2d weekend assignment

1.  Position vs. clock time data give us a function y = .01 t^2 - 40 t + 100, where t is time in seconds and y is position in cm.. 

 ·         Make a table for y vs. t, using t = 0, 100, 200, 300, 400.

·         For each time interval calculate the average rate at which position changes.

·         Plot average rate of position change vs. midpoint clock time and sketch a straight line through your graph points.

·         Your graph will show v vs. t.  Find the slope and v-intercept of your straight line.

·         What is the equation of your linear function?  Give the function in slope-intercept form, using v and t as your variables.

·         According to Bubba what should be the equation of your straight line? 

2.  Cooter figured out that when his car rolled off its blocks and coasted down the hillside in his unmowed front yard, its velocity function was v = -.27 t + 12.8.  According to Bubba what will be the position vs. clock time function corresponding to the velocity function?  If you think need the information to answer the question, you should know that Cooter's car was at position y = 12 with respect to the coon dog under his front porch.  

3.  The temperature of a bolled possum just brought out of the pressure cooker in Cooter's very warm kitchen is given by T = 95 + 130 * 2^(-.7 t), where t is clock time in hours and T is temperature in Fahrenheit.   

·         Evaluate the temperature at t = 0, .5, 1, 1.5 and 2 and construct a graph of T vs. t.

·         Find the average rate at which temperature changes during each interval.

·         Plot average rate of temperature change vs. midpoint clock time and sketch a straight line through your graph points.  How well do the graph points match the straight line?

·         If we call the rate of temperature change R, then you just plotted a graph of R vs. t.  Label the graph as such.

·         Determine the slope-intercept form of your linear equation.

·         How well does your linear equation match the actual rate of temperature change?

·         This function isn't quadratic.  Do you think the rate function is really supposed to be linear? 

4.  If y = a t^2 + b t + c, then suppose that t = t1 and t = t1 + `dt are two clock times. 

 ·         What is the difference between the first clock time and the second?  What therefore is the time interval between these two clock times?

·         What are the values of y at t = t1 and at t = t1 + `dt?

·         By how much does y change between these two clock times?

·         At what average rate does y change between these two clock times?

·         What is the clock time midway between the two given clock times?

·         What is the velocity function corresponding to the given position function?

·         What is the value of the velocity function at the midway clock time?

·         How are your various answers related to one another?

5.  The velocity of an automobile is given in meters / sec by the function v(t) = 3 t + 1, where t is clock time in seconds. 

·         Find the velocity at t = 4 and at t = 7, and use these velocities and the 3-second time interval to find the average rate of velocity change between t = 4 and t = 7, as well as the change in the position of the automobile.

·         On a graph of velocity vs. clock time, locate the t = 4 and t = 7 points.  From the t = 4 point sketch a projection line (a line straight from the point to the axis; this line will make a right angle with the axis)  to the t = 4 point on the t axis and label this line with its length, writing the length just to the right of the line and about halfway up.  Do the same for the t = 7 point of the graph.  Then connect the t = 4 and t = 7 points.

·         Find the slope of the line connecting the t = 4 and t = 7 points, and label that line segment with the slope, placing the slope in a rectangular box just above the middle of the line segment.

·         The figure you have constructed is a trapezoid, bounded below by the t axis, having as its vertical sides the projection lines you constructed, and the sloping line segment for its top.  Its area is equal to the product of its average altitude and its width.  What is its area?  Label the area by writing it in the center of the trapezoid and placing a circle around it.

·         What is the meaning, in terms of velocities, rate of change of velocity, change in position, of the slope and of the area of this graph?

·         What is the position function for the given velocity function (hint:  the velocity function is the rate-of-change function for the position function)?  By how much does this function change between t = 4 and t = 7?

6.  The velocity of an automobile is given in meters/sec by the function v(t) = .4 t^2 + 3 t + 1.  Sketch a v vs. t graph and locate the t = 4, t = 7 and t = 10 points of the graph.

·         For the t = 4 and t = 7 points, construct a trapezoid like the one you constructed in the preceding problem.  Find slope and area, and label everything as you did in that problem.

·         Do the same for the t = 7 and t = 10 points.

·         At what average rate did velocity change between the t = 4 and t = 7 points? At what average rate did velocity change between the t = 7 and t = 10 points?  At what specific clock time do you estimate that the rate is actually equal to the average rate?

