1 / 12

Bell Ringer

This lesson focuses on the concept of continuity in calculus. Learn about removable and essential discontinuities and how to create extended functions that are continuous. Explore examples and discover how the Intermediate Value Theorem applies to continuous functions.

minniehoward
Download Presentation

Bell Ringer

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. Bell Ringer Solve even #’s

  2. 2.3 Continuity Pg.

  3. 2 1 1 2 3 4 Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

  4. Removable Discontinuities: (You can fill the hole.) Essential Discontinuities: oscillating infinite jump

  5. has a discontinuity at . Write an extended function that is continuous at . Note: There is another discontinuity at that can not be removed. Removing a discontinuity:

  6. Note: There is another discontinuity at that can not be removed. Removing a discontinuity:

  7. Also: Composites of continuous functions are continuous. Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. examples:

  8. Because the function is continuous, it must take on every y value between and . Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between and .

  9. F2 Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5. Therefore there must be at least one solution between 1 and 2. Use your calculator to find an approximate solution. Is any real number exactly one less than its cube? Example 5: (Note that this doesn’t ask what the number is, only if it exists.) 1: solve

  10. Graph: Note resolution. Graphing calculators can sometimes make non-continuous functions appear continuous. CATALOG F floor( This example was graphed on the classic TI-89. You can not change the resolution on the Titanium Edition. The calculator “connects the dots” which covers up the discontinuities.

  11. Graph: GRAPH Graphing calculators can make non-continuous functions appear continuous. CATALOG F floor( If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct floor graph. The open and closed circles do not show, but we can see the discontinuities. p

  12. Homework: 2.3a p84 #’s 3,12,21,30 p76 #’s 45,59,69 2.3b p84 6,15,24,42,63

More Related