Page 1.

1. In terms of pay, pay rate and hours worked, explain the relationship between `dQ, Q'ave and `dt, and give one corresponding equation relating the three quantities. Explain how these quantities could be related using a trapezoid (there are two possible ways to construct the trapezoid—for a little extra credit explain both).

2. What is the first difference S' of the sequence S corresponding to the 6^th row of Pascal's Triangle?

3. If the depths of the water in a uniform cylinder at clock time t = 0, 10, 20, 30 and 40 sec are 100 cm, 81 cm, 64 cm, 49 cm and 36 cm, then at what average rate does the depth change during each of the corresponding 10-second intervals?

4. For the sequence a(1), a(2), a(3), ..., what is the 10th term? What is the 50th term? What is the nth term?

5. The period of a pendulum is close to T = .2 `sqrt(L), where T is in seconds and L in cm. A ball rolls down an incline, starting from rest as a pendulum is released. The pendulum, which has length 19 cm, strikes a wall at the instant it reaches its equlibrium position and rebounds to almost its original position, then swings back to its equilibrium position and rebounds again, repeating this process many times.

• At the instant the pendulum first strikes the wall the ball on the incline is 3 cm from its starting point. What is the average rate at which the position of the ball changed as it traveled this interval?
• At the instant of the second pendulum strike against the wall the ball is 27 cm from its starting point. What has been its average rate of position change since the previous strike?
• At the instant of the third pendulum strike against the wall the ball is 75 cm from its starting point. What has been its average rate of position change since the previous strike?
• At the instant of the second pendulum strike against the wall the ball is 147 cm from its starting point. What has been its average rate of position change since the previous strike?
• At what rate or rates does its average rate of position change seem to be changing?

Mathematical Modeling Test 1, Summer 2000

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6. If there is a 15% chance for each of the sane people in an unchanging population to become demented in a given year, and a 20% chance that for each demented person to become sane during the same period, then if there are initially 5000 sane and 5000 demented people in the population, how many of each will there be at the end of each of the first three years?

• What transition matrix can be used to model this situation?
• What will be the transition matrices for 2 years, 4 years, 8 years and 16 years?
• What do you think will be the stable populations, where the numbers of sane and demented will be constant?

7. In a bug diffusion model with three trees, there are always 100 bugs on the first tree, which is on the left. In each transition 30% of the bugs on each tree except the last move to the tree immediately to the right and 5% of the bugs on each tree except the first move to the tree immediately to the left, while 50% of the bugs on the last tree move to the right and are 'lost'. Initially there are no bugs on either the second or the third tree.

• How many bugs will there be on each of these trees after each of the next four transitions?
• How many bugs will be lost by the tree on the right during the second and third transitions?
• Describe the conditions for the biased random walk that is being modeled here, and explain how this random walk might correspond to bug diffusion on a windy day.
• In what ways does this model correspond to a model of the diffusion of nutrients across a cell wall?

8. Evaluate the function a(n) = 7n^3 – 40 n + 3 for n = 1, 2, 3, 4 and 5.

• Find the sequences S', S'', etc., until you obtain a constant sequence.
• Use this sequence to build up the next two terms of S (in the process you will also build up the next two terms of S', S'' etc.). Then compare with what you get when you evaluate a(n) for n = 4 and 5.
• What is the transition matrix for finding S' from S?

9. The function f(t) = 15 * .9^t + 21 gives the temperature in degrees of water in a container at clock time t minutes.

• Find the temperature at clock times t = 0, 3, 6, 9 and 12 minutes, and find the average rate at which temperature changes over each time interval.
• Write your temperatures as a sequence S and write the sequence S'. Then divide each member of S' by 3 to get the sequence S' / 3. What is the meaning of this sequence?
• If A stands for the transition matrix from the preceding problem and S for the sequence, then what does A * S give you? What would you do to A * S to get the average rates of temperature change?

10. A 3 x 3 neural starts at 31 and has goal 13, with a 'mine' at 33. The first three processes observed are

31 -> 32 -> 33

31 -> 32 -> 22 -> 23 -> 33

31 -> 21 -> 22 ->12 -> 13

• What will be the synapse strengths at this time?
• What is the most likely of the possible paths for the next process?
• Why do we say that this net defines a 2-dimensional biased random walk?

