AP Calculus BC 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 iff lim_{x→a⁻} f(x) = lim_{x→a⁺} f(x) = L.

Limit laws: lim(f ± g) = lim f ± lim g, lim(fg) = (lim f)(lim g), and lim(f/g) = (lim f)/(lim g) provided lim g ≠ 0.

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

Special limits: lim_{x→0} (sin x)/x = 1 and lim_{x→0} (1 − cos x)/x = 0.

lim_{x→0} (1 + x)^{1/x} = e and lim_{x→∞} (1 + 1/x)^x = e.

f is continuous at a iff lim_{x→a} f(x) = f(a), which requires f(a) defined, the limit existing, and the two being equal.

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

A removable discontinuity (hole) occurs when the limit exists but differs from f(a) or f(a) is undefined; a jump discontinuity occurs when one-sided limits differ.

Derivative definition: f'(x) = lim_{h→0} [f(x+h) − f(x)]/h = lim_{t→x} [f(t) − f(x)]/(t − x).

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

Differentiability implies continuity; continuity does not imply differentiability (e.g., f(x) = |x| at 0).

Power rule: d/dx x^n = n x^{n−1} for all real n; also d/dx (constant) = 0.

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

Derivatives of elementary functions: (sin x)' = cos x, (cos x)' = −sin x, (tan x)' = sec²x, (e^x)' = e^x, (ln x)' = 1/x.

Higher derivatives: f''(x) is the derivative of f'(x); notation f^{(n)}(x) or dⁿy/dxⁿ.

Average rate of change of f on [a,b] is [f(b) − f(a)]/(b − a); instantaneous rate is f'(a).

Chain rule: d/dx f(g(x)) = f'(g(x))·g'(x).

General power rule: d/dx [g(x)]^n = n[g(x)]^{n−1}·g'(x).

d/dx sin(g) = cos(g)·g', d/dx e^{g} = e^{g}·g', d/dx ln(g) = g'/g.

Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x (attach dy/dx to each y-derivative), then solve for dy/dx.

Inverse function derivative: if g = f⁻¹, then g'(x) = 1/f'(g(x)), valid where f'(g(x)) ≠ 0.

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

d/dx a^x = a^x ln a, and d/dx log_a x = 1/(x ln a).

Logarithmic differentiation: take ln of both sides before differentiating, useful for y = f(x)^{g(x)}.

Related rates: differentiate a relating equation with respect to t, then substitute known values and rates.

Linear approximation: f(x) ≈ f(a) + f'(a)(x − a); the differential is dy = f'(a) dx.

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

Other indeterminate forms (0·∞, ∞ − ∞, 0⁰, 1^∞, ∞⁰) must be rewritten or log-transformed before applying L'Hôpital.

Motion: velocity v(t) = s'(t), acceleration a(t) = v'(t) = s''(t); speed is |v(t)|.

A particle is at rest when v(t) = 0; it moves right when v(t) > 0 and left when v(t) < 0.

Marginal cost/revenue/profit is the derivative of the corresponding total function.

Exponential growth/decay model: y = y₀e^{kt}, with dy/dt = ky.

Extreme Value Theorem: a continuous f on a closed interval attains both an absolute max and min; check critical points and endpoints.

Critical point: c in the domain where f'(c) = 0 or f'(c) is undefined.

First derivative test: f' changes + to − at c gives a local max; − to + gives a local min.

Second derivative test: f'(c) = 0 and f''(c) > 0 gives a local min; f''(c) < 0 gives a local max.

f is increasing where f' > 0 and decreasing where f' < 0.

f is concave up where f'' > 0 and concave down where f'' < 0; inflection points occur where concavity changes.

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

Rolle's theorem is the special case of MVT where f(a) = f(b), giving some c with f'(c) = 0.

FTC Part 1: if F(x) = ∫_a^x f(t) dt and f is continuous, then F'(x) = f(x).

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

Leibniz rule: d/dx ∫_{a(x)}^{b(x)} f(t) dt = f(b(x))b'(x) − f(a(x))a'(x).

Power rule for antiderivatives: ∫ x^n dx = x^{n+1}/(n+1) + C for n ≠ −1; ∫ (1/x) dx = ln|x| + C.

