1 / 27

Math 180

Math 180. 3.2 – The Derivative as a Function. By modifying the definition slightly, we can consider the derivative as a function of : is _____________ at if exists. ______________ is the process of calculating the derivative .

isabel
Download Presentation

Math 180

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. Math 180 3.2 – The Derivative as a Function

  2. By modifying the definition slightly, we can consider the derivative as a function of : is _____________at if exists. ______________is the process of calculating the derivative.

  3. By modifying the definition slightly, we can consider the derivative as a function of : is _____________at if exists. ______________is the process of calculating the derivative. differentiable

  4. By modifying the definition slightly, we can consider the derivative as a function of : is _____________at if exists. ______________is the process of calculating the derivative. differentiable Differentiation

  5. Ex 1.Using the definition, differentiate .

  6. Ex 2.Using the definition, find the derivative of for . Now find the tangent line to the curve at .

  7. Ex 2.Using the definition, find the derivative of for . Now find the tangent line to the curve at .

  8. Note: There are many ways people write the derivative of : And here’s what it looks like to plug a value into the derivative:

  9. Note: There are many ways people write the derivative of : And here’s what it looks like to plug a value into the derivative:

  10. Graphing Derivatives Remember that the derivative is the slope of the tangent line.

  11. Note that only exists if both of the following limits exist: Left-hand derivative at Right-hand derivative at

  12. Note that only exists if both of the following limits exist: Left-hand derivative at Right-hand derivative at

  13. Note that only exists if both of the following limits exist: Left-hand derivative at Right-hand derivative at

  14. Ex 3.Compute the right-hand and left-hand derivatives as limits to show that is not differentiable at .

  15. Ex 3.Compute the right-hand and left-hand derivatives as limits to show that is not differentiable at .

  16. Ex 3.Compute the right-hand and left-hand derivatives as limits to show that is not differentiable at .

  17. Ex 3.Compute the right-hand and left-hand derivatives as limits to show that is not differentiable at .

  18. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  19. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  20. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  21. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  22. So, derivatives won’t exist at the following kinds of places: corner cusp vertical tangent discontinuity

  23. Theorem: If has a derivative at , then is continuous at . (That is, differentiable functions are continuous. And if a function is not continuous at a point, then it is not differentiable there.)

More Related