CLEP Calculus cheat sheet

Limit exists only if left-hand and right-hand limits agree: lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = L.

Direct substitution first: if f is continuous at a, then lim(x→a) f(x) = f(a).

0/0 or ∞/∞ is indeterminate: apply L'Hôpital's Rule, lim f/g = lim f'/g', provided the new limit exists.

For rational functions at x→±∞, compare degrees: numerator lower gives 0, equal gives ratio of leading coefficients, higher gives ±∞.

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.

lim(x→0) (1 + x)^(1/x) = e; equivalently lim(x→∞) (1 + 1/x)^x = e.

A function is continuous at a when lim(x→a) f(x) = f(a): the limit exists, f(a) is defined, and they match.

For a piecewise function, evaluate the one-sided limits at the break point separately and set them equal to enforce continuity.

Definition of derivative: f'(x) = lim(h→0) [f(x+h) − f(x)]/h; it equals the slope of the tangent line.

Power rule: d/dx x^n = n·x^(n−1) for any real n.

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

Chain rule: d/dx f(g(x)) = f'(g(x))·g'(x); the inner derivative multiplies the outer derivative.

Derivatives to memorize: (sin x)' = cos x, (cos x)' = −sin x, (tan x)' = sec²x, (eˣ)' = eˣ, (ln x)' = 1/x.

Critical points occur where f'(x) = 0 or f' is undefined; sign change of f' from + to − indicates a local maximum.

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

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).

Related rates: differentiate the relating equation with respect to time t, then substitute known values and solve for the unknown rate.

Antiderivative: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ −1; for n = −1, ∫(1/x) dx = ln|x| + C.

Fundamental Theorem of Calculus: if F' = f, then ∫ₐᵇ f(x) dx = F(b) − F(a).

Second FTC form: d/dx ∫ₐˣ f(t) dt = f(x); with an upper limit g(x) it becomes f(g(x))·g'(x).

Substitution: ∫f(g(x))g'(x) dx = ∫f(u) du with u = g(x); change limits accordingly for definite integrals.

Integrals of trig and exponential functions: ∫cos x dx = sin x + C, ∫sin x dx = −cos x + C, ∫eˣ dx = eˣ + C.

A definite integral gives signed area; area between curves is ∫ₐᵇ [top(x) − bottom(x)] dx.

For solids of revolution about the x-axis, disk method volume is V = π∫ₐᵇ [f(x)]² dx.

Riemann sums: the limit of Σ f(xᵢ*)Δx as Δx→0 equals the definite integral; left and right sums bracket the value for monotone f.

Even/odd symmetry: ∫₋ₐᵃ f dx = 2∫₀ᵃ f dx when f is even, and equals 0 when f is odd.