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Understanding Limits and Continuity: Average and Instantaneous Speed in Calculus

This chapter focuses on the concepts of limits and continuity in calculus, particularly in the context of rates of change. It introduces average speed during time intervals and instantaneous speed at a specific moment, using examples such as a falling rock. The text explains how to find limits using substitution, addresses indeterminate forms, and discusses one-sided limits. Various methods for determining limits, including algebraic, graphical, and tabular approaches, are also covered, providing students with a comprehensive understanding of these fundamental calculus concepts.

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Understanding Limits and Continuity: Average and Instantaneous Speed in Calculus

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  1. Chapter 2 Limits and Continuity

  2. Introduction • Economic Injury Level (EIL)- text p. 58

  3. 2.1: Rates of Change and Limits Average Speed The average speed during an interval of time is • Example: If a rock falls from a cliff, what is its average speed after the first 2 seconds? (Distance for a falling object equation: d = 16t2) = = = = 32 ft/s

  4. Instantaneous Speed (speed at an exact time) • Example: If a rock falls from a cliff, what is its instantaneous speed at 2 seconds? To answer this question, we will find the AVERAGE speed of the rock between 2 seconds and slightly more than 2 seconds (we’ll call it “2+h” seconds, where his approximately zero). (Recall: Distance for a falling object equation: d = 16t2) = = = = … =64 At 2 seconds, the rock is travelling at 64 ft/s.

  5. Instantaneous change using the table in the graphing calculator (Calculator may only be used to check.) • Enter the following in Y1: • TableSet: start at x =1 Δx= .1 (look at table) • Change to: start at x =.1 Δx= .01 (look at table) • Change to: start at x =.01 Δx= .001 (look at table) • What value is y getting close to? As h approaches 0, y has the limiting value of 64.

  6. Definition of LIMIT: L a

  7. The Definition of Limit L a

  8. Basic Limits (Substitution) • limx4 2x – 5 = • limx-3 x2 = • limxcos x = • limx1sin =

  9. Indeterminate Forms Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Notice form Ex. Factor and cancel common factors

  10. More Examples

  11. More Examples Evaluate the following:. • Existence of a limit does not depend on whether or not the function is defined at c.

  12. Example: Find using the graphing calculator.

  13. (Found on text p. 61) Limit Theorems

  14. More Examples

  15. Find the limit:

  16. One-Sided Limits If limxc+f(x) = limxc-f(x) = L then, limxcf(x)=L (Again, L must be a fixed, finite number.) Basically, the y-value the function approached from the left side of a particular x-value must be the same y-value the function approaches from the right!

  17. One-Sided Limit Example 1. Given Find Find

  18. One-Sided Limits Example: f(4) = f(2) =

  19. One-Sided Limits f(0) = f(4) = f(3) = f(6) =

  20. But

  21. One-Sided Limits

  22. Limits from Tables • For each of the following tables, find the right-hand, left-hand, and "regular" limits at the center value of the table. If no limit exists, then state that.

  23. Limits from Tables

  24. Summary • The limit of a function at x = c does not depend on the value of f(c). • The limit only exists when the limit from the right equals the limit from the left and the value is a FIXED, FINITE #! • A common limit you need to memorize: • Limits fail to exist: (ask for pictures) 1. Unbounded behavior – not finite 2. Oscillating behavior – not fixed 3. - fails def of limit • Limits can be found algebraically (substitution), graphically, and from a table.

  25. HOMEWORK: p.

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