2.1: Rates of Change & Limits

1. 2.1: Rates of Change & Limits Greg Kelly, Hanford High School, Richland, Washington

2. Then your average speed is: Suppose you drive 200 miles, and it takes you 4 hours. If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.

3. A rock falls from a high cliff. The position of the rock is given by: After 2 seconds: average speed: What is the instantaneous speed at 2 seconds?

4. for some very small change in t where h = some very small change in t We can use the TI-89 to evaluate this expression for smaller and smaller values of h.

5. 1 80 0.1 65.6 .01 64.16 .001 64.016 64.0016 .0001 .00001 64.0002 We can see that the velocity approaches 64 ft/sec as h becomes very small. We say that the velocity has a limiting value of 64 as h approaches zero. (Note that h never actually becomes zero.)

6. 0 The limit as h approaches zero:

7. Limit notation: “The limit of fof x as x approaches c is L.” LIMITS AT INFINITY (Page 608)and *If x is negative, xn does not exist for certain values of n, so thesecond limit is undefined.

8. 1. If f ( x ) becomes infinitely large in magnitude (positive or negative) as x approaches the number a from either side, we write or In either case, the limit does not exist. 2. If f ( x ) becomes infinitely large in magnitude (positive) asx approaches a from one side and infinitely large in magnitude (negative) as x approaches a from the other side, then does not exist. 3. If and , and L≠ M, then does not exist. EXISTENCE OF LIMITS: (see page 602) Means approach from the right Means approach from the left

9. The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

10. Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. (See page 603 for details.) For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

11. does not exist because the left and right hand limits do not match! 2 1 1 2 3 4 left hand limit right hand limit value of the function At x=1:

12. because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=2:

13. because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=3:

14. CONTINUITY AT x = c (see page 617) A function f is continuous at x = c if the following three conditions are satisfied: 1. f (c) is defined If f is not continuous at c, it is discontinuous there.

15. 2 1 1 is a point of discontinuity even though f (1) exists, thedoes not exist. (rule 2) 1 2 3 4 2 is a point of discontinuity even though f (2) exists andexists, f (2) ≠ (rule 3) 3 is a point of continuity because f (3) exists, exists, andf (3) = Examples: 4 is a point of discontinuity becausef (4) does not exist. (rule 1)

16. # # # # # ∞ – ∞ # #1 #2 # ∞ #1 #2 – ∞ # #1 #2 #1 #2 Interval notation Graph ( #, means to start just to the right of the number [ #, means to start exactly at the number # ) means to stop just to the left of the number (exclude the #) # ] means to stop exactly at the number (include the #) ( #1, #2 ) is an open interval ( #1, #2 ] is a half-open interval [ #1, #2 ) is a half-open interval [ #1, #2 ] is a closed interval

17. 2. Use the rules for limits, including the rules for limits at infinity, and to find the limit of the result from step 1. If f (x) = p(x)/q(x), for polynomials p(x) and q(x), q(x) ≠ 0, and can be found as follows. Finding Limits at Infinity (page 610) 1. Divide p(x) and q(x) by the highest power of x in q(x).

18. EXAMPLE # 1 0 0 0 0

19. EXAMPLE # 2 0 0 0 0

20. EXAMPLE # 3 0 0 ∞

21. IN SUMMARY If the degree of the numerator is larger than the degree of the denominator the limit is infinity. If the degree of the denominator is larger than the degree of the numerator the limit is zero. If the degree of the numerator is the same as the degree of the denominator the limit is the quotient formed by their coefficients a/b. However on a test, I want to see the steps involved. I do not want you to use this summary. You may use it only to verify your answers.