4.1.1 Creation, saving and running

A script file is a sequence of commands (often called a program) that should be run. MATLAB® executes commands in the order they were written. Corrections can be entered directly in the file and new commands can be added. The extension of the saved script file is ‘.m’, hence the term m-files.

A script file should be typed and edited into a MATLAB® Editor. To open the Editor, the command edit should be typed and entered in the Command Window, or the ‘New’ and ‘M-file’ options in the ‘File’ menu should be selected. After this the Editor Window appears (Figure 4.1):

The commands of the script file should be typed line by line. It is also possible to type two or more commands on the same line, dividing them by commas or by semicolons. A new line will be accessible after pressing the Enter key. Each new line is numbered automatically. A command can also be typed in another editor and then copied into the Editor Window.

Figure 4.2 on p. 97 shows an elementary script file typed into the Editor Window. The first lines are usually explanatory comments, preceded by the comments sign % and in green (light gray here); they are not part of the execution. The commands appear in black for better legibility.

In the latest MATLAB® version the Editor Window appears with a message bar (located on the right) of the so-called M-Lint analyzer, which detects possible errors, comments on them and recommends modifications for better program performance. The bar is topped by the message indicator. A green indicator means no errors, warnings or possibility of improvements; a red indicator means syntax errors have been detected; and an orange indicator means warnings or possibility of improvements (but no errors). When the analyzer detects an error or possibility of improvement, it underlines/highlights the text and a horizontal line appears on the message bar. When the cursor is placed at this line the comment message appears (see Figure 4.2). The changes it recommends – such as the one shown in this figure (addition of a semicolon) – need not be executed because we want to display the resulting value of the DNA volume, V_DNA. The M-Lint regime appears in the Editor Window by default and can be disabled by un-signing the ‘Enable integrated M-Lint warning and error messages’ check-box in the M-Lint option of the Preferences Window, which can be selected from the File in the Editor Window or in the Desktop menu.

The script file should be saved after it has been written into the Editor Window. For this the ‘Save As …’ option from the File menu should be chosen. After this the ‘Save file as …’ window appears and the desired file location and file name should be typed respectively into the ‘Save in:’ and ‘File name:’ positions. By default the ‘MATLAB’ subdirectory in the ‘My documents’ directory is automatically selected for the file location, or the desired directory has to be chosen from the pop-up menu which appears after pressing on the arrow on the right-hand side of the ‘Save in:’ field of the ‘Save file as …’ window. The name for a script file should begin with a letter, and should not repeat the user-defined or predefined variables, MATLAB® commands or functions. In addition, it is not recommended to introduce spaces into the name, which cannot be longer than 63 characters.

To execute a script file: first, the user should check the current MATLAB directory, and if the file is not in it a directory with this file should be set; secondly, the file name should be typed in the Command Window and then entered.

4.1.2 Input values to variables of a script file

To assign a value to the variables defined in the script file when the script is run the input command should be used; this command has the forms:

Numeric_Variable_Name = input('Displayed string') Character_Variable_Name = input('Displayed string','s')

where 'Displayed string' is a text that displays in the Command Window and prompts us to assign a number to the Numeric_Variable_ Name, or a string to the Character_Variable_Name; 's' indicates that inputting characters are string.

After running a script file, when the input command is started, the string written in this command appears on the screen and the user should type and enter a number or string; inputted values are assigned to the Numeric_Variable_Name or to the Character_Variable_Name depending on the command form used.

As an example is a command script file that convert a weight, g_lb, in pounds to kilograms, g_kg, by the expression g_kg = g_lb/2.2046:

The file is saved with the pound2kg name. After typing and entering this file name in the Command Window, the prompt ‘Enter your weight in pounds, G =’ appears on the screen, a weight value should be typed (in pounds) after the sign ‘=’ and enter pressed; the weight is converted to kilograms and displayed on the screen. The run command, prompts, and input and output numbers are:

With the input command the vectors and matrices can be inputted in the same way as for a variable – numbers in square brackets.

