AP Calculus AB cheat sheet

lim(x→a) f(x) = L means f(x) can be made arbitrarily close to L by taking x sufficiently close to (but not equal to) a.

One-sided limits: the two-sided limit exists if and only if the left-hand and right-hand limits both exist and are equal.

A function is continuous at x = a when lim(x→a) f(x) = f(a); this requires f(a) defined, the limit existing, and equality holding.

Removable discontinuity: limit exists but differs from f(a) or f(a) is undefined; non-removable: jump, infinite, or oscillating behavior.

Algebraic limit techniques: direct substitution, factoring and canceling, rationalizing, and combining complex fractions.

Squeeze theorem: if g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L.

lim(x→0) (sin x)/x = 1 and lim(x→0) (1 − cos x)/x = 0.

For rational functions, limits at infinity are governed by comparing degrees of numerator and denominator; divide by the highest power of x.

Intermediate Value Theorem: if f is continuous on [a, b] and k lies between f(a) and f(b), some c in (a, b) satisfies f(c) = k.

Derivative definition: f′(a) = lim(h→0) [f(a+h) − f(a)]/h = lim(x→a) [f(x) − f(a)]/(x − a).

f′(a) is the slope of the tangent line at x = a; the tangent line is y = f(a) + f′(a)(x − a).

Differentiability at a point implies continuity there; continuity does not imply differentiability (e.g., |x| at 0).

Power rule: d/dx xⁿ = n xⁿ⁻¹ for all real n; d/dx (constant) = 0.

Product rule: (fg)′ = f′g + fg′. Quotient rule: (f/g)′ = (f′g − fg′)/g².

d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x, d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x.

Higher derivatives are obtained by differentiating repeatedly; f″ gives concavity and acceleration.

Average rate of change over [a, b] is [f(b) − f(a)]/(b − a); instantaneous rate is the derivative.

Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x); identify outer and inner functions before differentiating.

d/dx [g(x)]ⁿ = n[g(x)]ⁿ⁻¹·g′(x); d/dx e^{g(x)} = e^{g(x)}·g′(x); d/dx ln(g(x)) = g′(x)/g(x).

Implicit differentiation: differentiate both sides with respect to x, applying the chain rule to y-terms, then solve algebraically for dy/dx.

Inverse function derivative: if g = f⁻¹, then g′(y) = 1/f′(g(y)) wherever f′ ≠ 0.

d/dx (arcsin x) = 1/√(1 − x²), d/dx (arccos x) = −1/√(1 − x²), d/dx (arctan x) = 1/(1 + x²).

Logarithmic differentiation: take ln of both sides before differentiating, useful for products, quotients, and variable exponents.

For y = aˣ, dy/dx = aˣ ln a; for y = log_a x, dy/dx = 1/(x ln a).

Chain rule applied to related quantities underlies implicit relations, e.g., differentiating x² + y² = 25 gives 2x + 2y·dy/dx = 0.

Velocity v(t) = s′(t) and acceleration a(t) = v′(t) = s″(t); speed is |v(t)|.

Particle is at rest when v(t) = 0; moving right/positive when v(t) > 0; moving left/negative when v(t) < 0.

Related rates: differentiate a geometric relation with respect to time t, then substitute known values and rates; all rates carry units per unit time.

Linear approximation (local linearity): f(x) ≈ f(a) + f′(a)(x − a) near x = a.

L'Hôpital's rule applies to 0/0 or ∞/∞ indeterminate forms: lim f/g = lim f′/g′ when the latter limit exists.

Marginal cost/revenue/profit approximate the change in total quantity per one additional unit.

Units of f′(x) are units of f per unit of x; interpretation must state both quantities.

A differentiable function's tangent line at a point gives the best linear estimate of nearby values.

Extreme Value Theorem: a continuous function on a closed interval attains both an absolute maximum and minimum.

Critical points occur where f′(x) = 0 or f′(x) is undefined; candidates for local extrema.

First derivative test: f′ changes + to − at c gives a local maximum; − to + gives a local minimum.

Second derivative test: f′(c) = 0 and f″(c) > 0 gives a local minimum; f″(c) < 0 gives a local maximum.

f is increasing where f′ > 0 and decreasing where f′ < 0; concavity is up where f″ > 0 and down where f″ < 0.

Inflection points occur where concavity changes sign (f″ changes sign), not merely where f″ = 0.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) satisfies f′(c) = [f(b) − f(a)]/(b − a).

Rolle's theorem is the MVT case where f(a) = f(b), guaranteeing f′(c) = 0.

To find absolute extrema on [a, b], evaluate f at all critical points in (a, b) and at both endpoints.

Riemann sum limit definition: ∫ₐᵇ f(x) dx = lim(n→∞) Σ f(xᵢ*)Δx over a partition of [a, b].

Left/right/midpoint Riemann sums and trapezoidal sums approximate definite integrals; more subintervals generally improve accuracy.

Fundamental Theorem of Calculus Part 1: if F(x) = ∫ₐˣ f(t) dt with f continuous, then F′(x) = f(x).

FTC Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f.

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫ (1/x) dx = ln|x| + C.

∫ₐᵇ f = −∫ᵦᵃ f; ∫ₐᵃ f = 0; ∫ₐᵇ f = ∫ₐᶜ f + ∫꜀ᵇ f.

Net change theorem: ∫ₐᵇ F′(x) dx = F(b) − F(a); the definite integral of a rate gives total accumulated change.

Accumulation function derivative with variable upper limit: d/dx ∫ₐ^{g(x)} f(t) dt = f(g(x))·g′(x).

Even/odd symmetry: ∫₋ₐᵃ f = 2∫₀ᵃ f for even f and 0 for odd f.

A differential equation relates an unknown function to its derivatives; a solution satisfies the equation on an interval.

General solutions include an arbitrary constant C; a particular solution uses an initial condition to determine C.

Separation of variables: rewrite as g(y) dy = h(x) dx, integrate both sides, then solve for y if possible.

Slope fields graph dy/dx at sample points; solution curves are tangent to the field segments.

Exponential growth/decay: dy/dt = ky has solution y = y₀e^{kt}; k > 0 growth, k < 0 decay.

Logistic model: dP/dt = kP(1 − P/L) has carrying capacity L and inflection at P = L/2.

Euler's method approximates solutions: y_{n+1} = yₙ + h·F(xₙ, yₙ) with step size h.

Verify a proposed solution by substituting it and its derivatives into the original differential equation.

Area between curves: A = ∫ₐᵇ [top(x) − bottom(x)] dx, with a and b the intersection x-values.

Area with respect to y: A = ∫ [right(y) − left(y)] dy when curves are easier to express as x = f(y).

Volume by disks: V = π∫ₐᵇ [R(x)]² dx when cross-sections perpendicular to the axis are solid disks.

Volume by washers: V = π∫ₐᵇ ([R(x)]² − [r(x)]²) dx for a region not touching the axis of rotation.

Volume by known cross-sections: V = ∫ₐᵇ A(x) dx, where A(x) is the cross-sectional area at x.

Average value of f on [a, b] is (1/(b − a))∫ₐᵇ f(x) dx; the Mean Value Theorem for Integrals guarantees f(c) equals this value for some c.

Arc length of y = f(x) on [a, b]: L = ∫ₐᵇ √(1 + [f′(x)]²) dx.

Accumulated displacement is ∫ v(t) dt; total distance traveled is ∫ |v(t)| dt.