2-1 Rates of Change and Limits

1. 2-1 Rates of Change and Limits

2. Average vs. Instantaneous Speed Position function  f (t) time  t time interval  h Average speed Ex 1) A rock breaks loose from the top of a cliff. What is the average speed during the first 2 seconds of fall? h Ex 2) What is the speed of the rock in EX 1 at the instant t = 2? 0  means nothing to us! 0 The idea of a limit comes about…

3. Definition of Limit Idea: How does a function behave as you approach a particular x-value? (what is the y-value that corresponds?) Precise Definition: Assume f is defined in a neighborhood of c and let c & L be real #s. The function f has a limit L as x approaches c if, given any positive number , there is a positive number such that for all x, Note: this limit is what the function approaches. It may or may not be the actual value of the function

4. Properties of Limits A) B) Limit of a constant is the constant sum sum difference difference product product quotient quotient constant × function constant × power of a function power The limit of of the limits is the (den  0)

5. Ex 3) Thm: Polynomial & Rational Functions f (x) is a polynomial function f (x) & g (x) are polynomials *substitute c in for x*

6. Ex 4b) Ex 5) (graph on calculator)  = 1 One Sided and Two Sided Limits Ex 6) (graph) window [–10, 10] × [–100, 100] Limit does not exist (DNE)

7. *Can have left & right hand limits from the right from the left Ex 7) • Thm: The function f (x) has a limit as x approaches ciff the right & left hand limits exist and are equal

8. Ex 8) Thm: The Squeeze Theorem / The Sandwich Theorem

9. homework Pg. 66 #2, 3 – 63 (mult of 3)