Theorems of calculus
The theorems that hold calculus together, stated precisely, explained in plain language and shown with graphs and examples.
Limits and continuity theorems
The theorems that make limits and continuous functions behave.
Squeeze theorem
g(x) \le f(x) \le h(x) \implies \lim_{x \to a} f(x) = L
The squeeze (sandwich) theorem: if g ≤ f ≤ h near a and g and h share a limit L, then f has limit L too. Statement, graph and examples.
Intermediate value theorem
f(c) = N \text{ for some } c \in (a, b)
The intermediate value theorem: a continuous function on [a, b] takes every value between f(a) and f(b). Statement, graph, counterexamples and examples.
Extreme value theorem
f(c) \le f(x) \le f(d) \text{ for all } x \in [a, b]
The extreme value theorem: a continuous function on a closed interval [a, b] has an absolute max and min. Statement, graph, counterexamples and examples.
Differential calculus theorems
Theorems about derivatives, extreme values and rates of change.
Rolle's theorem
f'(c) = 0 \text{ for some } c \in (a, b)
Rolle's theorem: if f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f'(c) = 0 for some c. Graph, counterexamples, examples.
Mean value theorem
f'(c) = \frac{f(b) - f(a)}{b - a}
The mean value theorem: a smooth function on [a, b] has a point where the tangent slope equals the average slope. Statement, graph, counterexamples, examples.
Fermat's theorem
f'(c) = 0
Fermat's theorem: if f has a local max or min at c and f'(c) exists, then f'(c) = 0. Statement, graph, why the converse fails, and worked examples.
Integral calculus theorems
The Fundamental Theorem of Calculus and the theorems built on it.
Fundamental theorem of calculus, part 1
\frac{d}{dx} \int_a^x f(t)\,dt = f(x)
FTC part 1: if f is continuous, the derivative of the integral from a to x of f(t) dt is f(x). Statement, graph, chain rule version and worked examples.
Fundamental theorem of calculus, part 2
\int_a^b f(x)\,dx = F(b) - F(a)
FTC part 2: the definite integral of f from a to b equals F(b) − F(a) for any antiderivative F. Statement, graph, a key counterexample and worked examples.
Mean value theorem for integrals
f(c) = \frac{1}{b - a} \int_a^b f(x)\,dx
The mean value theorem for integrals: a continuous function on [a, b] equals its average value at some point c. Statement, graph, counterexample, examples.
Series and multivariable theorems
Taylor's theorem, convergence results and the big theorems of Calculus 3.
Taylor's theorem
f(x) = P_n(x) + \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1}
Taylor's theorem with the Lagrange remainder: how close a Taylor polynomial is to f(x). Statement, error bound, graph and worked examples with e and sin x.
Clairaut's theorem
f_{xy}(a, b) = f_{yx}(a, b)
Clairaut's theorem: if the mixed partials f_xy and f_yx are continuous, they are equal. Statement, a classic counterexample, a graph and worked examples.
Green's theorem
\oint_C P\,dx + Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA
Green's theorem turns a line integral around a closed curve into a double integral over the region inside. Statement, hypotheses, area formula, examples.
Divergence theorem
\iint_S \mathbf{F}\cdot d\mathbf{S} = \iiint_E \nabla\cdot\mathbf{F}\,dV
The divergence theorem: the outward flux of a vector field through a closed surface equals the triple integral of its divergence. Statement and examples.
Stokes' theorem
\oint_C \mathbf{F}\cdot d\mathbf{r} = \iint_S (\nabla\times\mathbf{F})\cdot d\mathbf{S}
Stokes' theorem: the circulation of a field around a closed curve equals the flux of its curl through any surface the curve bounds. Statement and examples.
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