Calculus

$f(x)$ is injective if $\forall x_1, x_2 \in D_f : f(x_1) = f(x_2) \Rightarrow x_1 = x_2$

$f(x)$ is surjective if $D’_f=\R \Leftrightarrow \forall y \in \R, \exists x \in D_f : f(x) = y$

$f(x)$ is bijective if it is injective and surjective, i.e., $\forall y \in \R: \exists ! x \in D_f:f(x)=y$

Rolle’s Theorem

If $f$ is:

  • continuous in $[a, b]$,
  • differentiable in $]a, b[$,
  • $f(a) = f(b)$
  • then, $\exists c \in ]a, b[: f’(c) = 0$

Bolzano’s Theorem

If $f$ is:

  • continuous in the limited and closed $[a, b]$ interval,
  • $f(a) \neq f(b)$,
  • then, $\exists c \in ]a,b[: f(c) = k$

Hyperbolic Sines and Cosines

Weierstrass’s Theorem

Lagrange’s Theorem

Cauchy’s Theorem

Limits

L’Hôpital’s Rule

Common Limits

Derivation