4.2.1 Creating the function

In algebra a function is presented as the equation y = f(x), whose right-hand side contains one – x – or more parameters (arguments) – x or x, a, b, …. When these parameters are assigned, the value of y is obtained. Similarly to the above, many MATLAB® commands discussed in the preceding chapters are written in function form, sin(x), cos(x), log(x), sqrt(x), etc., and can be used for direct calculations or placed into more complicated expressions by typing their name with the argument. In MATLAB® it is possible to create any new function and re-use it with different argument values and in different programs; such functions are termed ‘user-defined’. Not only a specific expression, but a complete program created by the user can be defined as a function and saved as a function file. The MATLAB® definition of a function reads

function[output_parameters] = function_name(input_parameters)

The word function appears in blue and must be the first word in the file. The function name is written to the right of the ‘=’ sign and should obey the same rules as for variable names (see subsection 2.1.4). As input_ parameters we should write a list of arguments for transfer into the function, and as output_parameters a list of those we want to derive from the function. The input parameters must be written in parentheses and the output parameters in square brackets. (In the case of a single output parameter the brackets can be dispensed with.) Parameters must be divided by commas.

As with a script file, a function file should be written in the Editor Window. An example of a function file is presented in Figure 4.4 on p. 101. This function is named dilution and calculates the molar concentration of a solute after dilution. It has three input parameters – the molar concentration M₁ and the solution volume V₁ before dilution, and solution volume V₂ after dilution – and one output parameter – the molar concentration of the solute after dilution.

The requirements and recommendations regarding the structure and certain parts of the function file are:

4.2.2 Saving a function as a function file

Before a function is used, it must be saved in a file. This is done exactly as for a script file – selecting ‘Save As’ from the File menu and then entering the desired location and name for the file. It is recommended that the file be called by the function name, e.g. the dilute function should be saved in a file named dilute.m.

Examples of function definition lines and function file names:

When a program has two or more functions, the name of a function file should be identical to the name of the main function (from which the program starts).

4.2.3 Running a function file

The function file can be run from another file or from the Command Window as follows. The function file definition line should be typed without the word ‘function’ and the input parameters should be assigned their values. For example, the dilution function file (see Figure 4.4) can be run thus:

Or alternatively with pre-assignment of the input variable:

A user-defined function can be used in another mathematical expression or program; for example, dilution gives M₂ solute concentrations in mol/L for V₂ = 0.6 L of the solution; thus the solution has M2_mol = M2*V2 moles. For the calculations the following commands should be typed in the Command Window:

4.3.1 Interpolation and extrapolation

When data are available (by measurement or by tables) at certain points and we need to estimate intermediate values between them, this can be done by interpolation. When values outside the data points need to be evaluated, this can be done by extrapolation. For example, the values of measured radioactive decay R of a liquid substance are 400, 200, 100, 50 and 20 disintegrations per minute at times t = 1, 2, 3, 4 and 5 hours, respectively. The graph with the data points is presented in Figure 4.5. Finding the decay values at t = 1.2 and 2.4 hours (values between data points) is an interpolation problem, and finding them at t = 0.8 and 5.8 hours (values outside data points) is an extrapolation problem.

In MATLAB® both interpolation and extrapolation operations can be done with the interp1 function. This command has the forms

yi = interp1(x,y,xi,'method') or

yi = interp1(x,y,xi,'method','extrap')

The first version is used for interpolation, and the second for extrapolation or for simultaneous interpolation and extrapolation.

The output parameter for these MATLAB® functions is yi – the interpolated value or vector of values. The input parameters are x and y, vectors of data values – argument and function, respectively; xi, a scalar or a vector with the points for which the values of yi are sought; 'method' is the string that contains the name of the mathematical method to be used for interpolation or extrapolation; some available methods are 'linear', 'cubic' and 'spline'; a default method is 'linear', which need not be specified.

