Core Theorems of Calculus

Intermediate Value Theorem (IVT)

If f(x) is continuous on [a,b] and N is any number between f(a) and f(b), then there exists at least one c in (a,b) such that f(c) = N. Guarantees a value exists between two points.

Extreme Value Theorem (EVT)

If f(x) is continuous on [a,b], then f(x) has both a maximum and a minimum value on that interval. Guarantees an absolute max and min exist on a closed interval.

Mean Value Theorem (MVT)

If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists at least one c in (a,b) such that f'(c) = \frac{f(b) - f(a)}{b - a}. Guarantees a point where the instantaneous rate of change equals the average rate of change.

Rolle’s Theorem

If f(x) is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists at least one c in (a,b) such that f'(c) = 0. Guarantees a horizontal tangent between two equal y-values.

Fundamental Theorem of Calculus – Part 1 (FTC1)

If f(x) is continuous on [a,b] and F(x) = \int_a^x f(t) \, dt, then F'(x) = f(x). Differentiating an integral "undoes" the integral.

Fundamental Theorem of Calculus – Part 2 (FTC2)

If f(x) is continuous on [a,b], then \int_a^b f(x) \, dx = F(b) - F(a) where F'(x) = f(x). Computes definite integrals using antiderivatives.

Net Change Theorem

If F'(x) = f(x), then \int_a^b f(x) \, dx = F(b) - F(a). The integral of a rate of change gives total change.