Differential calculus
Vocabulary
O Critical point or stationary point: punto
O
O
O
O
O
O
stazionario
Difference quotient: rapporto incrementale
Derivative: derivata
Differentiable function: funzione derivabile
Local extrema: massimo e/o minimo locali o
relativi
Second derivative function: derivata seconda
Nth derivative function: derivata n-esima
Fermat’s Theorem
Let the function f be defined in the closed
interval [a,b]. If f(x) has a local extremum at a
point x0 in the interval (a,b) where it is also
differentiable, then f’(x0) = 0
De L’Hopital’s Theorem
Let f(x) and g(x) be differentiable at least in
(with x0 in I) moreover assume that
or
moreover assume
and that the limit
then the limit
and we have
exists,
exists
Exercise 1
O Given a differentiable function
define
.
O Show that each critical point of f is also a
critical point of g.
O Exhibit an f such that g(x) has more critical
points than f.
Exercise 2
O Use de L’Hopital’s rule to compute
Exercise 3
Show that the graph of the function
has a vertical tangent at the point x0 = 1