Team Homework
A How-To GuideHow To Write Mathematics
Writing Mathematics is a key part of the introductory precalculus and calculus courses at the University of Michigan (Math 105/115/116). The objective of this website is to prepare students to organize and present mathematics in the rigorous manner expected of them in these courses.
What Is Meant By A Conclusion
Conclusion is a buzz word often thrown around, so it is important you understand what this word means in the context of the introductory mathematics courses.
Conclusions are the results of your computations as well as insightful observations about these results.
Often this includes- discussing the validity of your results
- interpreting these results in the context of the problem
- comparing your results with your intuition when faced with a real world problem
Video Tutorial
Below is a video tutorial discussing how to present your conclusions.
All problems are from:
Deborah Hughes-Hallett, Andrew Gleason, et al.: Calculus: Single Variable, Fourth Edition, Wiley, 2004
Test Yourself
Instructions
Read the following team homework problem. Below it are four possible conclusions. Think about which one best fits the criteria described above and then check your answer.The Question
When Galileo was formulating the laws of motion, he considered the motion of a body starting from rest and falling under gravity. He originally thought that the velocity of such a falling body was proportional to the distance it had fallen. What do the experimental data in Table 1.3 tell you about Galileo's hypothesis? What alternative hypothesis is suggested by the two sets of data in Table 1 and Table 2?
| Distance (feet) | Velocity (feet/second) |
| 0 | 0 |
| 1 | 8 |
| 2 | 11.3 |
| 3 | 13.9 |
| 4 | 16 |
| Time (seconds) | Velocity (feet/second) |
| 0 | 0 |
| 1 | 32 |
| 2 | 64 |
| 3 | 96 |
| 4 | 128 |
Possible Conclusions
(a) Galileo's hypothesis that velocity of a falling body is proportional to the distance it has fallen says that V=k*D where V is velocity in ft/sec, D is distance fallen in ft and k is a constant of proportionality. If this were true then in Table 1, V/D should be constant. Here we compute the V/D for different entries in the table:
| V/D=8/1=8 |
| V/D=11.3/2=5.65 |
| V/D=13.9/3=4.63 |
| V/D=16/4=4 |
Clearly, V/D is not a constant so Table 1 suggests that Galileo's hypothesis is incorrect. Let's compute V/T where T is time in seconds for the entries in Table 2.
| V/T=32/1=32 |
| V/T=64/2=32 |
| V/T=96/3=32 |
| V/T=128/4=32 |
We see that for all these entries V/T is constant. These data suggest that the velocity of a falling body is proportional to the time that the body has been falling, in particular V=32*T. Coincidentally gravity in feet per second2 is approximately 32.
(b) Galileo is wrong because V is not a multiple of D by Table 1. Instead V is a multiple of T.
(c) Galileo's hypothesis that velocity of a falling body is proportional to the distance it has fallen says that V=k*D where V is velocity in ft/sec, D is distance fallen in ft and k is a constant of proportionality. If this were true then in Table 1, V/D should be constant. Here we compute the V/D for different entries in the table:
| V/D=8/1=8 |
| V/D=11.3/2=5.65 |
| V/D=13.9/3=4.63 |
| V/D=16/4=4 |
Clearly, V/D is not a constant so Table 1 suggests that Galileo's hypothesis is incorrect. Let's compute V/T where T is time in seconds for the entries in Table 2.
| V/T=32/1=32 |
| V/T=64/2=32 |
| V/T=96/3=32 |
| V/T=128/4=32 |
We see that for all these entries V/T is constant. So the velocity of a falling body is proportional to the time that the body has been falling, in particular V=32*T. Coincidentally gravity in feet per second2 is approximately 32.
(d) Galileo originally thought that the velocity of such a falling body was proportional to the distance it had fallen. This means V=kD. If we look at the data in table 1 we get k=8, 5.65, 4.63, and 4. But when you have proportionality k has to be the same always. So Galileo's hypothesis is incorrect. For Table 2 we get k=32, 32, 32 and 32. Since k is the same always here, instead of velocity being proportional to distance it is proportional to time.