Mathematical modelling and problem solving

Modelling with functions and equations

In this module we consider modelling simple relations using variables, functions and equations. These are the most fundamental mathematical tools.

Before the problems begin:
Remember In your solutions:
  1. (FORMULAS PROBLEM) A number of mathematical equations are listed below.

    A common thing to look at when you see an expression is its mathematical form, e.g. continuous/discrete, linear/nonlinear etc. This is extensively discussed in mathematics courses. However, such observations have to do only with the properties of the expression itself, and is like studying the grammar of a sentence rather than its meaning. In contrast, mathematical modelling is about how to say something about reality by using mathematical expressions, and this is what I want you to think about in this problem.

    For each equation discuss what the justification of the equation is (how do we know it is true or at least reasonable), and how well it can be expected to fit with reality (i.e. is it exact or is it some kind of approximation). If you can, try to categorize the expressions in different groups. Since this is the first problem in the course, a hint is not to spend all your time here since you have more problems in this set. A short comment on each equation and maybe a few general observations is sufficient.
    • a^2+b^2=c^2  (Pythagoras theorem)
    • stock index= 2045+0.0034 t  (trend analysis)
    • population= C * a^t  (C and a are constants, t is the time in years)
    • F= G m1 m2 / r^2  (gravity between two bodies)
    • F=ma (Newtons force equation)
    • 100*weight+length < 320  (max allowed parcel size for a postal service)
    • #presentStudents + #absentStudents = #allStudents  (for a class)
    • I1+I2+I3=0  (total electric current into a circuit knot)
    • insurance rebate in % = #insurances *2 + 0.2* min(7, #years as customer)   (lower rates for good customers)
    • (a+b)^2 = a^2 + 2ab + b^2
    • A >= 0.08 * L * I   (dimensioning requirement for 12V cables if you have a current I [A], cable length L[m] and cable area A[mm^2])
    • air drag force = C* v^2 * A  (air drag for a moving object, C is a constant depending on the shape but not the size, v velocity, A cross section area)
    • weight = C* length^3  (the constant C should be chosen depending on type of object e.g. persons, dogs, cars)
    • p(getting heads when tossing a coin) = 1/2

  2. (INTUITIVE CURVE FITTING PROBLEM) For two related physical variables the following relationship has been measured:

    T (time)
    D (distance)

    {{88.0, 57.9},{224.7,108.2},{365.3,149.6},{687.0,228.07},{4332, 778.434},{10760, 1428.74},{30684,2839.08},{60188,4490.8},{90467,5879.13}}

    Suggest a mathematical equation describing the relation between these two quantities. Feel free to use your creativity, any method to get to a good result is fine. Explain how you did to find the model, and motivate your choice. Is the fit of your equation good? How can deviations from the known table entries be justified? How could the model you found be used for points not in the table, e.g. for other values of T? Hint: Make sure you get started by doing something. Plot the points, plot some functions and explore all possibilities you can think of. What you definitely shouldn't do is to try to look up an answer, or use some software that searches automatically - it is your own exploration that is important.

    You don't have to use Mathematica for this problem but if you want to plot the data points in Mathematica you can use ListPlot. If you want to superimpose a function and the points in the same diagram you can use Show with Plot[your function] and ListPlot[the data] as arguments. Look in the Mathematica help system for extensive documentation and examples.

  3. (PENDULUM PROBLEM) Use dimensional analysis to find a mathematical expression for the period of a pendulum. Begin by setting up variables that could be expected to affect the period, and determine their dimension. Explain the solution process and how you arrive to the conclusion. Make a little experiment to verify your result, and estimate any unknown proportionality constant (don't forget this last step to make the task complete!).

    NOTE: Avoid getting involved in complicated physics here. This is not the task. The task is simply to guess a formula based on dimensional correctness only! If you don't understand what this is, stop and ask. If you already know the answer, part of the purpose of the problem is lost, then ask us and we will find an alternative for you. Please do the same also for other problems in the course if you realize that you already know the answer.

  4. (BICYCLE BRAKING PROBLEM) According to an experiment with a bicycle the following stopping distances were measured:

    speed [km/h]
    stopping distance [m]

    {{5, 0.65}, {10, 1.3}, {15, 2.7}, {20, 5.1}, {22, 5.6}, {25, 7.4}, {30, 10.4}}

    a) This problem is about how to best fit a curve to given data. To get the basic idea of something and improve understanding, it is often good to start with a very simple example. So first try to manually fit a quadratic function to the points (1,3), (2,2), (3,4). Start by writing the equations that you know should hold, and then solve for the unknown parameters. When you see how this is done, add one more point (4,5). What happens?

    Now consider the least squares method. Describe the idea of linear least squares as well as you can (see the paper available as additional course material, note that "linear least squares" is a special case of the least squares method). Why is the least squares method meaningful only in the case when you have more data points than parameters? What happens otherwise? What does the Mathematica function Fit do?

    b) Fit a straight line to the data using the least squares criterion (this is yet another special case called linear regression and is very common). Discuss the quality of the model.

    c) Propose and motivate a better model for the distance as a function of the speed, and fit it to the table entries. If you can, motivate your model theoretically, and not only empirically. How would you judge the quality of the fit? How can deviations from the known table entries be justified? Could this model be reliably used for new points not in the table? Try to draw as many conclusions of the data and your model as you can.

  5. (SPLINES PROBLEM) Now we will consider curve fitting that passes exactly through the data points. We will use the same stopping distance data as in the previous problem.

    a) Polynomial series can be used to approximate any function with arbitrary precision. Use a polynomial of sufficiently high degree to fit the given points in problem 4 exactly. What degree should that polynomial have and why? What does the result look like?

    b) A common technique for curve fitting that passes exactly through given data points is linear and cubic spline interpolation. Make sure you first understand the essential points, without getting lost in unnecessary details (see the lecture slide). If you feel you need to learn more about this look in a book or see for example
    • Brief mathematical descriptions: link1, link2, link3, link4 (look for others if you like!)
    • An app where you can construct cubic splines through a given set of points. Cubic splines are called "natural" splines in this app.
    Briefly explain the mathematical idea of spline interpolation (you are not required to actually compute a spline for the example). Why is spline interpolation often better than high degree polynomials for exact curve fitting?

    c) Compare exact curve fitting (splines) with approximate model fitting using the least squares method (as in problem 4). Try to indicate advantages and disadvantages with the respective methods. Consider both the situation when you have many accurate data points and when you have few data points and/or data of very low quality.

    d) All smooth computer graphics is based on splines. The letters you see on the screen, the curves in a graphics program. If we want to describe an arbitrary 2-D curve, could we use the cubic spline approach as in b) directly? Think for example of drawing an 8. (If you don't get it you may skip this question)

  6. (SURPRISE PROBLEM) Say that we would like to put a number on how surprised we get when we see something happen. For example, when the sun goes up in the morning we are not surprised at all, and if it didn't we would be very surprised. If you think about it you will surely come to the conclusion that there are many factors in your understanding of the world that influences your degree of surprise. However, an interesting idea is that all these factors can be summarized in a single number, namely your estimate of the probability that a particular event will happen, p(event). We will not consider how to estimate this number. However, if this can be done, we can ask ourselves which function surprise(p) we should then use to calculate our level of surprise?

    a) As a first step, use your intuition to suggest the extreme points of this function.

    b) Now consider the following intuitive argument. If we see two unrelated events A and B happen, it seems reasonable that the sum of the surprises of these two events independently should be equal to the surprise of the single combined event A AND B. If you also take this into account, what function do you propose?


Do not submit an incomplete module! We are available to help you, and you can receive a short extension if you contact us.