Initial Derive Exercise

1.  Write down a few sentences describing what you've learned so far about problem solving, mathematical modeling, and the way you personally approach both.

2.  Water flows out of a cylinder and into a tube of radius .5 centimeter.  The water is flowing at 200 centimeters / second. 

• What additional length of the tube will be filled in a second?
• What additional volume of the tube will be filled in a second?
• If the water flows out of a uniform cylindrical container of radius 5 centimeters, by how much will the water level in that cylinder go down during this 1-second interval?

3.  When a certain pendulum is at position x, in centimeters, its velocity changes at the rate - 2 x, in centimeters / second^2.

Suppose that the pendulum is at position x = 3 cm and is moving at velocity + 1.5 cm / s. 

• What is the rate at which its velocity changes?
• Assuming that this rate doesn't change for the next .1 second, by how much will its velocity change during that time?
• How far will the pendulum move during this .1 second interval?
• What therefore will be its approximate position and velocity at the end of the .1 second interval?
• Using analogous steps find its approximate position and velocity at the end of another .1 second interval.
• Repeat the process once more to find the approximate position and velocity at the end of a third .1 second interval.
• Graph the position vs. clock time and velocity vs. clock time.
• What aspects of the motion of a pendulum are represented by your graph?

4.  Suppose that period of a pendulum is 2 seconds.  The period is the time required for the pendulum which is released from rest to swing back to its equilibrium position, continue on to a point on the other side of equilibrium, then swing back to equilibrium, through equilibrium and back to its original point.

The pendulum is positioned so that it swings perpendicular to a ramp, passing just above the ramp just as the pendulum passes through its equilibrium point.

If a ball is released from rest the top of the ramp at the same instant the pendulum is released, and is struck by the pendulum and knocked off the track just as the pendulum reaches its equilibrium point, then

• How long did the ball spend rolling along the ramp? 
• If the collision occurred at a point 10 cm from the top of the ramp, what was the ball's average velocity for this interval?
• Assuming that the ball's velocity changed at a uniform rate with respect to clock time, sketch a graph of its velocity vs. clock time.  Find the slope and the area beneath the graph.

We wish in another trial to position the pendulum so that it swings past the track and strikes the ball on its next return to equilibrium. 

• How far down the track should it be positioned?

Challenging question:  If the ball has diameter 25 cm and the pendulum is moving at 100 cm / s, what is the uncertainty in the distance the ball traveled from release to collision?  For now, you can ignore the effects of the pendulum's diameter.

5.  If you measure a cube and find that its edges are each 2.0 cm, what then is its volume?

If you think the measurement of the sides might be off by as much as 0.1 cm, then the edge measurement would be between 1.9 cm and 2.1 cm.  What volumes would correspond to these measurements?

What is the difference in these edge measurements, as a percent of the original edge measurement?

What is the difference between the smallest and largest volume, as a percent of your original calculated volume?

How do the percents compare?

What is the rate of change of the volume result with respect to the measured edge length?

6.  For the glue stick:

• Report the volume of the glue stick, based on your best measurement.
• Report the minimum and maximum possible volumes, based on your minimum and maximum estimates of the measurements.
• What is the difference between your max and min diameter measurements, as a percent of your best diameter measurement?
• What is the difference between your max and min length measurements, as a percent of your best length measurement?
• What is the difference between your max and min volumes, as a percent of your first volume calculation?

Report your best volume estimate, and the max and min possible volumes, for the tube.  At this point you don't need to do percent calculations for the tube.

7.  Suppose you have a sane-demented situation, with 20% of the sane becoming demented and 30% of the demented becoming sane during any transition.

We have seen that if you start off with 500 sane and 500 demented, and graph the number of sane vs. the number of demented for several transitions, the points on the graph lie on a straight line with slope -1.

• If we start with 800 sane and 200 demented, will the resulting points again lie along a straight line?  If so, what will be its slope?  Note that you probably want to calculate at least one transition for this initial population.
• Answer the same assuming initial population 200 sane and 800 demented.

8.  Sketch a trapezoidal approximation graph for the function y = 12 / x, using x values 1, 3, 6 and 8.  Label slopes and areas.

9.  The possible outcomes for a 3-flip trial are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.  On the 'diffusion model', HHH would correspond to position 3 from the start, HHT, HTH and HTT would correspond to position 1, HTT, THT, TTH would correspond to position -1 and TTT to position -3.  If we write down the position for each possible outcome, we find that the mean position is 0, the average deviation from the mean position is 1.5, and the standard deviation (which is the square root of the mean of the squared deviations) is sqrt(3), approximately 1.732.  We did this in class.  We can also sketch a graph indicating the number of possible outcomes vs. position, and use this graph as a basis for sketching a bell-shaped curve.

List the possible outcomes for a 4-flip trial.  There are 16 such possible outcomes.  Repeat the above analysis for the case of 4-flip trials, and sketch a graph.

10.  I'm walking toward that wall again.

• If I travel half the remaining distance to the wall with every step, will I ever get to the wall?
• If I travel half the remaining distance to the wall with every step, and every step takes as long as the preceding step, is there a limit to how long it will take?
• If I travel half the remaining distance to the wall with every step, and every step takes half as long as the preceding step, will I ever get to the wall?
• If I travel half the remaining distance to the wall with every step, and the time required for every step is the square of half the time required for the preceding step, is there a limit to how long it will take?
• If with every step I travel proportion p of my previous distance, starting with proportion p of the distance to the wall, for what values of p do I eventually run into the wall?
• If with every step I travel proportion P of my distance on the preceding step, and if the time required every step is proportion q of the time required for the preceding step, then is there a limit to how much time I will spend stepping?