How much of each of the following statements do you recognize?  Of those which are in the form of questions, answer all you can.

 

a. The number of possible outcomes of a series of coin flips doubles with each additional flip.

 

b.  Pascal's Triangle models the numbers of ways of getting different numbers Heads and Tails given the number of flips

 

c.  Pascal's Triangle allows us to model the distance of a given random walk (ave distance = sum ( probability of distance * distance) ).

 

d.  The logistic map and Rule 30 are alternative ways of generating random numbers, one better than the other.  The logistic map tends to fall into repeating patterns, even after we get to the point where we can't see them, presumably even beyond our ability to identify them by sophisticated methods of analysis.  This could lead to bad random walk models.

 

e.  If we try a random walk with digits of 1/43 we get skewed results--never approach the correct limits.  With Rule 30 we do.  With logistic map at, say, 3.4, we don't.  As we approach 4 we lose the ability to distinguish but we expect that it's there, just not as obvious.

 

f.  The diffusion equation for trapezoidal approximations that the rate of change of the diffusing quantity is proportional to the rate of slope change of the graph of the quantity vs. position.

 

g.  Diffusion is equivalent to each bug random-walking a short distance; same distribution in the limit.

 

h.  The mean of a distribution is the square root of the 'average' of the squared deviations from the mean.

 

i.  The means of the x and y kinetic energies in the billiard model are equal within the limits of statistical significance.

 

j.  If a function y(x) is evaluated at equal intervals `dx and the results used to form a sequence then  

k.  A first pre-difference of a sequence is any sequence whose first difference is that sequence. 

 

l.  If a sequence represents a rate function r(t) evaluated at equal intervals `dt a possible quantity function y(t) is obtained by constructing a pre-difference sequence starting with 0 and multiplying the members of that sequence by `dt.

 

m.  If the graph of a sequence is increasing at an increasing rate what can be said about its first and second differences?  What about the other possibilities (e.g., decreasing at an increasing rate)?

 

n.  If a sequence is increasing what can be said about the sequence of first pre-differences?

 

Do the following problems:

 

 

1.  Every year the amount of money in an account increases by 10%.  The initial amount, at t = 0, is $1000.  How much money do we have at t = 1, t = 2, t = 3 and t = 4, where t is clock time measured in years?  Sketch a trapezoidal approximation graph of the value of the account vs. clock time.  Label slopes and areas.  For each of the four trapezoids determine the ratio of slope to average altitude.  What are your results?

 

2.  Every month the number of fish in a pond increases by 10%.  The initial population, at t = 0, is 50.  How many do we have at t = 1, t = 2, t = 3 and t = 4, where t is clock time measured in months?  Sketch a trapezoidal approximation graph of the number of fish vs. clock time.  Label slopes and areas.  For each of the four trapezoids determine the ratio of slope to average altitude.  What are your results?

 

3.  Compare the results of the last two problems and see if you can generalize this pattern to a Theorem.  If so see if you can prove the Theorem.

 

 

4.  Prove that it is impossible end up at an odd distance from the starting point on a 200-step random walk.  Can you generalize this result and prove it?

 

 

5.  The rate at which the speed of a skateboarder changes is -9 m/s^2 * slope of incline, provided the slope of the incline isn't too great.  Suppose that the altitude of an incline is given in meters by y = .01 (x-10)^2, where x is the distance in meters from the left end of the incline.  Sketch a trapezoidal approximation graph of this function, using interval `dx = 2, and label the slopes.  Imagine a skateboarder traveling down this incline. Determine the average rate at which the skateboarder's velocity changes on each segment of this graph.  On each segment the velocity at the end of the segment will be v = sqrt( v0^2 + 2 a `dx), where a is the rate at which velocity changes, v0 is the velocity at the beginning of that segment and `dx the width of the corresponding trapezoid. (Note again that we are assuming small slopes and that everything done here is an approximation based on that assumption, the accuracy of which depends also on the width `dx of the interval).  Assuming that the skateboarder starts from rest (i.e., initial velocity 0) find the velocity of the skateboarder at the end of each segment.  See if you can also determine approximately how long it takes the skateboarder to travel length of each segment, and the total time required to reach the x = 10 point.

