In this assignment you will not be presented with concrete situations to model and interpret. Instead, you are asked to speak about "how fast the value of a function changes with respect to changes in its argument." Think of this as being about an "abstract" situation where there are quantities and units, but they are not given to you.

Functions in this assignment are defined "piecewise." That means that they have one definition over part of their domain and another definition over another part.

I. Investigations

Start with the function . Enter it as y = 1+sin(x) if x<0 (ctrl-a) (tab) cos(x) if x ≥ 0. (See Piecewise Defined Functions in GC.)

Define a function that gives its average rate of change over intervals of length h. Use 2, 1, 0.5, and 0.01 as values of h.

After examinging the average rate of change function: Play the movies below (by double-clicking them), or play with the original GC graph, and then explain why the rate of change function behaves the way it does over large and small intervals.

Use the insights gained here to explain why the average rate function for y = |x| behaves as it does for large and small values of h.

h = 2.0

h = 1.0

h = 0.2

h = 0.05

I. Do the following for each of these two functions:

a. Behavior of the function

Enter the function in Graphing Calculator and graph it by entering y = f (ctrl-9) x or y = g (ctrl-9) x. Notice any behavior that merits close scrutiny.

b. Average rates of change

Enter the definition of a function whose values give the original function's average rate of change over every inteval of length h. Use values of 2.0, 1.0, 0.5, 0.1, and 0.00001 for h. Explain is going on.