Workshop Instructors Guide, Wed Oct 8

Note

If you prefer, bring in an overhead with your own function d(t) instead of having the students make up their own.

Solution

To be used for "closure" summary by Workshop leaders; followed by (brief) discussion or Q/A if appropriate.

Years of training has made this situation "obvious" to us, but it's worthwhile to consider into just how many separate steps the reasoning can be divided (carefully ennumerate each of the following to the students making sure there is compherension and consensus among them on each step):

  1. The display on the speedometer (at time t0)

    ...which equals:

  2. The instantaneous velocity of the car (at time t0)

    ...which equals:

  3. The limit of the average velocities for intervals of time around t0

    ...which equals:

  4. Graphically, the limit of the slopes of secants (why? see units on axes, and use geometry) around t0

    ...which equals:

  5. The slope of the tangent at time t0.

    Foreshadowing:

    ...which equals (as we defined it in the next section):

  6. The number d'(t0), which is the value of the derivative function d'(t) at t=t0.

    In closing, something like:

    "Hence if the graph actually represented a real-world car trip, we see now that the display of your speedometer (a very concrete thing!) would indeed actually match 5., the slope of the tangent, as well as 6., the value of the "derivative" function, which we will soon define.

  7. If there is time, the students can be given something short about continuity as well, such as, problem #29 in Stewart page 129 for section 2.4 (since most students will have a copy of the book, you a handout will not be given): 
    
For what value of the constant c is the function f continuous on
(-infinity,infinity) ?

	{ cx+1   if  x<= 3
f(x)  = {
	{ cx2-1   if x > 3