I.  Quiz

II.  Logistic Questions

III.  Homework Questions

IV.  Local graph experiment  The students will pair and each pair will write down an arbitrary function and an arbitrary value of x.  The that pair will enter the function into a graphing calculator and zoom in on the point.  Then the pair will select either line, parabola, circle, ellipse, or other according to what their screen shows.  Using two points on their graph, they will compute the slope of the curve produced.  

V.  Speed  Suppose that you are cycling on Highway 50 and clock the following times:

Time (t)	Distance (s)
1:00	0
1:15	4
1:30	9
2:00	16

Then your average velocity is defined by

(s(t_f )- s(t_i))/(t_f - t_i)

where t_f is the time at the end of the ride and t_i is the time at the beginning of the ride.  The above expression is called the difference quotient.

Using 1:00 as time 0 minutes, we can compute the average velocity

(16 - 0)/(60 - 0) = .267 miles per minute.

How fast is the bicycle moving when t = 15 minutes?

We can compute the average velocity from 0 to 15 minutes by

(4 - 0)/(15 - 0) = .267

or we can compute the average velocity from 15 to 30 by

(9 - 4)/(30 - 15) = .333

Both of these are estimates.  We can get a better estimate by measuring points closer to time 15, but we can still never get the exact velocity.

If we have a formula we can use the ideas learned from the calculator exercise and note that the slope of the tangent line is the the instantaneous velocity.  

VI.  Integral Calculus

How does one find the area under a curve?  The easiest area formula we know is the area of a rectangle (base)(height).  If we approximate the area under a curve by drawing several small rectangles (a diagram will be given), then we will have a close approximation.  We can never get the exact area, but we can come as close to the area as we wish.  At the end of this quarter we will learn to find the exact area if we are given a formula.