Mathematical Modeling Test 1, Summer 2000

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11. A random walk in two dimensions can move one unit right, left, up or down at each step, all with equal probability.

• If the random walker moves right, then right, how far does it end up from the starting point?
• If the random walker moves right, then up, how far does it end up from the starting point?
• If the random walker moves right, then left, how far does it end up from the starting point?
• List all the possible sequences of two moves, and the distance of each from the starting point.
• What is the average distance from the starting point?
• Would the average distance be affected if the probability of a move to the right was .3 which the probability of a move to the left was .2, while up or down remained at .25? Explain your answer.

12. Give the definition of average deviation and of standard deviation.

• Find the mean and use the definition of standard deviation to standard deviation of the number of heads expected on a large number of repeated trials of five flips of a fair coin.
• Compare to the large-n standard deviation .5 `sqrt(n).
• Explain why the average deviation and standard deviation might, give a first glance at their definitions, be expected to be equal, then explain why they are not.

13. Sketch a graph of the normal curve which approximates the frequency distribution of the number of heads expected on 100 coin flips.

• If 68% of the time the number of heads is expected to be within 1 standard deviation of the mean, while 96% of the time it is within two standard deviations and 99.8% of the time within three standard deviations, then what do you think is the likelihood of getting more than 65 heads on a given trial?
• If you have 100 randomly moving molecules in a room, what is the likelihood that at a given instant each side of the room has at least 35 molecules?

14. The standardized normal distribution has mean 0 and standard deviation 1. The equation for the curve is normalized, and is given by N(x) = 1 / `sqrt( 2 `pi) * e^(-x^2 / 2). Sketch and label a trapezoidal approximation graph for N(x) from x = 0 to x = 1, using two intervals, and find the total area of the trapezoids. Explain what your result has to do with the claim that 68% of the area of a normal curve lies within one standard deviation of the mean.

Extra credit: Sketch and label a trapezoidal approximation graph for N(x) from x = 4 to x = 5, using two intervals, and use your result to explain why it is very plausible that 98.8% of the distribution lies within 3 standard deviations of the mean.

15. Sketch and label according to the conventions stated in class a trapezoidal approximation graph for f(t) = 15 * 1.7^t + 21 for t = 0 to t = 12 with increment 4 between t values. Interpret everything that can be interpreted for each of the following situations:

• f(t) is the depth of water, in centimeters, in a container
• f(t) is the rate, in degrees per millisecond, at which the temperature of a chemical reaction is changing.

1. For n = 4 and then for n = 7 compare the standard deviation of the frequency distribution for row n + 1 of Pascal triangle to the expression 1/2 `sqrt(n).

2. The frequency distribution for row n+1 of Pascal's triangle has standard deviation very close to 1/2 `sqrt(n). Well over 99% of the occurrences for such a distribution will lie within 3 standard deviations of the mean.

• Sketch graphs for curves which approximate the n = 20, n = 50 and n = 100 frequency distributions, showing for each the interval within three standard deviations of the mean.
• For each n, what fraction of the domain from 0 to n will lie within 3 standard deviations of the mean?
• Make a similar sketch for n = 1 trillion.

3. For a large-n row of Pascal triangle, if we normalize the frequency distribution the total area under the approximating curve will be 1. The function we get in the limit as n approaches infinity is the Normal Distribution function f(x) = 1 / `sqrt(2 `pi ) * e^(-(x - mean)^2 / (2 * std dev) ). If we change our scale so that the mean is 0 and the standard deviation is 1, we obtain the standard normal distribution function N(x) = 1 / `sqrt(2 `pi) * e^(-x^2 / 2).

Evaluate this function for x = -4, -3, -2, -1 and 0 and sketch the corresponding Trapezoidal Approximation Graph.

• Determine the area in each trapezoid.
• Use your results to determine the approximate probability of an occurrences which lie between the mean and 1 standard deviation from the mean, between 1 and 2 standard deviations from the mean, between 2 and 3 standard deviations from the mean, and between 3 and 4 standard deviations from the mean.

