1 / 15

Applied Calculus (MAT 121) Dr. Day Fri day Feb 17, 2012

In this lesson, we will explore how the first derivative of a function can help us determine if a function is increasing or decreasing and locate the maximum and minimum values of the function. We will also discuss the significance of critical numbers and the first derivative test.

hbuckner
Download Presentation

Applied Calculus (MAT 121) Dr. Day Fri day Feb 17, 2012

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. Applied Calculus (MAT 121)Dr. Day Friday Feb 17, 2012 • Using the First Derivative • Is a function increasing or decreasing? • Where are the max and min values of a function? MAT 121

  2. What does the first derivative tell us? Big Ideas • The direction a function is moving at any point—how it is changing at that point—is revealed in its derivative, because a function’s derivative gives us instantaneous rate of change (IROC) at a point. This connects with the increasing and decreasing nature of a function. • The peaks and valleys of a function—its maximum and minimum values and where they occur—can often be detected using the derivative. When a smoothly flowing function reaches a max or a min, its tangent line slope can be nothing except 0, because the tangent line is horizontal. MAT 121

  3. The First Derivative Reveals Whethera Function is Increasing and Decreasing • Connection: Sign of Derivative Direction of Change MAT 121

  4. The First Derivative Reveals Whethera Function is Increasing and Decreasing Example: Determine the intervals on which the function f(x) = x3 – 3x2 – 24x + 32 is increasing and on which it is decreasing. MAT 121

  5. The First Derivative Reveals Whethera Function is Increasing and Decreasing Example: Determine the intervals on which the function f(x) = x2/3 is increasing and on which it is decreasing. MAT 121

  6. The First Derivative Reveals Whethera Function is Increasing and Decreasing Example: Determine the intervals on which the function f(x) = x+ 1/xis increasing and on which it is decreasing. MAT 121

  7. A Function’s Maximum and Minimum Values • What do we mean by maximum and minimum? • How does the derivative connect with the existence of these extreme values? MAT 121

  8. A Function’s Maximum and Minimum Values • What does RELATIVE mean? MAT 121

  9. A Function’s Maximum and Minimum Values • How does the derivativeconnect with the existence of these extreme values? MAT 121

  10. A Function’s Maximum and Minimum Values • What if the derivative doesn’t exist at some point? MAT 121

  11. A Function’s Maximum and Minimum Values • First: Identify a function’s critical numbers. • Then: Look at the sign of the derivative near each critical number MAT 121

  12. A Function’s Maximum and Minimum Values Example: Determine the relative extrema for the function f(x) = x3 – 3x2 – 24x + 32. MAT 121

  13. A Function’s Maximum and Minimum Values The FirstDerivative Test MAT 121

  14. A Function’s Maximum and Minimum Values Example: Determine the relative extrema for the function f(x) = x + 1/x. MAT 121

  15. Assignments WebAssign 4.1 due Monday 2/20 Reminder: Check results for Test #2 MAT 121

More Related