Unit 1

Applied Calculus

Continuity checklist at x=a
Left/right derivatives at a corner
Geometric meaning of Rolle (horizontal tangent)
Geometric meaning of LMVT (chord = tangent slope)

$l im x - > 0 sin \frac{x}{x} = 1$
$l im x - > 0 (1 - cos x) / x^{2} = \frac{1}{2}$
$D^{n} (e^{(} a x)) = a^{n} e^{(} a x)$
$D^{n} (sin a x) = a^{n} sin (a x + n * \frac{p i}{2})$
$D^{n} (cos a x) = a^{n} cos (a x + n * \frac{p i}{2})$
$D^{n} (l o g x) = (- 1)^{(} n - 1) (n - 1)! / x^{n}$
$D^{n} (uv) = S u m C (n, r) u^{(} n - r) v^{r}$
Rolle: f'(c) = 0 when f(a)=f(b)
$L M V T : f^{'} (c) = [f (b) - f (a)] / (b - a)$
Taylor expansion about x=a with remainder