The 'extrap' string is required for the 'linear' method; the word can be omitted for the 'cubic' and 'spline' methods.

For the example in Figure 4.5 the commands for interpolation (at points 1.2 and 2.4 hours) and extrapolation (at points 0.8 and 5.8 hours) are:

4.3.2 Solution of non-linear algebraic equations

The matrix solution for a set of linear algebraic equations was discussed in chapter 2 (subsections 2.2.2, 2.3.4.1). Here we deal with non-linear equations with iterations in which the computer finds the x value at which f(x) = 0. The function for this process is fzero and its general form is:

x = fzero('fun', x0)

The fzero function seeks an x value representing the solution near the guess point x₀; and 'fun' is the string function to be solved. In this string the equation itself can be written, or the name of the user-defined function that contains it.

One possible way to determine the approximate value of x₀ is to plot the graph of the equation and check the x-value at which the function is zero. For example, consider the Arrhenius equation

where the value k/A = 8.7271 × 10⁻¹⁴ was previously determined at T = 300 K and R = 8.314 J/kmol. The unknown variable is the activation energy E₀ and should be named x, and the equation presented in the form . The value of x₀ must be positive for physical reasons and for many reactions it is around 5 × 10⁴ J/mol.

The solution command is:

The string with the solved equation cannot include pre-assigned variables; for example, it is not possible to define k/A = 8.7271 × 10⁻¹⁴, R = 8.314, T = 300 K and then write 'k_A-exp(− x/(R*T))'.

For substitution of pre-assigned parameters into the solving expression the following form of fzero is available:

x = fzero(@(x) fun(x, preassigned_variables1, preassigned_variables2,… ), x0)

@(x) fun¹ indicates a user-defined function that contains the equation with the additional arguments preassigned variablesl, preassigned variables2, ….

Using this form, we can write the above example as a function file with the name actener, input parameters k/A, R, T and x₀, and output parameter E₀. In this function file the fzero command reads x = fzero (@ (x) myf(x,k_A,R,T), x0) and the function definition line for the myf function should appear as function f = myf(x,k_A,R,T). The full text of the actener function is

function Eo = actener(k_A, R, T,x0)

% activation energy calculation

% k_A is reaction constant

% R – gas constant, J/mol/K

% T – temperature, K

% x0 – initial Eo

Eo = fzero(@ (x) myf(x,k_A,R,T), x0);

function f = myf(x,k_A, R, T)

f = k_A-exp(− x/(R*T));

This function should be saved as actener.mfile and run into the Command Window with assigned parameters k/A, R, T and x₀:

4.3.3 Integration

The area beneath a segment of a function f(x) or beneath data points (Figure 4.6) can be determined by numeric integration. For this the area is subdivided into small geometrical elements, e.g. rectangles and trapezoids. The integral is the sum of the areas of these elements. Below, two MATLAB® functions for integration are described – quad and trapz. The first is used when f(x) is presented as an analytical expression, and the second when it is presented as data points.

4.3.4 Derivatives

The derivative dy/dt of the function y(t) is characterized by the infinitesimal change in the function at some given point; in geometrical representation (Figure 4.7, p. 111) it is the slope of the tangent to the curve at the ith point (x,t).

Following these definitions, the derivative can be presented for numerical calculations as:

Thus, the derivative can be calculated at each i-point as the ratio of y- and x-differences between neighboring points. The diff command for these differences has the forms:

dy = diff(y) or dy_n = diff(y,n)

where y is the vector of y-values at points i = 1, 2, …; the n argument indicates how many times the diff command should be applied and the order of the particular difference. For example, if n = 2 the diff command should be applied twice; the dy and dy_n output parameters are vectors of the determined values of the first- and nth-order differences.

The dy vector is one element shorter than y and correspondingly the dy_n vector is n elements shorter than y; for example, if y has 10 elements dy has nine elements and, say, the third-order derivative vector dy_3 has only seven elements.

