Video Notes: Limits, FTC, Antiderivatives, Substitution

Vocabulary flashcards covering core concepts from limits, the Fundamental Theorem of Calculus, continuity, and basic integration rules based on the video notes.

Limit

The value that f(x) approaches as x approaches a given input; the limit exists if the left and right limits agree and are finite.

Integrable

A function for which the definite integral over an interval exists (the area under the curve can be computed).

Right-hand limit

The limit of f(x) as x approaches a from the right (x > a).

Left-hand limit

The limit of f(x) as x approaches a from the left (x < a).

Definite integral

The accumulation of quantities over [a,b], denoted ∫_a^b f(x) dx; represents the net area under f on the interval.

Indefinite integral

The family of antiderivatives: ∫ f(x) dx = F(x) + C, where F'(x) = f(x).

Antiderivative

A function F whose derivative is f, i.e., F′(x) = f(x).

Fundamental Theorem of Calculus (FTC)

If f is continuous on [a,b], then ∫_a^b f(x) dx = F(b) − F(a) for any antiderivative F of f.

Continuity

A function with no breaks, jumps, or holes; it has limits at every point that equal the function value.

Continuous on [a,b]

f is continuous at every point in the interval [a,b].

Linearity of definite integrals

∫a^b [αf(x) + βg(x)] dx = α∫a^b f(x) dx + β∫a^b g(x) dx; also ∫a^b c·f(x) dx = c∫_a^b f(x) dx for constant c.

Substitution (u-substitution)

A method to simplify integrals by setting u = g(x); du = g′(x) dx, transforming ∫ f(g(x))g′(x) dx into ∫ F(u) du.

Chain Rule

If y = f(g(x)), then dy/dx = f′(g(x))·g′(x).

Constant Multiple Rule (Integrals)

You can pull constants out of an integral: ∫ c·f(x) dx = c∫ f(x) dx.

Sum Rule (Integrals)

The integral of a sum is the sum of the integrals: ∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx.