Intermediate Value Theorem

1. Intermediate Value Theorem Section 3.7

2. 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

3. 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.

5. 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.

6. 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.

7. Example 1: • Sketch a graph to decide if the cosecant function, f(x) = csc (x) is continuous over the domain [-π, π].

8. 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.

9. 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)?

10. Homework Pages 188 – 189 4, 6-9, 11, 12, 15