Intermediate Value Theorem

Section 3.7

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Intermediate Value Theorem: Intuition

• Traveling on France’s TGV trains, you reach speed of 280 mi/hr.

• How do you know at some point of train ride you were traveling 100 mi/hr?

• To go from 0 to 280, must have passed through 100 mi/hr since speed of train changed continuously

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Intermediate Value Theorem

• Suppose that f is continuous on the closed interval [a,b]. If L is any real number between f(a) and f(b) then there must be at least one number c on the open interval (a,b) such that f(c) = L.

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Limitations of IVT• If d(0) = 100 and d(10) = 35, where t is measured in seconds. • d is a continuous function, the IVT tells you that at some point

between t=0 and t =10, the decibel level reached every value between 35 and 100.

• It does NOT say anything about: • When or how many times (other than at least once) a particular

decibel was attained. • Whether or not decibel levels bigger than 100 or less than 35

were reached.

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The Difference Between VROOOOOOOOM and VROOOOOOOM.

These graphs of PC's noise illustrate that very different behaviors are consistent with the hypothesis that d(t) is continuous and that its values at t=0 and t=10 are 100 and 35 respectively.

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Example 1:

• Sketch a graph to decide if the cosecant function, f(x) = csc (x) is continuous over the domain [-π, π].

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Example 2

• Consider the equation sin x = x – 2 . Use the intermediate Value Theorem to explain why there must be a solution between π/2 and π.

Example 3

• Consider the function ,

• Calculate f(6), f(-5.5), f(0)

• Can you conclude that there must be a zero between f(6) and f(-5.5)?

Homework

Pages 188 – 1894, 6-9,

11, 12, 15