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Intermediate Value Theorem

Intermediate Value Theorem. 1.4. 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

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  1. Intermediate Value Theorem 1.4

  2. 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 Intermediate Value Theorem: Intuition

  3. 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. Intermediate Value Theorem

  4. 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. Limitations of IVT

  5. 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.

  6. 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)? Example

  7. Verify the IVT applies to the given interval & find the value of c guaranteed by the theorem. Therefore So IVT applies. Then find f(c) by plugging it in and graphing to find the zero in the interval. So f(2)=4 because y=0 at 2, which is in the interval. Example

  8. p. 81 #83-86, 91-94 Homework

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