∫ e^x dx = e^x + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ sec²x dx = tan x + C.

u-substitution reverses the chain rule: ∫ f(g(x))g'(x) dx = ∫ f(u) du; change limits when evaluating definite integrals.

Integration by parts: ∫ u dv = uv − ∫ v du, commonly choosing u by LIATE order.

Net change theorem: ∫_a^b F'(x) dx = F(b) − F(a); the definite integral of a rate gives total accumulation.

Properties: ∫_a^a f = 0, ∫_a^b f = −∫_b^a f, and ∫_a^b f = ∫_a^c f + ∫_c^b f.

A separable equation dy/dx = f(x)g(y) is solved by separating: (1/g(y)) dy = f(x) dx, then integrating both sides and adding C.

Exponential model: dy/dt = ky has solution y = Ce^{kt}; sign of k distinguishes growth from decay.

Slope fields: at each point (x,y) draw a short segment of slope dy/dx; solution curves follow the field.

Logistic equation: dy/dt = ky(1 − y/L) has solution y = L/(1 + Ae^{−kt}), with carrying capacity L.

Euler's method: y_{n+1} = y_n + h·f(x_n, y_n), stepping with fixed step size h.

Equilibrium solutions occur where dy/dx = 0; stability is read from the sign of dy/dx on either side.

Initial conditions determine the constant C after integrating a separable equation.

Area between curves: A = ∫_a^b [top(x) − bottom(x)] dx, with a, b at intersection points.

Disk method (axis-aligned): V = π∫_a^b [R(x)]² dx.

Washer method: V = π∫_a^b ([R(x)]² − [r(x)]²) dx.

Cylindrical shells about a vertical axis: V = 2π∫_a^b (radius)(height) dx.

Average value of f on [a,b] is (1/(b − a))∫_a^b f(x) dx.

Arc length: L = ∫_a^b √(1 + [f'(x)]²) dx.

Volume by cross sections with area A(x): V = ∫_a^b A(x) dx.

Parametric slope: dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

Second derivative: d²y/dx² = [d/dt(dy/dx)]/(dx/dt).

Parametric arc length: L = ∫_a^b √((dx/dt)² + (dy/dt)²) dt.

Vector-valued position r(t) = ⟨x(t), y(t)⟩ has velocity r'(t) and acceleration r''(t); speed is |r'(t)|.

Polar-to-rectangular: x = r cos θ, y = r sin θ; polar slope dy/dx = (r' sin θ + r cos θ)/(r' cos θ − r sin θ).

Polar area: A = (1/2)∫_α^β r² dθ.

Polar arc length: L = ∫_α^β √(r² + (dr/dθ)²) dθ.

Distance traveled by a particle on [a,b] is ∫_a^b |r'(t)| dt.

Geometric series: ∑ ar^n converges to a/(1 − r) iff |r| < 1; diverges if |r| ≥ 1.

nth-term test: if lim a_n ≠ 0, then ∑ a_n diverges; lim a_n = 0 does not imply convergence.

p-series ∑ 1/n^p converges iff p > 1; harmonic series (p = 1) diverges.

Integral test: for positive, decreasing, continuous f with f(n) = a_n, ∑ a_n and ∫_1^∞ f(x) dx share convergence behavior.

Comparison and limit comparison tests: compare a_n to a known benchmark series; limit comparison uses lim a_n/b_n = c with 0 < c < ∞.

Ratio test: let L = lim |a_{n+1}/a_n|; converges if L < 1, diverges if L > 1, inconclusive if L = 1.

Alternating series test: ∑ (−1)^n b_n converges if b_n is positive, decreasing, and → 0; error ≤ first omitted term.

Absolute convergence implies convergence; conditional convergence means ∑ a_n converges but ∑ |a_n| diverges.

Taylor series about a: f(x) = ∑ f^{(n)}(a)(x − a)^n/n!; Maclaurin series uses a = 0.

Known Maclaurin series: e^x = ∑ x^n/n!, sin x = ∑ (−1)^n x^{2n+1}/(2n+1)!, cos x = ∑ (−1)^n x^{2n}/(2n)!, 1/(1 − x) = ∑ x^n for |x| < 1.

Radius of convergence R from the ratio test; the interval must be checked separately at endpoints x = a ± R.

