Instructor's Answers to Questions

  1. Pascal's Triangle & Probability
  1. "The position-r number in the number-n row of Pascal's Triangle tells us how many of these outcomes will have r heads."
  1. I am not absolutely certain what this is saying.
  2. What does "r" represent?

    3.  

    For example if you flip a coin 4 times, you could get 2 'heads' in 6 ways. The number 6 is in the number-4 row of Pascal's Triangle (actually the fifth row, since the first is #0) in position 2 (actually the third number since the first is in position #0).
  3. What does "r" represent?
  4.  
  1. Random Walks
  1. Sane & Demented
  2. Please define "stochastic process".

    3. A stochastic process is one in which you have probabilities of transitions between states, like the sane-demented model or the sane-borderline-demented model. The columns of the transition matrix represent the probabilities of the different possible transitions of one category—e.g., the first column represents the probabilities of transitions of the sane grouop—and thus must add to 1.
  3. Trapezoidal Graphs
  4. I know what altitude is. However, I do not understand how to find it for the trapezoid.

 

Altitude for a trapezoidal graph is how far you are above the horizontal axis. Each trapezoid has two sides.

  1. Interpreting Trapezoidal Graphs
  1. I think that I understand what this is saying - but I am not perfectly comfortable with the topic.

 

This question isn't specific enough to answer. Actually it's a comment, not a question.

  1. Velocity Functions & What They Tell Us
  1. Analyzing Sequences
  2. Please define "asymptotic".

 

A curve is asymptotic to a line if the curve gets closer and closer to the line, approaching to any possible degree of closeness, but never reaching the line. For example 1 / x gets smaller and smaller, without bound, but never reaches 0. So the graph of 1/x is asymptotic to the x axis.

  1. Cellular Automata
  1. Is this referring to the Christmas Tree problem?

    2. The Christmas Tree problem is an example of a cellular automaton.
  2. Exponential Population Growth
  3. What does "k" represent?
  4. Also, I am having a hard time reading this. It's a little confusing.

    5.  

    k represents a number that can have any value required by the situation. e^(.01 t) and e^(3 t) are both exponential functions, but the graph of the first is much 'flatter' than the graph of the second.

     

  5. Last Week Questions
  6. I understand how to set up the Trapezoidal Graphs and how to graph then, but I still don't understand how to find the slope and the average area. Could you please explain how to find these things?

    7. Slope is rise / run from point to point. The run is the 'horizontal' displacement from the first point to the second; the rise is the 'vertical' displacement from the first point to the second. The area of a trapezoid is the product of the average altitude and the width of the trapezoid. The trapezoid has two altitudes—from the x axis up to the first point, and from the x axis up to the second point.
  7. I don't understand how to find the rate of change of a function. I don't really understand how to approach the problem. I don't understand where you got the x1 and x2 for your formula. Could you please explain how to work this and use the formula?

    8. First you gotta have a function, say y = f(x) = 3 x^2 + 4 x – 7, and you want to find the average rate of change between two x values. So you have to pick two x values. Say the values are x1 = 2 and x2 = 5. Then you find y1 = f(2) and y2 = f(5). Subtract to find the change in y, subtract to find the change in x, and find (change in y) / (change in x).
  8. On the topic Cellular Automata, I don't get any of it. Could you please work a problem that has to do with this so I can feel comfortable working this problem?

    9. It's the Christmas Tree Lights problem, as worked in class last week.
  9. Random walks
  10. I do not understand what C(n, r) means, I understand what the n and the r means, but the C confuses me.

    11. C(n,r) is the number in row n at position r of Pascal's Triangle, and tells you how many of the possible outcomes of a toss of n coins will have r 'heads'.
  11. Rate of change of a quadratic
  12. `dt->0 what does this mean

    13. `dt stands for delta-t with the Greek Delta. `dt is the time interval between which the values of the quadratic are calculated. If we want to find the rate of change of a function at a specific value of t, we need to find the average rate of change over smaller and smaller intervals in the vicinity of that t value. We find the average rate of change between t and t + `dt, then we see what happens at `dt shrinks to 0.

     

  13. Rate-of-change relationships and rate of change equations

    14. What is the question?

  14. Can we use the function y`=k/y^(3/2) for the cone problem on the test since it represents the flow from a cone.

    15. If you know how to find k from the given information you can use this function.
  15. Need to know more about the understanding of trapezoidal graphs.
  16. Need to know more about the rate of change of a quadratic.
  17. What is an asymptotic?

 

  1. I understand part of the rate of change in a function. I understand how to set up the problem and how to figure out the change, but I would like to see another problem just to fix it into my brain.
  2. I think I understand the Cellular Automata. Is it like the magnet things, or is it something different?

    3. Like the magnets, and even more like the Christmas Tree Lights.
  3. What is the Logistic Population Model? I don't recall seeing it in my notes.

 

See page 5 of handout when you get it

  1. What does the C(n, r) notation for random walks using Pascal's triangle mean? It says there are C(n, r) ways to take r steps to the right. We take r out of n steps to the right so there are n-r steps to the left. Where does that come from? Also where does |2r-n| come from to find the distance?

