangle, a² + b² = c².
Distance & Midpoint	Measuring segments in the coordinate plane.
Similar Triangles	Same shape, different size: corresponding sides are proportional.
Perimeter & Area	Measuring around and inside basic shapes.
Circles: Area & Circumference	C = 2πr and A = πr².
Volume: Prisms & Cylinders	Volume = base area × height.
Volume: Cones, Pyramids & Spheres	Pointed solids hold one third of the matching prism; spheres use 4/3 πr³.
Special Right Triangles	The 45-45-90 and 30-60-90 side ratios.

Functions & Algebra II · 16 topics

Function Notation & Evaluation	Reading f(x) notation and plugging in inputs.
Domain & Range	Which inputs a function accepts, and which outputs it can produce.
Function Composition	Feeding one function's output into another: f(g(x)).
Inverse Functions	The function that undoes f: f⁻¹(b) is the input that f sends to b.
Piecewise Functions	Functions defined by different rules on different intervals.
Nonlinear Systems	Where a line meets a parabola: set the two formulas equal.
Polynomial Division	Dividing a polynomial by (x − a) with long or synthetic division.
Remainder & Factor Theorems	The remainder when p(x) is divided by (x − a) is simply p(a).
Zeros of Polynomials	Finding all the roots of a cubic by factoring it down.
End Behavior of Polynomials	Far from the origin, only the leading term matters.
Simplifying Rational Expressions	Factor top and bottom, then cancel the common factor.
Operations on Rational Expressions	Multiplying and dividing algebraic fractions.
Logarithms	log_b(x) asks: to what power must b be raised to get x?
Properties of Logarithms	Logs turn products into sums, quotients into differences, powers into multiples.
Arithmetic Sequences	Sequences that grow by a constant difference each step.
Geometric Sequences	Sequences that grow by a constant ratio each step.

Trigonometry · 5 topics

Right-Triangle Trigonometry	SOH-CAH-TOA: the three trig ratios of an acute angle in a right triangle.
Degrees & Radians	Two ways to measure the same angle: 180° equals π radians.
The Unit Circle	Exact sine, cosine, and tangent values at the special angles.
Trig of Any Angle	Reference angles plus quadrant signs extend trig beyond 90°.
Inverse Trig Functions	arcsin, arccos, and arctan undo the trig functions on restricted ranges.

Precalculus · 8 topics

Asymptotes of Rational Functions	The vertical asymptotes and the end behavior (horizontal or slant asymptote) of a rational function.
Graphs of Rational Functions	Holes, asymptotes, and intercepts tell the whole story of the graph.
Vectors: Components & Magnitude	A vector is a displacement: components ⟨Δx, Δy⟩ and a length.
Vector Operations	Scaling, adding, and dotting vectors - all component by component.
Parametric Equations	Describing a moving point by giving x and y as functions of time.
Polar Coordinates	Locating points by distance from the origin and angle from the x-axis.
Sigma Notation & Series	Σ compresses a sum: read the limits, add up the terms.
Average Rate of Change	The slope of the secant line: (f(b) − f(a)) / (b − a).

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