·         By approximately how much did the position change during each of the two time intervals?  Do you think your estimates are overestimates or underestimates of the actual changes in position?

·         You could more closely estimate changes in position using t intervals from 4 to 4.1, then to 4.2, then to 4.3, etc..  It wouldn't be practical to use an increment of .1 to find the changes from t = 4 to t = 7 by hand, but it wouldn't be any problem to use Excel to do this.  How could set up Excel to find these results, and the total change from t = 4 to t = 7?

·         The velocity function is the rate-of-change function for the position function.  Can you speculate on what the position function might be for the given velocity function? 

·         If you found a position function, how close are its changes to the changes predicted by the trapezoidal graph?

7.  The normal curve is given by the function N(z) = 2 / sqrt(2 `pi)  * e^-(z^2 / 2).  For a graph of N(z) vs. z, find and plot the graph points corresponding to z = -2, -1, 0, 1, 2.  Sketch the trapezoids corresponding to these points, and label heights, slopes and areas according to the conventions of the preceding problems.

 ·         What proportion of the total area lies between z = 0 and z = 1?

·         What proportion of the total area lies between z = 1 and z = 2?

·         How do these proportions compare with the proportions originally given for the normal curve?

·         Refine your graph between z = 0 and z = 1 by using the graph points corresponding to z = 0, .5 and 1.  Construct the three trapezoids defined by these points, and find the total area lying between z = 0 and z = 1.

8.  There are 10 Christmas trees is a circle.  Every two seconds, each tree looks at each of its nearest neighbors to see whether their lights are on or off.  If exactly one of its neighbors has its lights on, it turns its light switch to the 'on' position.  Otherwise it turns its light switch to the 'off' position.

 ·         If all the trees have their lights on at the beginning, what will happen during the next 20 seconds?

·         If all the trees except the one in the middle have their lights on at the beginning, what will happen during the next 20 seconds?

·         If only the middle three trees have their lights on at the beginning, what will happen during the next 20 seconds?

·         Answer the same questions if the rule is that a tree will place its switch in the 'on' position if at least one of the neighbors has its lights on.

·         Answer the same questions if the rule is that a tree will place its switch in the 'on' position if at most one of the neighbors has its lights on.

·         Answer the same questions if the rule is that a tree will place its switch in the 'on' position if at least one of the neighbors is in a different state than the tree itself (i.e., at least one neighbor is off when the tree is on, or on when the tree is off).

9.  A string with a beads located at x = 0, 3 cm, 6 cm, ..., 18 cm is originally stretched out along the x axis.  The end beads are 'nailed down' so they can't move.  The other beads are free to move a little ways up and down; then they do so they stretch the string out a bit and the string stretches a bit tends to pull them back toward the x axis—unless of course a neighboring bead is further out, in which case the string tends to pull closer beads away from the x axis.

The beads are pulled to positions +2 mm (for the bead at 3 cm), -1 mm, -4 mm, +1 mm, +3 mm (that's the bead at 15 cm).  Sketch a graph of this configuration, and determine the slopes of the six segments between the beads. 

Each bead is inially stationary, and each bead experiences an rate of velocity change which is equal to the slope to its right, minus the slope to its left, with the acceleration in cm / s / s. 

Where will each bead be, and what will be the velocity of each, 2 seconds later?

Starting from the velocities and positions you just found, determine the new slopes, and use these new slopes to get the new rates of velocity change.  Calculate the velocities and positions after another 2 seconds.

10.  Repeat the problem with the Christmas trees using 9 trees in a circle instead of 10.  How does the behavior of this system differ from that of the 10-tree system for different rules, and why?

11.        Make up the transition rule that gives the Christmas tree systems the most interesting behavior.

12.  Sketch and label the trapezoidal graph for y = x^2 + 1 for x varying from 0 to 1 by increment .5—that is, evaluate y for x = 0, .5 and 1.  What is your total area?  Do you think this area is greater or less than the area under the actual y = x^2 + 1 curve?

Repeat for the same function on the same interval, but use increment .25

·         Do you think your area is closer or less close to the actual area under the y = x^2 + 1 curve than before?

·         Write your slopes as a sequence and predict the next slope.

