Important
Theorems in Calculus (continued)
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The Intermediate Value Theorem:
i.e. follows from continuity of
a function Suppose that f is
continuous on the closed interval [a,b] and W is any number between f(a) and f(b). Then there is a number c in
[a,b] for which
f(c) = W. The graph below provides a
geometric interpretation of the Intermediate Value Theorem.
This theorem is helpful in
determining if roots exist in certain ranges for continuous functions. |
|
Example applying the Intermediate Value Theorem. Problem Statement: Use the Intermediate Value
Theorem to verify that f(x) has a zero in the given interval for f(x) = x3 -
4x - 2
in [-2, -1]. Now
f(˗ 2) = ˗ 2,
f(-1) = +1, Since f(˗ 2) < 0 and f(˗ 1) > 0 there exists a root (say x = c) such that f(c)
= 0. |
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with other important theorems. |
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