Average Rates of Change
When an object falls from a resting start, the distance (in feet) it has fallen x seconds after being released is given by the function d(t) = 16t2 (assuming no air resistance). Remember that in GC, to enter a function definition you type d ctrl-9 t = etc. (See Defining Functions in GC.)
1. Define the function d, as given above, in Graphing Calculator.
a. Graph the function g(x) = d(x +
1.2) - d(x). [That is, enter the expression g ctrl-9 x = d ctrl-9 x+1.2 etc. Then graph g by entering y = g(x)].
What does the point (0.7, g(0.7)) represent? What does each point on the graph
of g represent?
b. Graph the function h(x) = d(x + .01)-d(x). What does the point (0.7, h(0.7)) represent? What does each point on the graph of h represent?
2. Define the function d, as given above, in Graphing Calculator.
a. Graph the function 
. What does each point on the graph of k represent? Give an example.
b. Graph the function 
, where h is a
parameter. What does each point on the graph of m represent? Give an example.
c. Expand and simplify the expression 
to explain why
the graph of m(x) is always a straight line, regardless of the value
of h.
d. What does this simplified expression tell you about changes in the object's average speeds over time?
3. Suppose that a weight is hanging from a spring, and that you "push" it vertically. The weight's distance from it's starting point can be expressed by the function d(t) = asin(bt), where d is in some unit of distance and t is in some unit of time. Suppose that a=2.3, b=1.4, t is in seconds, and d(t) is in inches.
a. Define a function that expresses this weight's average velocity over every interval of time of length 0.5 seconds.
b. Graph this function in Graphing Calculator. Describe this function, then interpret it. That is, what does each point on the graph represent? Give an example.
c. Express your function in (a) using h for the length of the interval of time over which to calculate an average rate of change. Then make a movie (using h as a slider value) that shows the behavior of your function as the interval of time is made smaller and smaller. What is happening?
d. Expand and simplify your function in (c)†. Use your simplified expression, together with anything you have learned from previous assignments, to explain why the "average velocity" function's graph approximates y=2.3cos(1.4x)(1.4) when it determines average velocities over small intervals of time. Do NOT use limits in your explanation. Nor should you make ANY reference to the idea of derivative. Rather, expand and simplify, then substitute 0.00001sec for the increment of
time in your simplified expression.
See comments on infinitesimals for a discussion of the conceptual issues involved here.