·         If your next slope is correct, how could you use it to predict the next value of y?

 ·         Write out a strategy for using Excel to find the total area for this problem using an increment of .01.  If you have Excel, see if you can find this area.

13.  The rate at which water level y changes in a uniform cylinder is -.3 cm / sec * `sqrt( depth in cm), abbreviated by

dy / dt = -.3 `sqrt ( y ).

If the water level is originally y = 100 cm, then

 ·         At what rate will the water level be changing?

·         If it continues changing at this rate for 5 seconds what will be the new level?

 Starting from the new level:

 ·         At what rate will the water level be changing?

·         If it continues changing at this rate for 5 seconds what will be the new level?

 Starting from the new level:

 ·         At what rate will the water level be changing?

·         If it continues changing at this rate for 5 seconds what will be the new level?

 Continue this process for 3 more steps.

 Make a table and a graph of water level vs. clock time.  What function models your results?

14.  If the rate at which the temperature of your thermometer changes is given by rate = -.8 deg / sec * (difference between thermometer temperature and room temperature), abbreviated

 dT / dt = -.048 ( T – Tr)

  then if the initial temperature is 100 degrees and the room temperature is 50 degrees:

 Starting from the initial temperature: 

·         At what rate will the temperature be changing?

·         If it continues changing at this rate for 5 seconds what will be the new temperature?

 Starting from the new temperature:

 ·         At what rate will the temperature be changing?

·         If it continues changing at this rate for 5 seconds what will be the new temperature?

Repeat this process for three more steps.

 Make a table and a graph of temperature excess (this is the excess temperature above room temperature—i.e., subtract room temperature from actual temperature) vs. clock time and see what function models your results.

15.  When a pendulum is at displacement x from its equilibrium position, the rate at which its velocity changes with respect to clock time is given by

 rate of velocity change = -20 cm/s/s * displacement from equilibrium

 which is abbreviated

 dv / dt = -20 * x.

 If the pendulum is originally 10 cm from its equilibrium position and at rest then:

 ·         At what rate is the velocity changing with respect to clock time?

·         If the velocity continues changing at this rate then what will be the velocity after .1 second?

·         What then will be the average velocity for this .1 second time interval?

·         How much will the position of the pendulum therefore change during the .1 second interval?

·         What will be the new displacement of the pendulum?

 Starting from the new position:

 ·         At what rate is the velocity changing with respect to clock time?

·         If the velocity continues changing at this rate then what will be the velocity after .1 second?

·         What then will be the average velocity for this .1 second time interval?

·         How much will the position of the pendulum therefore change during the .1 second interval?

·         What will be the new displacement of the pendulum?

 Repeat this process for three more steps.

 Make a table and a graph of pendululm position vs. clock time.

16.  A uniform cylinder has radius 4 cm and is initially filled to a depth of 90 cm above a hole .4 cm in diameter.  Water flows through the hole at a speed given by v = `sqrt( 2 * 980 * y), where y is the depth in cm and v is the velocity in cm / sec.

Initially:

·         At what speed will the water be flowing from the hole?

·         How many cubic cm of water will flow from the hole per second?

·         By how much must the depth change in a second to accomodate this flow?

·         If depth keeps changing at this rate, what will be the depth after 5 seconds?

Starting from this depth:

·         At what speed will the water be flowing from the hole?

·         How many cubic cm of water will flow from the hole per second?

·         By how much must the depth change in a second to accomodate this flow?

·         If depth keeps changing at this rate, what will be the depth after 5 seconds?

Repeat this calculation for three more 5-second time intervals.  Make a table and sketch a graph of depth vs. clock time.

17.  If water depths of 90 cm, 50 cm and 20 cm are observed at clock times t = 10 sec, 20 sec and 30 sec, then what quadratic model y = a t^2 + b t + c satisfies these conditions?

18.  For the water depth vs. clock time data of the preceding exercise, at what average rate is water depth changing with respect to clock time for each of the two 10-second intervals defined by the data?

19.  For your model of depth vs. clock time what is the average rate of depth change with respect to clock time between t = 14 and t = 16, between t = 24 and t = 26 and between t = 34 and t = 36?

20.  Using a coin flip perform 10 different random walks, each consisting of 10 coin flips, and record the number of steps you moved along the x axis.