When submitting your work electronically, show the details of your work and give a good verbal description of your graphs.
One very important goal of the course is to learn to communicate mathematical thinking and logical reasoning. If you can effectively communicate mathematics, you will be able to effectively communicate a wide range of important ideas, which is extremely valuable in your further education and in your career.
When writing out solutions, self-document. That is, write your solution so it can be read without reference by the reader to the problem statement. Use specific and descriptive statements like the following:
Here are some data for the temperature of a hot potato vs. time:
Time (minutes)
Temperature (Celsius)
0
119
15
119
30
94.75276
45
85.03294
60
76.62978
75
69.36494
90
63.08422
105
57.6543
Graph these data below, using an appropriate scale:
Pick three representative points and circle them.
Write the equations that result from the assumption that the appropriate mathematical model is a quadratic function y = a t^2 + b t + c.
Eliminate c from your equations to obtain two equations in a and b.
Solve for a and b.
Write the resulting model for temperature vs. time.
Make a table for this function:
Time (minutes)
Model Function's Prediction of Temperature
0
15
30
45
60
75
90
105
Sketch a smooth curve representing this function on your graph.
Expand your table to include the original temperatures and the deviations of the model function for each time:
Time (minutes)
Temperature (Celsius)
Prediction of Model Deviation of Observed Temperature from Model 0
119
15
105.9956
30
94.75276
45
85.03294
60
76.62978
75
69.36494
90
63.08422
105
57.6543
Find the average of the deviations.
1. If you have not already done so, obtain your own set of flow depth vs. time data as instructed in the Flow Experiment (either perform the experiment, as recommended, or E-mail the instructor for a set of data).
Complete the modeling process for your own flow depth vs. time data.
Use your model to predict depth when clock time is 46 seconds, and the clock time when the water depth first reaches 14 centimeters.
Comment on whether the model fits the data well or not.
2. Follow the complete modeling procedure for the two data sets below, using a quadratic model for each. Note that your results might not be as good as with the flow model. It is even possible that at least one of these data sets cannot be fit by a quadratic model.
Data Set 1
In a study of precalculus students, average grades were compared with the percent of classes in which the students took and reviewed class notes. The results were as follows:
Percent of Assignments Reviewed
Grade Average
0
.9392691
10
1.399793
20
1.803491
30
2.157374
40
2.467591
50
2.739528
60
2.977911
70
3.186877
80
3.370059
90
3.530637
100
3.671401
Determine from your model the percent of classes reviewed to achieve grades of 3.0 and 4.0.
Determine also the projected grade for someone who reviews notes for 80% of the classes.
Comment on how well the model fits the data. The model may fit or it may not.
Comment on whether or not the actual curve would look like the one you obtained, for a real class of real students.
Data Set 2
The following data represent the illumination of a comet by a certain star, reasonably similar to our Sun, at various distances from the star:
Distance from Star (AU)
Illumination of Comet (W/m^2)
1
1230
2
307.5
3
136.6667
4
76.875
5
49.2
6
34.16667
7
25.10204
8
19.21875
9
15.18519
10
12.3
Obtain a model.
Determine from your model what illumination would be expected at 1.6 AU from the star.
At what range of distances from the star would the illumination be comfortable for reading, if reading comfort occurs in the range from 25 to 100 Watts per square meter?
Analyze how well your model fits the data and give your conclusion. The model might fit, and it might not. You determine whether it does or doesn't.