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.
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:
You saw in part I that you can define a function piecewise over an interval. Suppose a function is partially defined as
Experiment by filling in the blank with various polynomials that will continue from 0 where cos(x) stops. Then answer parts (a), (b), and (c) below. When responding to them, simply change the definiton of h. All the functions you define in terms of h will update automatically.
a) Fill in the blank with a polynomial function so that
III. Extension - 2
Graph f(x) = x2 mod 2 [enter as y = mod(x2, 2)]. Then graph the average rate of change function for f using intervals of length 1, 0.5, and 0.001.
Explain what is represented by any point on the average rate function's graph. (Don't say "average rate of change." Say what "average rate of change" means.)
Notice that the graph of f is discontinuous, but that the graph of its derivative (i.e., its average rate of change function over extremely small intervals of change) appears to be continuous. Are appearances deceiving? Explain.