4. Sketch the approximate distribution corresponding to the n = 30 row of Pascal's Triangle. Using the best technique you can, determine the fraction of the time you can expect at least 90 percent of the molecules in a billiard ball simulation to all be located on the left side of the rectangle. Repeat for n = 100.

5. Suppose you have a long row of trees, all bug-free except for one tree, somewhere near the middle of the row, on which reside 1000 bugs. If the bugs all do a 5-step random walk, what do you then expect the distribution to look like? Make a sketch. If the random walk is biased by a slight breeze, so that there is a 60 percent probability on any step of moving to the right and only a 40 percent probability of moving to the left, what will the expected distribution look like? Make a sketch.

6. If a row of trees contains bugs which randomly walk from tree to tree, then what transition matrix could be used to model the diffusion of the bugs?

7. What matrix would we use with a sequence S of 6 numbers in order to obtain the first-difference sequence S'? What matrix would we use with the original sequence S to obtain the second-difference sequence S''?

8. What is the first difference of the sequence corresponding to the 6^th row of Pascal's Triangle?

9.  Find the 16-transition matrix for a sane-demented model with transition probabilities 10% (sane to demented) and 20% (demented to sane). What does this matrix tell you about the results of a large number of transitions?

10.  The sane-demented model gives an unchanging population distribution if the number of sane becoming demented is equal to the number of demented becoming sane. For 1000 people, with transition rates of 12 % and 5%, what are the numbers of sane and of demented that result in an unchanging distribution?

11.  A cell membrane is 3 microns thick. We divide it into 3 individual microns of thickness. Due to contact with a nutrient solution the leftmost micron, which is that the outer surface of the membrane, maintains a saturation population of 100 nutrient molecules. The molecules undergo a biased random walk which results in the migration during 1 transition time of 20% of the molecules in each micron to the micron to its right, and a 5% migration to the left. The right most micron will transport 30% of its molecules into the cell. Initially there are no nutrients in any but the leftmost micron. What will be the cell populations after the first, second, third and fourth transitions, and how many nutrient molecules will be transported into the cell during this time? What will be the steady-state population (i.e., the population after many transitions) of each micron and what will be the steady-state rate at which nutrient molecules will be transported?

12.  In a sane/demented model, the probability of a sane person becoming demented during a given year is 18% while the probability of a demented person becoming sane is 12%. If the initial population consists of 1000 sane individuals, then what will be the number of sane individuals after each of the first four transitions? Write these populations as a sequence and determine a rule for the sequence. Determine also the steady-state configuration of the population, and the 16-transition matrix.

13.  Find the first five numbers (n=1, 2, 3, 4, 5) of the sequence defined by the function a(n) = 3 n^2 - 7n + 1. Find the sequences S, S', S'', etc., until one of the sequences becomes a nonzero constant. Repeat for the sequence defined by a(n) = 3 n^2 - 2n + 5, then again for the sequence defined by a(n) = 5 n^2 - 7n + 1. What do you conclude about the effects of the each of the coefficients a, b and c of the function a n^2 + b n + c on the constant obtained by this process? What further conclusions can you draw from the functions a(n) = 3 n^4 - 7n + 1 and a(n) = 3 n^5 - 7n + 1?

14.  What is the sequence S' corresponding to the sequence S: a(1), a(2), a(3), a(4), ... . What is the nth term of the sequence S'? What matrix A would operate on the column matrix [ a(1), a(2), a(3), a(4), ... ]{T} in order to obtain the sequence S' (the {T} tells you that the matrix, which is written as a row matrix, should be transposed into a column matrix)? What is the sequence S''? What matrix B would operate on the column matrix [ a(1), a(2), a(3), a(4), ... ]{T} in order to obtain the sequence S''? Show that the matrix B is identical to the square of the matrix A.

15.  Suppose that you earn 10% interest on a $1000 investment, and that interest compounds every year. How much money will you therefore have after each of the first 5 years? Write your amounts as a sequence, and find the rule, in terms of ratios and/or differences, that governs this sequence. Suppose now that you have, in addition, $10,000 buried in an old chest in the back pasture. Write the total amount of money in the chest and in your investment as a sequence, and explain how you would obtain the rule for this sequence in terms of ratios and and/or differences.

