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Derivatives Lesson Oct 19
1.
Continuity
2.
Graph the function:
x(x 1) f(x) = (x 1)
3.
Graph the function: g(x) = x + 1
x 1
4.
A function f is continuous at x = a if: 1)
lim f(x) exists x a 2) f is defined at a 3) lim x a f(x) = f(a)
5.
Determine if the function f defined by:
{ f(x) = x2 4 , x = 2 x 2 4 , x = 2 is continuous.
6.
7.
{ f(x) =
3 x , x > 1 x2 + 2x , x < 1 Is this function continuous at x = 1?
8.
9.
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
lim x a+ f(x) = f (a) and lim f(x) = f(b) x b right continuous left continuous
10.
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.
11.
f(b) f(c) = k
f(a) a c b
12.
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],
f(c) < f(x) and f(d) > f(x)
13.
f(d)
f f(c) a c d b
14.
f
g a a
15.
h
a If a function f has a derivative at x = a, then f is continuous at x = a.
16.
Exercise 2.8 Questions: 1, 5, 7, 13, 15, 21
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