For example, a species concentration c changes with time t due to a chemical reaction, as shown here:

These data have to be used to determine the reaction rate r = − dc/dt [mol/ (L min)]. The commands to be applied in the Command Window are:

The second-order derivative, representing the acceleration rate of the reaction, ,

is:

> > a = − diff(r)./diff(t(2:end)) % t(2:end) is used as r is one element shorter

a =

− 0.0416 -0.0167 -0.0083 -0.0048 -0.0030 -0.0020

Negative values of a indicate deceleration of the reaction. The same results can be reached via the second-order c differences:

> > a = − diff(c,2)./diff(t(2:end)).^{^}2

a =

− 0.0416 -0.0167 -0.0083 -0.0048 -0.0030 -0.0020

Note, that if the step h of argument t is constant, it can be used in changing diff(t).

The derivatives of a function given by an equation can be determined in the same way as tabulated data; the step in t in such a case can be smaller and lead to more exact derivative values. For example, assuming that the data in the above table are very accurately described by the equation c = 0.25/ (1 + 05 t), the reaction rate can be determined with the following commands:

Significant deviations in the values of r and t are evidence that the number of points in the table is not enough for exact determination of the derivatives. They also confirm that for an analytically given function, more points apparently provide more exact results.

Frequently, a problem involving a derivative is formulated so that the latter (reaction rate, population growth, etc.) needs to be determined for one specific point. In such a case the minimal range of t (or any other possible argument –x, z, etc.) should be assigned one step after that point. For example, if the required value of a rate is needed at t = 1 hour, the minimal interval for the numerical differentiation is t … t + h, where h is the time step, which should be specified.

4.3.5 Curve fitting

The process of matching an expression or curve to the data points is called curve fitting or regression analysis. The fitting expression may be based on a theory or chosen empirically. This subsection describes MATLAB® curve fitting involving polynomials.

The process consists of finding the values of the coefficients of the mathematical expression which should fit the experimental data. In the case of a polynomial expression

the coefficients are the a_i values, and n here is both the length of the polynomial and the number of coefficients in it.

The highest exponent, n – 1, is called the degree of the polynomial; for example, the strait line y = a₁x + a₀ is a first-degree polynomial that has two coefficients, a₁ and a₀, to be determined.

When the experimental data contain n points (x,y) the n coefficients can be determined by solving the set of y(x) equations written for each of the points. Polynomial fitting where the number of coefficients is the same as that of points is not always efficacious and might lead to significant discrepancies in regions between data points. Another case of possible inefficiency is a linear y(x) dependence where two points suffice for determination but the experimental data contain a larger number, which means more equations than unknowns. The commonly used method for better fit is least squares, in which the coefficients are determined by minimizing the sum of squares of the differences between the polynomial and data values; these differences are called residuals and are denoted R. The minimization is done by taking the partial derivative of R with respect to each coefficient and imposing their equality to zero, for example for the straight line and and the set of two equations should be solved to determine a₁ and a₂.

Polynomial fitting in MATLAB® can be done with the polyfit function, whose simplest form is

a = polyfit(x,y,m)

where the input arguments x and y are the vectors of the x- and y-coordinates of the data points and m is the degree of the polynomial, which in the notation used above is equal to n – 1; the output argument a is the (m + 1)-element vector of the fitting coefficients, in which the vector of the first element a(1) is a_n–1, the second a(2) is a_n-2, …, and the final a(m + 1) is a₀.

With the fitting coefficients the y-values can be calculated at any x in the fitted interval with the polyval command; this command has the following simplest form

y_polynomial = polyval(a,x_polynomial)

where a is the vector of polynomial coefficients as per the polyfit, and x_polynomial is that of the coordinates at which the y_polynomial values are calculated.

