derivatives lesson oct 19
TRANSCRIPT
Continuity
Graph the function: f(x) = x(x 1)(x 1)
g(x) = x + 1x 1 Graph the function:
A function f is continuous at x = a if:
1) limx a
f(x) exists
2) f is defined at a
3) limx a
f(x) = f(a)
Determine if the function f defined by:
4 , x = 2{ x 2=x2 4
x 2,
f(x) = is continuous.
f(x) =x2 + 2x 1x <,3 x , x 1>{
Is this function continuous at x = 1?
Continuity on an interval
A function f is continuous on the closed interval [a, b] if it is continuous at every number in the open interval (a, b) and
limx a+
f(x) = f (a) and limx b
f(x) = f(b)
right continuousleft continuous
The Intermediate Value Theorem: If the function f is continuous on the closed interval [a, b], with f(a) f(b), and k is a number between f(a) and f(b), then there exists at least one number c in (a, b) for which f(c) = k.
=
f(b)
f(c) = k
f(a)
a bc
The Extreme Value Theorem: If the function f is continuous on the interval [a, b], then there exist numbers c and d in [a, b] such that for all x in [a, b],
and<f(c) f(x) >f(d) f(x)
f(d)
f(c)
a bc d
f
a
fg
a
a
h
If a function f has a derivative at x = a, then f is continuous at x = a.
Exercise 2.8
Questions: 1, 5, 7, 13, 15, 21