class 050831

preliminary questions

(refer to matching basic graphs, from 0829)

How do we recognize the three basic points of a linear  function based on y = x?

 

How do we recognize the three basic points of a quadratic function based on y = x^2?

 

How do we recognize the three basic points of an exponential function based on y = 2^x?

 

How do we recognize the three basic points of a square root function based on y = sqrt(x)?

 

According to the depth vs. clock time model:

Is the depth vs. clock time function most clearly related to the linear function y = x, the quadratic function y = x^2, the exponential function y = 2^x or the square root function y = sqrt(x)?

 

What is the depth at clock time t = 25 seconds?

 

What is the depth at clock time t = 200 seconds, and what is wrong with this answer?

 

At what clock time is the depth equal to 25 cm?

 

What is the minimum depth and when does this occur?

In class we obtained the following information for the depth vs. clock time experiment conducted Monday.  The results were averaged over the results obtained by students and represent an approximate weighted mean for each depth:

clock time (sec) depth (cm)   depth according to 3-point model
0 90   94
3.8 80   80.4
7.2 70   69.2
10.4 60   59.5
14 50   49.7
19 40   37.7
24.5 30   27
30.5 20   18.1
41 10   9.58

Using the points (4, 80), (14, 50) and (41, 10) and the form y = a t^2 + b t + c we obtained a system of 3 equations in the unknowns a, b, c and solved the equations, ending up with a = .041, b = -3.74 and c = 94.6.  This gives us the model

According to this model the depths at the clock times given in the above table are as given in the fourth column of the table.  Each depth was obtained by substituting the corresponding t valud into the 3-point model y = .041 t^2 - 3.74 t + 94.6.  We note that the predictions of the model in the fourth column are very close to the observed values.

Excel gives us the graph below.  The function

was obtained by Excel as the best degree-2 polynomial fit to all the points.

This model takes into account all the points, not just 3 selected points, and fits the data even better than our 3-point model did.

Homework:

Work the problems assigned near the end of the worksheet Completion of the Introductory Flow Model, as handed out it class.  If you didn't get the handout or if you have misplaced it, simply go to the webpage http://www.vhcc.edu/pc1fall9/, click on Assts and go to Assignment 2.  Then click on the Overview and Introduction: ...  link, read and take notes on everything you see there (very similar to what we did in class Monday).  Then click on the Completion of the Introductory Flow Model link, read everything and do the exercises at the end.

Assignment 2

q_a_ Assignment 2.

Suggested Review: Text, Section 1.5

Overview and Introduction:  The Modeling Process applied to Flow From a Cylinder (see also Class Notes #01).  Read note below first.

Completion of the Introductory Flow Model (Class Notes #02) Distance students: submit your solutions to the exercises. 

Distance Students:  View Introduction on CD #2.

When you have completed the entire assignment run the Query program.