For example, the height of five seedlings and the amount of nutrients for each of them are 4.7, 7, 15.2, 22.3 and 27.2 cm and 12, 28, 60, 95, 120 mg. For fitting these data by the first-degree polynomial (m = 1), and for generating the next curve and data plot, the following script file, named fitexample, can be created:

After typing and entering the file name in the Command Window, the following coefficients are displayed and the graph in Figure 4.8 is generated:

Many bio-applications require an exponential or a logarithmic fitting function:

In the first case the fitting function should be rewritten as ln y = a₁x + ln a₀ and the polyfit function can be used with argument log(y) instead of y, e.g. a = polyfit(x, log(y),1) where in the vector a the a(1) element is a₁ and a(2) is ln a₀, thus a₀ = exp(a(2)).

In the second case the polyfit function can be used with argument log(x) or log10(x) instead of x, e.g. a = polyfit(log(x), y,1) or a = polyfit(log10(x),y,1) where in vector a the a(1) element is a₁ and a(2) is a₀.

4.4.1 Body mass index

Body mass index (BMI) I is a measure that relates the body weight G in kg and height h in m; it is defined as

This index is frequently used to assess how much an individual’s body weight departs from what is normal or desirable. A BMI between 20 and 25 kg/m² is normal, below 20 underweight, between 25 and 30 overweight, and above 30 obese.

Problem: Compose a script program that calculates and displays the BMI of the user and writes a conclusion about the user’s weight.

In the script presented below, the user enters his/her weight and height, and the program then calculates the BMI, determines the user’s category and displays the results.

% BMI calculator

clear

w = input('Enter your weight in kg'),

h = input('Enter your height in m'),

I = w/h^{^}2;

if I < 20

S = 'too small';

elseif I > = 20&I < = 25

S = 'normal';

elseif I > = 25&I < = 30

S = 'large';

else

S = 'too large';

end

fprintf(' Your BMI is %5.1f kg/m^2, thus your weight is %s ',I,S)

With the script file name BMI entered into the Command Window, the program displays the prompt to input the weight in kg; with the weight typed and the Enter key pressed, a new prompt for height in m appears; with the latter typed and entered, the BMI is calculated and displayed together with a conclusion.

The following shows the Command Window when the file was run:

4.4.2 Basal metabolic rate

The basal metabolic rate (BMR) P in kcal/day is the human minimal daily energy requirement (in calories). For women this index is given by the expression:

and for men by:

where w is weight in kg, h is height in cm and a is age in years.

Problem: Write a script program for calculating and displaying the BMR. The script file solving this problem is written below.

% BMR calculator

clear

sex = input('Enter the sex: 1 - male, 2 - female'),

w = input('Enter your weight in kg'),

h = input('Enter your height in m '),

a = input('Enter your age in years '),

if sex < 1

P = 66 + 13.7*w + 58*h-6.8*a;

else

P = 655 + 9.6*w + 1.8*h-4.7*a;

end

fprintf(' Your BMR is %5.1f cal/day ',P)

After entering into the Command Window the script file name BMR, the program displays the prompt to sex, type 1 or 2 (male or female). After pressing the enter key, prompts then appear successively about weight in kg, height in m and finally age. The BMR is then calculated and displayed.

The following shows the Command Window when the file was run:

4.4.3 Conversion of concentration units

The concentration of components in solutions is expressed in different forms, among them molarity, molality and molar fraction. Molarity M is the solute amount per liter of solution, mol/L; molality m is the solute amount per solvent mass, mol/kg; and molar fraction χ is the mole solute to mole solution ratio. The expression for conversion from M to m and to x is

where d is the solution density (g/L), M₁ the molecular weight of the solute (g/mol) and M₂ = 18 g/mol is the molecular weight of water.

Problem: Write a function file with name molfrac that calculates and displays solute molality and molar fraction for a given solute molarity and molar mass, and solution density. Calculate the molality and the molar fraction for these data: M = 3 mol/L, d = 1.5 × 10³ g/L and M₁ = 40 g/mol (sodium hydroxide).

