Class Opener:

1. Class Opener: • A rectangular package to be sent by the U.S. Postal Service can have a maximum combined length and girth(perimeter of cross section) of 108 inches. • Write the volume V of the package as a function of x. What is the domain of the function?

2. Example: • Find the domain of each function: • Volume of a Sphere:

3. Example: • Find the domain of the given function:

4. Put Technology to Work • Using the graphing calculator find the domain and range of the following function:

5. Example: • Use a graphing calculator to find the domain and range of the following functions.

6. Real World Connections The number N (in thousands) of employees in the cellular communications industry in the U.S. increase in a linear pattern from 1998 – 2001. In 2002, the number dropped, then continued to increase through 2004 in a different linear pattern . These two patters can be approximated by the function: Where t = years, and 8 = 1998. Use this function to approximate the number of employees for each ear from 1998 to 2004 .

7. Physics Connection A baseball is hit at a point 3 feet above the ground at a velocity of 100 ft/s and at an angle of 45 degrees. The path of the baseball is given by the function: Will the baseball clear a 10 foot fence located 300 feet from home plate? Left Side of Room Work it by Hand Right Side of Room work it graphically on a calculator

8. Calculus Connection • One of the basic definitions for calculus employs the ratio: This is known as the difference quotient.

9. Evaluating with Difference Quotient For find the difference Quotient.

10. Find the Difference Quotient

11. assignment • Pg. 11 – 15 • Exs. 12 – 32 even, 39 – 46, 52 – 62 even, 68 – 74 even, 79 – 82, 85 – 87, 91 – 102, 113 – 116

12. Review: Vertical Line Test • Is this a Function?

13. Increasing and Decreasing Functions • A function f is increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2) • A function f is decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2) • A function f is constant on an interval if for any x1 and x2 in the interval f(x1) = f(x2)

14. Increasing and Decreasing Function

15. Example: • On your calculator graph • Determine the open intervals on which each function is increasing, decreasing, or constant.

16. Student Check: • Determine the open intervals on which each function is increasing ,decreasing, or constant.

17. Relative Minimum and Maximum • A function value f(a) = is called a relative minimum of f if there exists an interval (x1,x2) that contains a such that: x1 < x < x2 implies f(a) f(x) A function value f(a)is called a relative maximum of f if there exists an interval (x1,x2) that contains a such that: x1 < x < x2 implies

18. Relative Minimum and Maximum

19. Approximating Relative Minima and Maxima • Using a calculator approximate the relative minimum of the function given by

20. Student Check • Using your calculator approximate the relative minimum and relative maximum of the function given by