 

6.  If you stop a billiard-ball simulation with 10 very small balls about 10 minutes after starting it from an unknown randomizer, the chance of any given ball being on the left-hand side is 50% and the chance of its being on the right-hand side is also 50%.  Assuming that the simulation is stopped without your looking, the expected distribution of balls between the left- and right-hand sides of the screen will therefore be the same as if you had flipped 10 coins, in the sense that your chance of having, say, 3 balls on the left-hand side and 7 on the right the the same as your chance of obtaining 3 heads and 7 tails.  What is the probability that when the simulation is stopped all the balls will be on the right-hand side?  If a snapshot of the screen was taken every second for an hour, how many of the snapshot would you expect to show all 10 balls on the right-hand side?  What is the probability that anyone of these all-right-hand-side snapshots will show all 10 balls in the upper-right-hand quarter of the screen?  How many snapshots of the screen would you expect to have to take in order to observe this upper-right-hand-quarter configuration?  How would your results change if there were 100 balls on the screen?  There are about 1,000,000,000,000,000,000,000,000,000 air molecules in this room, each moving at about the speed of a jet plane.  Do you think you have to worry about all of them ending up on the other side room and suffocating you?

 

 

7.  If by adding a 10-pound weight on her shoulders, an athlete increases the number of pushups she can do from 17 to 24, and by adding another 10-pound weight the number increases from 24 to 28.  At what rate is the number of pushups increase per pound of added weight for the first 10 lb of added weight?  What is the rate for the next 10 lbs of added weight?  Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?

 

8.  If the evaporation rates for 500 cm^2 areas of salt water at a certain temperature are 250 mg/hr for a 3% solution, 370 mg/hr for a 6% solution and 300 mg/hr for a 9% solution, at what average rate does the evaporation rate change with respect to the percent of the solution between 3% and 6%, and at what average rate does the evaporation rate change with respect to the percent of the solution between 6% and 9%?  Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?

 

9.  If the yield of a wheat field is 30 bushels/acre when the field receives 10 inches of rain during its growing season, 37 bushels/acre when the field receives 12 inches of rain, and 41 bushels/acre when the field receives 14 inches of rain, then at what average rate does the yield change with respect the amount of rain between 10 and 12 inches of rainfall, and between 12 and 14 inches of rainfall.  Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?

 

10.  If an engine under certain specified conditions of use lasts for 2500 hr if its oil is changed once every 500 hours 10,000 miles, 3000 hr if its oil is changed 3 times every 500 hours, and 3100 hr if its oil is changed 5 times every 500 hours, between 1 changes and 3 changes every 500 miles what is the average rate of change of lifetime in hours with respect to number of changes in 500 miles?  Between 3 changes and 5 changes every 500 miles what is the average rate of change of lifetime in hours with respect to number of changes in 500 miles?  Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?

 

11.  If the velocity of a car changes from 30 mph to 40 mph as clock time changes from t=43 seconds to t = 47 seconds, and from 40 mph to 50 mph as clock time changes from t = 47 seconds to t = 55 seconds, then at what average rate does the velocity change with respect to clock time between t = 43 seconds and t = 47 seconds?  At what average rate does the velocity change with respect to clock time between t = 47 seconds and t = 55 seconds?  What would be the average rates as determined by another clock which reads exactly 36 seconds earlier than the first?  Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?

 

12.  If between 12 and 15 hours of weekly training a marathoner's time decreases by 6 minutes, and between 15 and 20 hours of weekly training the time decreases by 3 minutes, then at average rate does the marathoner's time change between 12 and 15 hours of weekly training, and between 15 and 20 hours of weekly training?  Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?

 

7.  If between 12 and 15 hours of weekly training a miler's time decreases at an average rate of 2 seconds per weekly hour of training, and between 15 and 20 hours of weekly training the miler's time decreases at an average rate of 1.2 seconds per weekly hr of training, then if with 12 hours of weekly training the miler's time is 4 minutes and 34 seconds, what will the miler's time at 15 and at 20 weekly hours of training?    Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?

 

13.  If between 10 and 12 in. of evenly spaced rain during the growing season the yield of a wheat field increases at an average rate of 4 bushels/acre per additional inche of rain, and between 12 and 14 in. the average rate of increase is 2 bushels/acre per additional inch of rain, then if the field produces 25 bushels/acre when it receives 10 inches of rain what will be the yield when 14 inches are received?    Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?

 

14.  Suppose the average evaporation rate for a certain container changes by 70 mg/hr per percentage point between solution concentrations of 2% and 5% and by an average of -30 mg/hr per percentage point between solution concentrations 5% and 10%.  If the evaporation rate for a 2% solution is 200 mg/hr then what evaporation rate would be expected for a 5% solution and for a 10% solution?  Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?

 

15.  If the maximum weight lifted by a weight lifter increases at a rate of .2 lb of lifting weight per pound of additional training weight for the first ten pounds of additional lifting weight and by .1 lb of lifting weight per pound of additional training weight for the next 10 pounds of additional training weight then if the lifter is currently capable of a 240 lb lift, how much could she lift by adding 10 lb training weight and how much by adding 20 pounds to her training weight?    Sketch and label of a trapezoidal approximation graph depicting this information.  How does the graph confirm your result?