The input parameters in the function definition line are: solution density d, solute molarity M and molar mass of the solute M1; the output parameters are molality and molar_fraction. The function file is:

The following shows the Command window when the file was run:

4.4.4 Population of Europe

Population statistics for Europe are:

Problem: Write a function file with name europes that predicts and displays the population in 2011. Use for prediction the interp1 function with three extrapolation alternatives provided: 1, linear; 2, cubic; 3, spline.

The first line of the function should be the function definition line with the function name europes and the input and output parameters. Thus, the following input parameters are specified: yi, the target year, and num, the chosen method alternative: 1, linear; 2, cubic; 3, any other number (except 1 and 2) for the spline. The specified output parameter is pei – the predicted population.

The function file written to solve the problem is:

function pei = europes(yi,num)

% Europe population prediction

% yi - year for prediction

% num - mehod number: 1-linear,2-cubic, 3-spline

y = [1750:50:1950 1999 2008];

p = [163 203 276 408 547 729 732];

if num==1

method = 'linear';

elseif num==2

method = 'cubic';

else

method = 'spline';

end

pei = interp1(y,p,yi,method, 'extrap'),

The following appears in the Command Window when the file was run:

4.4.5 Reactant breakdown

In the two-stage consecutive reaction A → B → C, the concentration of intermediate reactant B can be expressed as

where [A]₀ is the initial concentration of reactant A, k₁ and k₂ are the reaction rate constants for the A → B and B → C reactions, respectively, and t is time.

Problem: Write a function file reactime.m that evaluates and displays time t in the course of which the intermediate reactant B appears, accumulates and breaks down to the current concentration. Calculate the time for: [B] = 0.01 mol/L at initial concentration [A]₀ = 2.5 mol/L, and constants k₁ = 2.6 × 10⁻³ s⁻¹ and k₂ = 9.5 × 10⁻³ s⁻¹; use for solution the fzero function with initial (guess) value x₀ = 2 s.

The input parameters for the function definition line are: the concentrations [B] and [A]₀, the reaction rate constants k₁ and k₂, and the initial value x₀; the output parameter is time t. In using fzero, the variable t in the expression above should be named x. The function file for solving this problem is:

The following appears in the Command Window when the file was run:

4.4.6 Amount of compound

The decay rate v of a compound is given by v(t) = − 0.69*e^−2.3t, where t is time in h, and v is in mol/h. The amount m of the compound after time t is:

where m₀ = 0.3 mol is the initial mass of the compound.

Problem: Write a script that calculates the amount of the compound every 30 minutes over 3 hours and plot the graph m(t).

The solution is organized in the following steps:

The script to solve this problem is:

q(i) = m0 + quad('-0.69*exp(− 2.3*t)',0,t(i));

end

fprintf(' Compound amount change Hour Mole ')

122This script was named and saved into the Integr2_Ex.m file. The following appears in the Command Window when the file was run:

> > Integr2_Ex

Compound amount change

And the resulting graph is:

4.4.7 Dissociation rate

Species dissociation is described by the equation

where m₀ and m are the amounts of the species initially and at time t, respectively, and k is the dissociation constant.

Problem: Write and save a function that calculates the reaction rate r = − dm/dt at t = 0, 0.1, …, 10 min, k = 1.8 min⁻¹ and m₀ = 5 mg, and plots the m(t) and r(t) curves in one graph.

The program is realized in the form of a function named DissRate with the following input arguments: m0, the species amount at starting time; k, the reaction constant; tstart, the starting time; tend, the end time; and h, the time step.

The solution is carried out in the following steps:

The function file to solve this problem is:

After running this file, the following appear in the Command and Figure windows:

> > DissRate(5,1.8,0,2,0.1)

The data are plotted in the figure overleaf:

4.4.8 Fitting of cancer incidence data

A high level of a particular substance in the atmosphere may increase the incidence of cancer. Assume that the function that relates cancer incidence Y to the amount of carcinogen X is polynomial. Observed cancer incidences are 10, 10, 30, 35, 55 and 120 at carcinogen concentrations respectively of 10, 40, 100, 200, 300 and 400 (dimensionless).

