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If $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$ on $[a, b]$, then $\frac{d}{dx} \left(\int_{a}^{x} f(t) \, dt\right) = f(x)$.

If $f$ is continuous on the interval $[a, b]$ and $d$ is any number between $f(a)$ and $f(b)$, then there exists at least one $c$ in $(a, b)$ such that $f(c) = d$.

For a positively oriented, simple closed curve $C$ and a continuously differentiable vector field $\mathbf{F} = (M, N)$ on a plane, $\oint_{C} (M dx + N dy) = \iint_{D} (\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}) dA$, where $D$ is the region bounded by $C$.

If $f: \mathbb{R} \to \mathbb{R}$ is $k$ times differentiable at some point $a$, then for some $x$ near $a$, the function $f(x)$ can be approximated as $f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + ... + \frac{f^{(k)}(a)}{k!}(x - a)^k + R_k(x)$, where $R_k(x)$ is the remainder term.

If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists at least one $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

If $f(x)$ and $g(x)$ are differentiable and $\lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0$ or $\pm\infty$, and $g'(x) \neq 0$ near $c$ (except possibly at $c$) and $\lim_{x \to c} \frac{f'(x)}{g'(x)} = L$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = L$.

For a smooth oriented surface $S$ with smooth boundary curve $C$ and a vector field $\mathbf{F}$, $\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$.

If $f$ is continuous on $[a, b]$ and $F$ is the antiderivative of $f$ on $[a, b]$, then $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$.

If function $f$ 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 $f'(c) = 0$.