16.  Of which do you think would better confirm the hypothesis that the x energies are inherently greater than the y energies, an x mean of 500 and a standard deviation of 50 with a y mean of 430 and a standard deviation of 30 based on 5 trials, or an x mean of 500 and a standard deviation of 70 with a y mean of 440 and standard deviation 50 based on 500 trials. Give a clearly reasoned argument for your conclusion.

17.  Give the expected distance traveled in a biased 1-dimensional random walk after 2, 4 and 6 steps, if the probability of a step to the right is 2/3 while the probability of a step to the left is 1/3. Does the rate at which expected distance changes with respect to the number of steps of increase, decrease, or remain constant?

1. Find the next three numbers in each of the following sequences, and state a rule for the sequence:

• **241248...
• **100908172.9...
• 4712203249...
• 1000110011901271...
• **1251229...

2. A 3 x 3 neural net has an obstacle at neuron 23 with initial synapse strengths for any neuron all being equal. Probability is incremented up or down by .05 depending on success or failure. The following sequences of moves are observed:

• 31 -> 32 -> 33 -> 23
• 31 -> 21 -> 22 -> 12 -> 13

What will be the synapse strengths after this sequence of moves?

What problem tends to arise for a neural net using increment.2 instead of .05?

You may use the diagrams below to save you some time

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

31 32 33 31 32 33 31 32 33

3. If the depth of the water in a uniform cylinder is changing at -7 cm / sec at clock time t = 30 sec and at a rate of -4 cm / sec at clock time t = 35 sec, then what is the approximate depth change between these two clock times?

4. If the rate at which the depth of water in a uniform cylinder changes is given by the function r(t) = -4 + t / 25, for 0 < t < 100, with r(t) in cm / sec when t is in sec, then what are the rates at clock times t = 0, 20, 40, 60, 80 and 100 sec?

• What then would be the average rate over each of the corresponding 20-second intervals?
• What would be the resulting depth change over each of these 20-second intervals?
• What would be the total depth change?

2. A 3 x 3 neural net has an obstacle at neuron 23 with initial synapse strengths for any neuron all being equal. Probability is incremented up or down by .05 depending on success or failure. The net executes the following sequence of moves:

• 31 -> 32 -> 22 -> 23
• 31 -> 21 -> 11 -> 12 -> 22 -> 21 -> 22 -> 23
• 31 -> 32 -> 33 -> 23
• 31 -> 32 -> 22 -> 12 -> 11 -> 21 -> 22 -> 23
• 31 -> 21 -> 22 -> 12 -> 13

What will be the synapse strengths of the net after this sequence of moves?

What problem would arise if the increment was .2 instead of .05?

You may use the diagrams below to save you some time

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

31 32 33 31 32 33 31 32 33

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

31 32 33 31 32 33 31 32 33

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

31 32 33 31 32 33 31 32 33

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

31 32 33 31 32 33 31 32 33

3. Answer the following questions about flow from a uniform cylinder:

• If the depth of the water in a uniform cylinder is changing at an average rate of -4 cm / sec, then what will be the depth change in a 12 second time interval?

• If the depth of the water in a uniform cylinder is changing at an average rate of -2.5 cm/sec between clock times t = 5 sec and t = 14 sec, then how much does the depth change between these clock times?

• If the depth of the water in a uniform cylinder is changing at -2 cm / sec at clock time t = 30 sec and at a rate of -1.4 cm / sec at clock time t = 40 sec, then what should be the depth change between these two clock times?

4.  If the depths of the water in a uniform cylinder at clock time t = 0, 10, 20, 30 and 40 sec are 100 cm, 81 cm, 64 cm, 49 cm and 36 cm, then at what average rate does the depth change during each of the corresponding 10-second intervals?

5. If the depth of water in a uniform cylinder is given by the function y(t) = .01 (100 - t)^2, then

• What are the depths at clock times t = 0, 20, 40, 60, 80 and 100 sec?
• At what average rate is the depth of water changing during each of these 20-second intervals?

