Intermediate Value Theorem

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Intermediate Value Theorem

• 2.4 cont.

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(No Transcript)
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Examples

• If between 7am and 2pm the temperature went from
  55 to 70.
• At some time it reached 62.
• Time is continuous
• If between his 14th and 15th birthday, a boy went
  from 150 to 165 lbs.
• At some point he weighed 155lbs.
• It may have occurred more than once.

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Show that a c exists such that f(c)2 for
f(c)x2 2x-3 in the interval 0, 2
f(x) is continuous on the interval
f(0) -3
f(2) 5
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Determine if f(x) has any real roots
f(x) is continuous on the interval
f(1) -
f(2)
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Is any real number exactly one less than its cube?
(Note that this doesnt ask what the number is,
only if it exists.)
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Max-Min Theorem for Continuous Functions
If f is continuous at every point of the closed
interval a, b, then f takes on a minimum value
m and a maximum value M on a, b. That is, there
are numbers a and ß in a, b such that f(a) m,
f(ß) M, and m f(x) M at all points x in a,
b
Max and mins at interior points
Min at interior point and max at endpoint
Max and mins at endpoints
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Why does the IVT fail to hold for f(x) on -1, 1?
Not Continuous in interval! Point of
discontinuity at x 0
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Show why a root exists in the given interval
Continuous in interval
f(-2) -10
f(-1) 1
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Show why a root exists in the given interval
Continuous in interval
f(1) 1