1 / 14

Differentiability

Photo by Vickie Kelly, 2003. Greg Kelly, Hanford High School, Richland, Washington. Differentiability. Arches National Park. RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE HADAPSAR, PUNE-411028. PRESANTATION BY Prof . DESAI S.S Mathematics department Subject – Real analysis

ohensley
Download Presentation

Differentiability

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, 2003 Greg Kelly, Hanford High School, Richland, Washington Differentiability Arches National Park

  2. RAYAT SHIKSHAN SANSTHA’SS.M.JOSHI COLLEGE HADAPSAR, PUNE-411028. PRESANTATION BY Prof . DESAI S.S Mathematics department Subject – Real analysis Topic - Differentatiable Function

  3. To be differentiable, a function must be continuous and smooth. Is the function continuous? = 2 Answer: Yes = 2

  4. But by definition, what is a derivative? A LIMIT or This is another reason why we need to keep the definition of the derivative in mind before moving on to the short cuts for derivatives.

  5. Differentiable also means that the left and right limits of the derivative are equal. Is the function Differentiable at x = 1? = 3 But a limit only exists if it’s left and right limits are equal.

  6. Differentiable also means that the left and right limits of the derivative are equal. Is the function Differentiable at x = 1? = 2 Answer: No ≠

  7. To be differentiable, a function must be continuous and smooth. Is the function Differentiable at x = 1? Notice that on the left side, the slope is approaching 3 ≠ While on the right side, the slope is 2

  8. To be differentiable, a function must be continuous and smooth. Is the function differentiable at x = 1? First of all, is the function continuous? Yes Now can we show if it is differentiable at x = 1?

  9. To be differentiable, a function must be continuous and smooth. Is the function differentiable at x = 1? =2 =2

  10. To be differentiable, a function must be continuous and smooth. Is the function differentiable at x = 1? Notice that on the left side, the slope is approaching 2 Therefore, the answer is YES. While on the right side, the slope is 2

  11. To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: corner cusp discontinuity vertical tangent

  12. If f has a derivative at x = a, then f is continuous at x = a. Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

  13. If a and b are any two points in an interval on which f is differentiable, then takes on every value between and . Between a and b, must take on every value between and . Intermediate Value Theorem for Derivatives p

  14. Thank you

More Related