Barrons AP Calculus

(Marvins-Underground-K-12) #1
that    interval.

(2) The Intermediate Value Theorem. If f is continuous on the closed interval
[a,b], and M is a number such that f (a) ≤ M ≤ f (b), then there is at least one
number, c, in the interval [a,b], such that f (c) = M.
Note an important special case of the Intermediate Value Theorem:
If f is continuous on the closed interval [a,b], and f (a) and f (b) have opposite
signs, then f has a zero in that interval (there is a value, c, in [a,b] where f (c) =
0).
(3) The Continuous Functions Theorem. If functions f and g are both
continuous at x = c, then so are the following functions:
(a) kf, where k is a constant;
(b) f ± g;
(c) f · g;
(d) , provided that g(c) ≠ 0.


Example 32 __

Show that has a root between x = 2 and x = 3.


SOLUTION: The rational function f is discontinuous only at ,
and f (3) = 1. Since f is continuous on the interval [2,3] and f (2) and f (3) have
opposite signs, there is a value, c, in the interval where f (c) = 0, by the
Intermediate Value Theorem.

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