1 / 12

Intermediate Value Theorem

Intermediate Value Theorem. Section 2.3b. Intermediate Value Property – describes a function that never takes on two values without taking on all the values in between. The Intermediate Value Theorem for Continuous Functions.

chibale
Download Presentation

Intermediate Value Theorem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Intermediate Value Theorem Section 2.3b

  2. Intermediate Value Property – describes a function that never takes on two values without taking on all the values in between. The Intermediate Value Theorem for Continuous Functions A function y = f (x) that is continuous on a closed interval [a, b] takes on every value between f (a) and f (b). In other words, if y is between f (a) and f (b), then y = f (c) for some c in [a, b]. 0 0 y x 0

  3. Exploration 1: Removing a Discontinuity Let 1. Factor the denominator. What is the domain of f ? Domain: 2. Investigate the graph of f around x = 3 to see that f has a removable discontinuity at x= 3. It appears that the limit of f as x 3 exists and is a little more than 3. 3. How should fbe redefined at x = 3 to remove the discontinuity? Use zoom-in and tables as necessary. should be defined as

  4. Exploration 1: Removing a Discontinuity Let 4. Show that (x – 3) is a factor of the numerator of f, and remove all common factors. Now compute the limit as x 3 of the reduced form for f. for Thus,

  5. Exploration 1: Removing a Discontinuity Let 5. Show that the extended function is continuous at x = 3. The function g is the continuous extension of the original function f to include x = 3. so g is continuous at x = 3

  6. Guided Practice Give a formula for the extended function that is continuous at the indicated point. For

  7. Guided Practice Give a formula for the extended function that is continuous at the indicated point. The extended function:

  8. Guided Practice Sketch a possible graph for a function f that has the properties stated below. exists, but does not. One possible answer: y x 3

  9. Guided Practice Sketch a possible graph for a function f that has the properties stated below. exists, , but does not exist. y One possible answer: x –2

  10. Guided Practice Sketch a possible graph for a function f that has the properties stated below. exists, exists, but f is not continuous at x = 4. One possible answer: y x 4

  11. Guided Practice Sketch a possible graph for a function f that has the properties stated below. is continuous for all x except x = 1, where f has a non-removable discontinuity. One possible answer: y x x = 1

  12. Guided Practice Find a value for a so that the given function is continuous. What might the graph look like? We require that Substitute: Does this answer make sense graphically?

More Related