Math 445 lab 3: scripts, log-linear relations

Write a single Matlab script file that solves the following problems, and use the Matlab ''publish'' function to create a nice-looking PDF of your code and graphs to turn in. Be sure to label each problem with comments like %% Problem 1%% so that the problems are separated and labeled in the PDF.

Problem 1: Redo problem 15 from lab 2 using a Matlab script. The problem is make a plot of the polynomial $f(x) = x^3 -5x^2 + 2x + 3$ over the range $-1 \leq x \leq 5$. Give the plot square axes and the title “math 445 lab 3 problem 1”.

Problem 2: Make a plot of $y(x) = 147 \times 10^{-2x}$ over the range $0 \leq x \leq 4$ using a dashed magenta line. Chose an appropriate plotting function that best displays the functional relationship between $x$ and $y$. Label the axes and place a coordinate grid within the plot. Title the plot “math 445 lab 3 problem 2”.

Problem 3: Make a plot of $y(x) = 0.3 \log x + 2$ over the range $1 \leq x \leq 10^5$ using a dot-dashed cyan line. Chose an appropriate plotting function that best displays the functional relationship between $x$ and $y$. For this plot, it's best to create the x vector with logspace rather than linspace; e.g. x = logspace(0,5,20) will set x to 20 points between $1 = 10^0$ and $10^5$ which are uniformly spaced on a logarithmic plot. Label the axes, set the $y$ range of the plot to $2 \leq y \leq 6$, and place a coordinate grid within the plot. Title the plot as in previous problems.

Problem 4: Make a plot of $y(x) = 10 x^{-3}$ over the range $10 \leq x \leq 10^4$ using a solid green line. Chose an appropriate plotting function that best displays the functional relationship between $x$ and $y$. Use the logspace function for x as in problem 3. Label the axes, set the $x$ range of the plot to $10^0 \leq x \leq 10^5$, and place a coordinate grid within the plot. Title the plot as in previous problems.

Instructions for problems 5,6,7: For these problems you will deduce the functional relationship between variables in data sets using graphical analysis. The data sets are given as N x 2 matrices with x as the first column and y as the second. For each data set, you will find a function y(x) that fits the data, using the following steps:

  1. Download or cut & paste the data set to a text file with an appropriate name, e.g. earthquake_magnitude.asc for problem 1.
  2. Load the dataset to Matlab with load. For example, data = load('earthquake_magnitude.asc');.
  3. Extract the two columns of the loaded data into two appropriately named vectors, e.g. M = data(:,1); and N = data(:,2); will extract the M (magnitude) and N (number) columns of the earthquake-magnitude data matrix into vectors M and N for problem 5. For the remaining generic instructions I'll call these vectors x and y.
  4. Experiment with plot, semilogy, semilogx, and loglog to determine the functional relationship between y and x.
  5. Estimate the constants in the log-linear relationship graphically to determine the specific functional relation between y and x.
  6. Plot the estimated function and the data together, and fine-tune your function by adjusting the constants until there is a good fit between the estimated function and the data.

Once you have good fit between the data and the function, make a plot that shows

and provide an explicit formula for $y(x)$.

Problem 5: The distribution of earthquake magnitudes, by Moment Magnitude scale. Big earthquakes are rare, and little earthquakes are frequent. In fact, there is a very clean empirical law that governs how many earthquakes of a given magnitude typically occur world-wide in a given year. Your job is to deduce that law from the following historical data.

earthquake_magnitude.asc
% M N
8 2
7 18
6 120
5 800
4 6200
3 49000
2 365000
1 2920000

The first column is the moment magnitude M, and the second column is the number of earthquakes N of that magnitude that occur, on average, in a year. The last two entries are estimates, since it's impossible to detect every small earthquake around the world. The data are obtained from Earthquake Statistics and Earthquake Prediction Research by Stefan Wiemer, Institute of Geophysics, Zurich.

Using Matlab plotting commands, deduce the form of the functional relationship N(M). Estimate the constants in the relationship by estimating the slope and the y-intercept, and then fine-tuning by matching the plot of your estimate against the plot of the data. Your final answer should be an explicit formula for N(M).

Problem 6: The distribution of earthquake magnitudes, by energy. The moment magnitude scale is logarithmic, in that an earthquake of magnitude M+1 releases about 32 times more energy than an earthquake of magnitude M. The following dataset gives the number N of earthquakes in a given year of energy E measured in Joules.

earthquake_energy.asc
% E  N
6e16 2
2e15 18
6e13 120
2e12 800
6e10 6200
2e09 49000
6e07 365000
1e06 2920000

Deduce the form of the functional relation E(N) using Matlab plotting, then estimate and fine-tune the constants in the relation, and provide an explicit formula for E(N).

Problem 7: World population. The following data set provides the human population P of the earth at a given time t, measured in years A.D.

world_population.asc
% t   P
1927 2e09
1960 3e09
1974 4e09
1987 5e09
1999 6e09
2011 7e09

Deduce the form of the functional relation P(t), determine the constants graphically, and provide the explicit function P(t).

Assume that the formula you derived for P(t) is valid indefinitely into the future and the past. What year will the population of the earth reach one trillion? What year were the first humans born? Do you believe these answers? If not, why not?