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 iff lim_{x→a⁻} f(x) = lim_{x→a⁺} f(x) = L.

Limit laws: lim(f ± g) = lim f ± lim g, lim(fg) = (lim f)(lim g), and lim(f/g) = (lim f)/(lim g) provided lim g ≠ 0.

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

Special limits: lim_{x→0} (sin x)/x = 1 and lim_{x→0} (1 − cos x)/x = 0.

f is continuous at a iff lim_{x→a} f(x) = f(a), requiring f(a) defined, the limit existing, and equality.

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

A removable discontinuity (hole) occurs when the limit exists but differs from f(a) or f(a) is undefined; a jump discontinuity occurs when one-sided limits differ.

Derivative definition: f'(x) = lim_{h→0} [f(x+h) − f(x)]/h = lim_{t→x} [f(t) − f(x)]/(t − x).

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

Differentiability implies continuity; continuity does not imply differentiability (e.g., f(x) = |x| at 0).

Power rule: d/dx x^n = n x^{n−1} for all real n; also d/dx (constant) = 0.

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

Derivatives of elementary functions: (sin x)' = cos x, (cos x)' = −sin x, (tan x)' = sec²x, (e^x)' = e^x, (ln x)' = 1/x.

Higher derivatives: f''(x) is the derivative of f'(x); notation f^{(n)}(x) or dⁿy/dxⁿ.

Average rate of change of f on [a,b] is [f(b) − f(a)]/(b − a); instantaneous rate is f'(a).

Chain rule: d/dx f(g(x)) = f'(g(x))·g'(x).

General power rule: d/dx [g(x)]^n = n[g(x)]^{n−1}·g'(x).

d/dx sin(g) = cos(g)·g', d/dx e^{g} = e^{g}·g', d/dx ln(g) = g'/g.

Implicit differentiation: differentiate both sides with respect to x, treating y as a function of x (attach dy/dx to each y-derivative), then solve for dy/dx.

Inverse function derivative: if g = f⁻¹, then g'(x) = 1/f'(g(x)), valid where f'(g(x)) ≠ 0.

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

d/dx a^x = a^x ln a, and d/dx log_a x = 1/(x ln a).

Logarithmic differentiation: take ln of both sides before differentiating, useful for y = f(x)^{g(x)}.

Related rates: differentiate a relating equation with respect to t, then substitute known values and rates.

Linear approximation: f(x) ≈ f(a) + f'(a)(x − a); the differential is dy = f'(a) dx.

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

Other indeterminate forms (0·∞, ∞ − ∞, 0⁰, 1^∞, ∞⁰) must be rewritten or log-transformed before applying L'Hôpital.

Motion: velocity v(t) = s'(t), acceleration a(t) = v'(t) = s''(t); speed is |v(t)|.

A particle is at rest when v(t) = 0; it moves right when v(t) > 0 and left when v(t) < 0.

Marginal cost/revenue/profit is the derivative of the corresponding total function.

Exponential growth/decay model: y = y₀e^{kt}, with dy/dt = ky.

Extreme Value Theorem: a continuous f on a closed interval attains both an absolute max and min; check critical points and endpoints.

Critical point: c in the domain where f'(c) = 0 or f'(c) is undefined.

First derivative test: f' changes + to − at c gives a local max; − to + gives a local min.

Second derivative test: f'(c) = 0 and f''(c) > 0 gives a local min; f''(c) < 0 gives a local max.

f is increasing where f' > 0 and decreasing where f' < 0.

f is concave up where f'' > 0 and concave down where f'' < 0; inflection points occur where concavity changes.

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

Rolle's theorem is the special case of MVT where f(a) = f(b), giving some c with f'(c) = 0.

FTC Part 1: if F(x) = ∫_a^x f(t) dt and f is continuous, then F'(x) = f(x).

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

Leibniz rule: d/dx ∫_{a(x)}^{b(x)} f(t) dt = f(b(x))b'(x) − f(a(x))a'(x).

Power rule for antiderivatives: ∫ x^n dx = x^{n+1}/(n+1) + C for n ≠ −1; ∫ (1/x) dx = ln|x| + C.

∫ e^x dx = e^x + C, ∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ sec²x dx = tan x + C.

u-substitution reverses the chain rule: ∫ f(g(x))g'(x) dx = ∫ f(u) du; change limits when evaluating definite integrals.

