1 / 24

Section 2.1

Section 2.1. Rates of Change and Limits. Average and Instantaneous Speed. Definition 1: A moving body’s average speed during an interval of time is found by dividing the distance covered by the elapsed time. Example 1: Finding an average speed.

Leo
Download Presentation

Section 2.1

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. Section 2.1 Rates of Change and Limits

  2. Average and Instantaneous Speed Definition 1: A moving body’s average speed during an interval of time is found by dividing the distance covered by the elapsed time.

  3. Example 1: Finding an average speed A rock breaks loose from the top of a tall cliff. What is its average speed during the first 2 seconds of fall?

  4. Example 2: Finding an instantaneous speed Find the speed of the rock in Example 1 at the instant t = 2.

  5. Definition of Limit Definition 2: Let c and L be real numbers. The function f has a limit L as x approaches c if,

  6. TOTD • An object dropped from rest from the top of a tall building falls y = 16t² feet for the first t seconds. • Find the average speed during the first 4 seconds of fall.

  7. Properties of Limits

  8. Example 3

  9. Example 4

  10. Theorem 2: Polynomial and Rational Functions • If any polynomial function and c is any real number, then • If f(x) and g(x) are polynomials and c is any real number, then

  11. Example 5: Determine the limit by substitution

  12. Example 6: Determine the limit by substitution

  13. Example 7: Determine the limit by substitution

  14. Example 8: Using the Product Rule:

  15. Example 9: Explain why you cannot use substitution to determine the limit. Find the limit if it exists.

  16. Example 10: Explain why you cannot use substitution to determine the limit. Find the limit if it exists.

  17. TOTD • Determine the limit by substitution.

  18. Example 11: Determine the limit graphically. Confirm algebraically.

  19. Example 12: Determine the limit graphically. Confirm algebraically.

  20. One-sided and Two-sided Limits Right-hand: Left-hand:

  21. Example 13: Function values approach two numbers

  22. Theorem 3: One-sided and Two-sided Limits A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal. In symbols,

  23. Example 14: Exploring right- and left-handed limits

  24. TOTD • Determine the limit algebraically.

More Related