1 / 11

2.3 Continuity

Photo by Vickie Kelly, 2002. Greg Kelly, Hanford High School, Richland, Washington. 2.3 Continuity. Grand Canyon, Arizona. 2. 1. 1. 2. 3. 4.

woodring
Download Presentation

2.3 Continuity

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. Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington 2.3 Continuity Grand Canyon, Arizona

  2. 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. Where is this function continuous?

  3. The analytic definition of Continuity at a point is: Closed circle Two sides of a curve leading to the open circle

  4. Closed circle Closed circle Two sides of a curve leading to the open circle Two sides of a curve leading to the open circle …and, therefore, f (x) is continuous at x = 3

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

  6. 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:

  7. has a discontinuity at . Write an extended function that is continuous at . Removing a discontinuity: Another way to express the solution could be:

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

  9. 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:

  10. 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 .

  11. 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 p

More Related