Integration by parts: ∫ u dv = uv − ∫ v du, commonly choosing u by LIATE order.

Net change theorem: ∫_a^b F'(x) dx = F(b) − F(a); the definite integral of a rate gives total accumulation.

Properties: ∫_a^a f = 0, ∫_a^b f = −∫_b^a f, and ∫_a^b f = ∫_a^c f + ∫_c^b f.

A separable equation dy/dx = f(x)g(y) is solved by separating: (1/g(y)) dy = f(x) dx, then integrating both sides and adding C.

Exponential model: dy/dt = ky has solution y = Ce^{kt}; sign of k distinguishes growth from decay.

Slope fields: at each point (x,y) draw a short segment of slope dy/dx; solution curves follow the field.

Logistic equation: dy/dt = ky(1 − y/L) has solution y = L/(1 + Ae^{−kt}), with carrying capacity L.

Euler's method: y_{n+1} = y_n + h·f(x_n, y_n), stepping with fixed step size h.

Equilibrium solutions occur where dy/dx = 0; stability is read from the sign of dy/dx on either side.

Initial conditions determine the constant C after integrating a separable equation.

Area between curves: A = ∫_a^b [top(x) − bottom(x)] dx, with a, b at intersection points.

Disk method (axis-aligned): V = π∫_a^b [R(x)]² dx.

Washer method: V = π∫_a^b ([R(x)]² − [r(x)]²) dx.

Cylindrical shells about a vertical axis: V = 2π∫_a^b (radius)(height) dx.

Average value of f on [a,b] is (1/(b − a))∫_a^b f(x) dx.

Arc length: L = ∫_a^b √(1 + [f'(x)]²) dx.

Volume by cross sections with area A(x): V = ∫_a^b A(x) dx.

Parametric slope: dy/dx = (dy/dt)/(dx/dt), provided dx/dt ≠ 0.

Second derivative: d²y/dx² = [d/dt(dy/dx)]/(dx/dt).

Parametric arc length: L = ∫_a^b √((dx/dt)² + (dy/dt)²) dt.

Vector-valued position r(t) = ⟨x(t), y(t)⟩ has velocity r'(t) and acceleration r''(t); speed is |r'(t)|.

Polar-to-rectangular: x = r cos θ, y = r sin θ; polar slope dy/dx = (r' sin θ + r cos θ)/(r' cos θ − r sin θ).

Polar area: A = (1/2)∫_α^β r² dθ.

Polar arc length: L = ∫_α^β √(r² + (dr/dθ)²) dθ.

Distance traveled by a particle on [a,b] is ∫_a^b |r'(t)| dt.

Geometric series: ∑ ar^n converges to a/(1 − r) iff |r| < 1; diverges if |r| ≥ 1.

nth-term test: if lim a_n ≠ 0, then ∑ a_n diverges; lim a_n = 0 does not imply convergence.

p-series ∑ 1/n^p converges iff p > 1; harmonic series (p = 1) diverges.

Integral test: for positive, decreasing, continuous f with f(n) = a_n, ∑ a_n and ∫_1^∞ f(x) dx share convergence behavior.

Comparison and limit comparison tests: compare a_n to a known benchmark series; limit comparison uses lim a_n/b_n = c with 0 < c < ∞.

Ratio test: let L = lim |a_{n+1}/a_n|; converges if L < 1, diverges if L > 1, inconclusive if L = 1.

Alternating series test: ∑ (−1)^n b_n converges if b_n is positive, decreasing, and → 0; error ≤ first omitted term.

Absolute convergence implies convergence; conditional convergence means ∑ a_n converges but ∑ |a_n| diverges.

Taylor series about a: f(x) = ∑ f^{(n)}(a)(x − a)^n/n!; Maclaurin series uses a = 0.

Known Maclaurin series: e^x = ∑ x^n/n!, sin x = ∑ (−1)^n x^{2n+1}/(2n+1)!, cos x = ∑ (−1)^n x^{2n}/(2n)!, 1/(1 − x) = ∑ x^n for |x| < 1.

Radius of convergence R from the ratio test; the interval must be checked separately at endpoints x = a ± R.