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Rates of Change and Limits

Rates of Change and Limits. Chapter 2. Limits and Continuity. Section 2.1. Quick Review. Quick Review. Quick Review Solutions. Quick Review Solutions. What you ’ ll learn about. Interpretation and expression of limits using correct notation

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Rates of Change and Limits

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  1. Rates of Changeand Limits Chapter 2 Limits and Continuity Section 2.1

  2. Quick Review

  3. Quick Review

  4. Quick Review Solutions

  5. Quick Review Solutions

  6. What you’ll learn about • Interpretation and expression of limits using correct notation • Estimation of limits using numerical and graphical information • Limits of sums, differences, products, quotients, and composite functions • Interpretation and expression of one-sided limits • The Squeeze Theorem …and why Limits can be used to describe continuity, the derivative and the integral: the ideas giving the foundation of calculus.

  7. Average and Instantaneous Speed

  8. Average and Instantaneous Speed The average speed or average rate of change of a moving body during an interval of time is found by dividing the change in distance or position by the change in time. The speed of a falling rock is always increasing. If we know the position as a function of time, we can calculate average speed over any given interval of time. But we can also talk about its instantaneous speed or instantaneous rate of change, the speed at one instant of time. As we will see after the next example, we need the idea of limit to make precise what we mean by instantaneous rate of change.

  9. Definition of Limit

  10. Definition of Limit continued

  11. Definition of Limit continued

  12. Properties ofLimits

  13. Properties ofLimits continued

  14. Properties ofLimits continued

  15. Example Properties ofLimits

  16. Polynomial and Rational Functions

  17. Example Limits

  18. Evaluating Limits As with polynomials, limits of many familiar functions can be found by substitution at points where they are defined. This includes trigonometric functions, exponential and logarithmic functions, and composites of these functions.

  19. Example Limits

  20. Example Limits [6,6] by [10,10]

  21. One-Sided and Two-Sided Limits

  22. One-Sided and Two-Sided Limits continued

  23. Example One-Sided and Two-Sided Limits Find the following limits from the given graph. 4 o 3 1 2

  24. Squeeze Theorem

  25. Squeeze Theorem

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