class opener:

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.

a) Write the volume V of the package as a function of x. What is the domain of the function?

Example:Find the domain of each function:

1. Volume of a Sphere:

Example:Find the domain of the given function:

Put Technology to WorkUsing the graphing calculator find the

domain and range of the following function:

Example:Use a graphing calculator to find the domain

and range of the following functions.

Real World ConnectionsThe 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 .

Physics ConnectionA 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

Calculus ConnectionOne of the basic definitions for calculus

employs the ratio:

This is known as the difference quotient.

Evaluating with Difference QuotientFor find the difference Quotient.

Find the Difference Quotient

𝑓 (𝑥 )=𝑥2−𝑥+1,𝑓 (2+h )− 𝑓 (2)

h

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

Review: Vertical Line TestIs this a Function?

Increasing and Decreasing FunctionsA 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)

Increasing and Decreasing Function

Example:On your calculator graph

Determine the open intervals on which each function is increasing, decreasing, or constant.

Student Check:Determine the open intervals on which each

function is increasing ,decreasing, or constant.

Relative Minimum and MaximumA 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

Relative Minimum and Maximum

Approximating Relative Minima and MaximaUsing a calculator approximate the relative

minimum of the function given by

Student CheckUsing your calculator approximate the

relative minimum and relative maximum of the function given by