• Use Excel to obtain the same results in an efficient and systematic manner.

6.  If the rate at which the depth of water in a uniform cylinder changes is given by the function r(t) = -4 + t / 25, for 0 < t < 100, with r(t) in cm / sec when t is in sec, then what are the rates at clock times t = 0, 20, 40, 60, 80 and 100 sec?

• What then would be the average rate over each of the corresponding 20-second intervals?
• What would be the resulting depth change over each of these 20-second intervals?
• What would be the total depth change?

• Use Excel to obtain the same results in an efficient and systematic manner.

7.  Suppose that the temperature of a warm object in a constant-temperature room is given by T(t) = 24 + 55 * 2 ^ (-.03 t), where T(t) is in Celsius when t is in minutes.

3 4 6 9 13 ...

1 2 4 8 16 ...

50 40 32 26 22 ...

70 38 22 14 10 ...

1 3 4 7 11 18 ...

13 20 39 76 137 ...

As best you can state the rule that allows you to predict the next number for each sequence.

2 ** delay if lab is not ready **. Write each of the sequences of the preceding problem as a data table using Excel, and do a curve fit to each. Every sequence but one can be exactly modeled by an appropriate function.

3. Suppose that water depth vs. clock time is given by the function y(t) = .010 (t - 100)^2. Expand the square of the binomial to obtain the y = a t^2 + b t + c form of this function. Determine the water depths at t = 0, 10, 20, ..., 100. Write these depths as a sequence and find the rule for this sequence.

4. Suppose that the temperature of a certain object in a room gets 1/3 of the way closer to the room temperature every 10 minutes. If the object starts at a temperature of 6.71 cm on a temperature scale, while room temperature on the same scale is 3.16 cm, then what will the temperatures be after each of the first four 10-minute time intervals?

Write your temperatures as a sequence of numbers and given a rule for calculating the next member of the sequence from the present number.

Sketch a graph of temperature vs. clock time and without doing the calculations for the next three 10-minute intervals, sketch the estimated curve for the graph.

Use Excel to calculate the temperature in excess of the room temperature at each clock time, and do a curve fit for this temperature excess vs. clock time. One of the functions will fit exactly.

5. On a certain island live an immortal race of 1000 non-reproducing people. Using a standard psychiatric evaluation developed and refined over a great number of years, every individual can be clearly classified as either sane or demented. Due to various physical, social and psychological factors, in a period of one year every sane individual has a 10% chance of becoming demented, and every demented individual has a 20% chance of becoming sane.

If on New Year's Day in the year 2000, there are 900 sane and 100 demented individuals, then how many sane and how many demented individuals will there be on New Year's Day in the year 2001? How many sane and how many demented individuals will there be on New Year's Day in each of the subsequent three years?

Sketch a graph of the number of sane individuals vs. the number of years since the year 2000. Does your graph seem to share more characteristics with a graph of pendulum frequency vs. length, a graph of water depth vs. length (flow from a uniform cylinder out of a uniform hole), or a graph of temperature vs. clock time (a warm object in a cooler room)?

6. There are five trees planted in a row. On the second three there suddenly appears an infestation of 1000 bugs. The spacing of the trees and the behavior of the bugs is such that in any given hour, 20% of the bugs on any tree will migrate to each adjacent tree, while 10% of the bugs on an end tree will wander away and starve to death. Find the number of bugs on each of the five trees after each of the next six hours.

Number the trees, in order, from 1 to 5. For each hour sketch a graph of the number of bugs vs. the number of the tree. Describe how the shape of the graph changes from hour to hour.

See if you can figure out how to use Excel to solve this problem. See if you can use Excel to plot the number of bugs on the second tree vs. clock time and see if you can find a way of fitting the curve.

What does the cardboard model have to do with this problem (if you haven't seen the cardboard model ask about it)?

7. Make up your own 3 x 3 neural net simulation and instead of increasing or decreasing synapse probabilities by .05, use .20. Investigate how this large value could cause the simulation to fail to 'find' a good solution, no matter how long it was run. Note: Add one rule to the rules for your probabilities: if the probability comes out less than 0, make it 0.