A
Antiderivative — A function F(x) such that F'(x) = f(x). Also called the indefinite integral. Example: x³/3 is an antiderivative of x².
Asymptote — A line that a curve approaches but never reaches. Horizontal asymptotes describe behaviour as x → ±∞; vertical asymptotes occur where functions blow up.
Average Rate of Change — [f(b) − f(a)] / (b − a). The slope of the secant line. As the interval shrinks to zero, this becomes the derivative.
C
Chain Rule — d/dx[f(g(x))] = f'(g(x)) · g'(x). Used to differentiate composite functions.
Concavity — f is concave up where f'' > 0, concave down where f'' < 0.
Continuity — f is continuous at a if lim(x→a) f(x) = f(a). No breaks, jumps, or holes.
Critical Point — Where f'(x) = 0 or f'(x) is undefined. Candidates for local maxima and minima.
D
Definite Integral — ∫ₐᵇ f(x) dx. The signed area under f between a and b. Equals F(b) − F(a).
Derivative — The instantaneous rate of change. f'(x) = lim(h→0) [f(x+h)−f(x)]/h.
Differentiable — f is differentiable at a if f'(a) exists. Differentiability implies continuity.
E
e — Euler's number ≈ 2.71828. The unique base where d/dx[eˣ] = eˣ.
Epsilon-Delta — The formal definition of a limit. lim f(x) = L if for every ε > 0 there exists δ > 0 such that |f(x)−L| < ε whenever 0 < |x−a| < δ.
F
Fundamental Theorem of Calculus — Part 1: d/dx[∫ₐˣ f(t)dt] = f(x). Part 2: ∫ₐᵇ f(x)dx = F(b)−F(a).
I
Implicit Differentiation — Differentiating both sides with respect to x when y is not isolated, using the chain rule on y terms.
Inflection Point — Where concavity changes; f''(x) = 0 and changes sign.
L
Limit — lim(x→a) f(x) = L means f(x) → L as x → a. The foundation of calculus.
Local Maximum — f(a) ≥ f(x) for all x near a. Requires f'(a) = 0 and f''(a) < 0.
M
Mean Value Theorem — There exists c ∈ (a,b) where f'(c) = [f(b)−f(a)]/(b−a).
P
Product Rule — d/dx[f·g] = f'g + fg'.
Quotient Rule — d/dx[f/g] = (f'g − fg') / g².
R
Riemann Sum — An approximation of ∫ₐᵇ f(x)dx using rectangles. The integral is the limit as rectangles become infinitely thin.
Related Rates — Problems where two changing quantities are linked; the chain rule connects their rates.
T
Tangent Line — Touches a curve at one point with slope f'(a): y − f(a) = f'(a)(x−a).
Taylor Series — Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ. Represents a function as an infinite polynomial.
U
U-Substitution — Integration technique: let u = g(x), du = g'(x)dx, then ∫f(g(x))g'(x)dx = ∫f(u)du.
Notation Quick Reference
Calculus uses notation from multiple traditions. Newton used dots (ẋ for velocity, ẍ for acceleration). Leibniz used d/dx and ∫. Both notations are still in use. Understanding both is essential:
More Essential Terms
Antiderivative — F(x) where F'(x) = f(x). Written ∫f(x)dx = F(x)+C.
Chain Rule — d/dx[f(g(x))] = f'(g(x))·g'(x). For composite functions.
Critical Point — x-value where f'(x) = 0 or f'(x) is undefined. Candidates for extrema.
Concavity — Concave up when f'' > 0 (bowl shape), concave down when f'' < 0 (arch shape).
Continuity — f is continuous at a if lim(x→a)f(x) = f(a). No gaps or jumps.
Divergence — A series or sequence that does not converge. Its partial sums grow without bound or oscillate.
Domain — The set of all x-values for which f(x) is defined.
Implicit Differentiation — Differentiating both sides of an equation with respect to x when y is not isolated.
Inflection Point — Where concavity changes; f''(c) = 0 and f'' changes sign at c.
L'Hôpital's Rule — If lim f/g gives 0/0 or ∞/∞, then lim f/g = lim f'/g'.
Local Maximum — f(a) ≥ f(x) for all x near a. The top of a local hill.
Monotonic — Always increasing (f' ≥ 0) or always decreasing (f' ≤ 0) on an interval.
Partial Derivative — ∂f/∂x: derivative of a multivariable function treating all other variables as constants.
Power Rule — d/dx[xⁿ] = nxⁿ⁻¹. The most used differentiation rule.
Product Rule — d/dx[f·g] = f'g + fg'. For products of two functions.
Quotient Rule — d/dx[f/g] = (f'g − fg')/g². For fractions where both parts depend on x.
Radius of Convergence — The value R such that a power series converges for |x−a| < R and diverges for |x−a| > R.
Related Rates — Problems connecting rates of change of two linked quantities using implicit differentiation.
Squeeze Theorem — If g(x) ≤ f(x) ≤ h(x) near a and lim g = lim h = L, then lim f = L.
Taylor Series — f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ. Represents a function as an infinite polynomial.
U-Substitution — Integration technique reversing the chain rule: let u = g(x), du = g'(x)dx.
Velocity — v(t) = ds/dt. The derivative of position with respect to time.