Excel Project in Calculus part 3

If you need background, see my first post:

First Excel in Calculus post.
** If you want the Word and Excel copies, @me (@robebymathdude) on Twitter **

For this third Excel project, I wanted to stress to the students how much they assume continuity, and how big an impact that assumption plays in their expectations.  I got at this by having them conjecture about important places for the function, and then go back and revise those guesses depending upon some other work. 

This was in the section where we first encountered local extrema, and had spent a good bit of time with rates of change, so I wanted to expand on those ideas.

For the first set, I had displayed some (x,y) pairs, and then had the students guess where the local max occurred after first finding Average Rates of Change (ARC) between each pair of points.  Almost all said it had to be around x=3, and no one mentioned the assumption of continuity.  I then showed them the function with the vertical asymptote (1e below).  About half of the students when answering part (g) mentioned the assumption of continuity, but the other half did not.  

IDSUM is the sum of the digits in their ID number.  This gives related but different problems. 

1.       For the function F-1 (light orange background), we will do some data analysis.

a.       For each pair, find the Average Rate of Change (ARC) between the pair. (The first row is “x” and the bottom row is “y”)

b.       Where (between what x-values) does it appear the ARC becomes zero?  Why?

c.       Find the ARC for the entire data set.  Between what pair of x-values (if anywhere) does it appear the function has the same ARC as the ARC for the entire data set?  Why?

d.       Now find the sum of the digits in your Blinn ID, and call that IDSUM.

e.       Graph the function  

 Include the graph in your write up.  (Desmos, fooplot, or some other graphing device)

f.        Now compute   f(0), f(1), f(2), f(2.5), f(3.5), f(4), f(5), f(6)

g.       How does this change your answers to parts b and c of this question?  What were you assuming about the function F-1 when you answered parts b and c?

For the next parts, I had them repeat the ARC calculations, and then make some projections for future values.  They then compared their guesses with the actual function values.  For the first, the function was

But for the last pair, the function was not linear, but instead exponential:

I then asked the students how the estimates compared to the original estimates, and what assumptions they made.