    2. Break it down to a specific example. If n = 10 and r = 4, what does that mean? How far do we end up from the starting point? What if n = 10 and r = 8? Where do we have to use the absolute value and why?
  2. I'm not exactly sure how to find the change in slope for a trapezoidal graph. For a rate function graph isn't the slope the rate in which the rate changes? So wouldn't the rate of the change in slope for trapezoidal be the rate of which the rate of the rate changes. Ex. meters per second^3. Isn't that rather confusing.

    3. It's confusing but it's an important—actually essential—topic. You are correct in your interpretation. For example if y = x(t) is the position function for an object, then v = y ' is the rate at which position changes, or the velocity, and is approximated by the slope of the trapezoidal graph. Then a = v ' = y '' is the rate at which the velocity v changes, and is called the acceleration. So a ' is the rate at which the accleration changes. You can identify with this: a car's acceleration is what makes it feel like you're pushed back in your seat. That sensation can change, and the rate at which it changes is related to the rate at which acceleration changes. Now, acceleration is the rate at which velocity changes and velocity is itself a rate. So the rate at which acceleration changes is the rate at which the rate of velocity change changes. That's a complex statement, but it can be understood.
  3. What is meant by row number i for transition matrices? And what is a column vector?

    4. A transition matrix has rows. They can be numbered 1, 2, 3, ..., depending on the size of the matrix. If we want to say something about what happens in any single row of the matrix we don't want to confine ourselves to row 2, or row 5, or any specific row. So we need a way to refer to a general row. So we let i stand for the number of the row. (The reason for the letter i is that it stands for 'index'—the number of a row is its index).
  4. What is it mean when it says an exponential curve is asymptotic to the x axis?
  1. Random Walk
  2. What is a random walk?

    3. It's what you did in the hall with the coins. The direction of each step is randomly determined by a coin flip.
  3. I understand it involves walking. But what kind of walking is it?
  4. Diffusion
  5. What is diffusion?

    6. Diffusion is experienced as the natural movement of things from greater concentration to lesser concentration. It's a 'spreading out' process. It generally occurs as a result of the fact that things like species concentrations in an ecosystem, chemicals in a solution, bugs on trees, and lots and lots of other things are randomly walking around, which causes them to diffuse outward from areas of high concentration.
  6. Trapezoid Graph
  7. How is the connecting line of a trapezoidal graph measured?

    8. You could find its length using the Pythagorean Theorem, and that's an important property in itself, but its main property for most applications is its slope, which is rise / run from point to point.
  8. Still don't understand how to get the rate of change of a function. Can't determine water loss from a uniform cylinder or from a cone.

    9. ** If you know how fast water comes out and if you know the cross-sectional area of the stream, you can find the volume of a 1-second section of the stream, which translates immediately into the rate at which volume is being lost from the cylinder or cone. **
  9. Question
  10. I do not understand any of Rate-of-Change?
  11. What is "cells" or lattice point in diffusion?

    12. A cell or a lattice point is just a more formal way of referring to a 'tree'. Actually, a 'tree' is just an easy-to-understand version of a cell or a lattice point.
  12. What is a single tri-diagonal?

    13. The matrix you get for the bug transition model is tri-diagonal: Everything is zero except for the main diagonal of the matrix and the diagonals directly above and below it.
  13. I do not know how to use e on the calculator.

    14. How you do that depends on your calculator. Ask one of us.
  14. I don't understand trapezoidal graphs.
  15. I don't understand Rate-of-change relationships and rate-of-change equations.

    16. This questions isn't specific enough.
  16. Pascal's triangle:
  17. I understand how to get the different rows of the triangle and how to obtain the various rows and how to use them to predict the probability of random walks.
  18. Random Walks:
  19. A random walk is the same as using pascal's triangle and coin flips its just probabilities.

    20. You use those things to analyze a random walk.
  20. Rate of change:
  21. The rate of change is represented by 'd and what is changing divided by the 'dt or change in time.

    22. right

  22. Diffusion:
  23. After a certain number of transitions the bugs will eventually be evenly distributed amongst the trees.

    24. Right
  24.  
  25. ?what I understand and don't understand?
  26. Pascal's Triangle and Probability
  27. Don't understand- how you can use the list for 2 flips to make a list of 3 flips.
  28. Don't know how to count the number of ways to get heads.

    29. Possibilities for 2 flips: HH, HT, TH, TT.

    To get possibilities for 3 flips write down the possibilities for 2 flips twice:

    HH, HT, TH, TT

    HH, HT, TH, TT

    Then add H to the beginning of each possibility in the first row and T to the beginning of each possibility in the second:

    HHH, HHT, HTH, HTT

    THH, THT, TTH, TTT.

    That's all that can happen for three flips.

  29. Diffusion
  30. I don't understand how diffusion works if you have more than one dimension.

 

On a regular 2-dimensional lattice bugs can travel to nearest neighbors above and below, or to the right or left.

On a circular configuration like the one on the last test bugs travel from ring to ring and then get equally distributed over each individual ring.