1. Friday, February 24, 2012MAT 121

2. Friday, February 24, 2012MAT 121 How calculate limits? (a) Try subbing in value! (b) Simplify if possible and return to (a). (c)Use a numerical approach: Sub in values closer and closer to the limiting input value. (d)Look at a graph of the function and visualize the limit. (e) Play the S & S Game: Identify SIZE and SIGN

3. Friday, February 24, 2012MAT 121

4. Friday, February 24, 2012MAT 121

5. Friday, February 24, 2012MAT 121

6. Friday, February 24, 2012MAT 121 We have determined relative max and relative min values using first derivatives. We now focus on identifying the MOST EXTREME values among those RELATIVE possibilities. Often, this involves LIMITING THE DOMAIN, and, therefore, considering what occurs at the END POINTS. Here are examples.

7. Friday, February 24, 2012MAT 121 (A)For f(x) = 3x 4 + 4x 3 + 1, determine its absolute extreme values over the interval [−2,1]. (B)If g(x) = x – 2√x: (i) determine its absolute extreme values over the interval [0,9]. (ii) determine its absolute extreme values for all x. (C)For 2006, operating rate of factories, mines, and utilities in a certain region of the country, expressed as a % of full capacity, is given by the function shown below, where 0 ≤ t ≤ 250 is measured in days. On which of the first 250 days of 2006 was the rate the highest?

8. Friday, February 24, 2012MAT 121

9. Friday, February 24, 2012MAT 121

10. Friday, February 24, 2012MAT 121

11. Friday, February 24, 2012MAT 121