Problem: Write and save a function that requires degree of polynomial as input parameter, and then fits the cancer incidence–carcinogen concentration data, displays fitting coefficients and plots y(x) points with defined polynomial curve. Take the degree of polynomial to be no more than 5.

The solution is:

This function file with this function has the name cancer. For running, the function requires the value of polynomial degree, m. The first line of cancer represents the definition of the function and the next three lines represent function help. After running, the function determines the fitting coefficients with the polyfit command, displays the latter and the polynomial degree with the disp commands, and plots the fit results together with the original data. If the inputted degree is more than 5 (the number of coefficients in the polynomial is m + 1 and it cannot be more than the number of points), the program displays ‘Error: Polynomial degree cannot be more than 5’, and the run should be repeated with the correct m value.

The Command Window and generated plot for polynomial degree 3 are:

> > cancer(3)

Polynomial coefficients are:

Polynomial degree is:

3

1. The user-defined MATLAB® function has in the first line: (a) the function definition; (b) help comments; or (c) definition of global variables?

2. Local variables are: (a) all variables defined, recognized and used only within the function file; (b) input/output parameters only, written in the function definition line; or (c) the variables appearing in the workspace after the function file has completed its work?

3. The unterp1 function is used: (a) for interpolation only; (b) for extrapolation only; or (c) for both interpolation and extrapolation?

4. In which form should the equation ae^−kx = b be written for solution with the aid of the fzero function: (a) just as it is written: b = ae^−kx; (b) ae^−kx – b = 0; or (c) ln(b/a) = − kx?

5. The function f(x) is given by data points; which MATLAB® function should be used for its integration: (a) quad, (b) trapz or (c) fzero?

6. The expression

    is used to convert degrees Fahrenheit F into degrees Centigrade C. Write the script that can be used as a Fahrenheit to Centigrade converter; display the result to two decimal places.

7. The death time t of spores in a thermal sterilization process is described by the equation

    t = 3.5 × 10⁻⁵e^0.083T

    where T is temperature in degrees Celcius, and t is time in seconds. Write the script that calculates and displays the death time of the spores at 100, 110, …, 180 °C. Plot the graph T as function of t, and add axis labels, grid and captions to the graph.

8. The van der Waals gas equation of state is:

    where P is pressure in atmospheres, V is volume in liters, T temperature in Kelvin, n is the number of moles, and a and b are constants that should be specified for each gas. Write the function file that calculates the volume of the gas at given P, T, n, a and b; use the fzero function to obtain V at P = 6 atm, T = 50 °C (323.15 K), n = 2 mol, R = 0.08206 L-atm/mol-K, a = 5.43 L²atm/mol², b = 0.0304 L/mol (water) and initial V = 5.

9. The waist-to-hip ratio (WHR) R = waist/hip is used as an indicator or measure of the state of health of a person; when this ratio is in the range 0.7–0.85 for women and 0.76–0.9 for men, it is ranked as normal; persons with higher WHRs (‘apple-shaped’ body) face a higher health risk than those with lower WHRs (‘pear-shaped’ body). Write a function file with waist and hip as input parameters, and R and information about the person’s state of health (normal, high health risk, low health risk) as output parameters.

10. New York city population between 1900 and 2000:

    Write a user-defined function with the table data, and enter into it the interp1 command with the 'linear' method to calculate the population within and outside the table range; save the defined function in the function file and use it to calculate the population in 1899, 2010 and 2012.

11. The viscosity of a solution η is 1.75, 1.5, 1.27, 1.3, 1.15 and 1.1 cP at temperatures t of 20, 25, 30, 35, 40 and 45 °C, respectively. Write a user-defined function that describes the graph η(t) together with interpolated values, and add to the graph axis labels, grid, legend and a caption; use the interp1 command with the ‘cubic’ method to obtain the viscosity at temperatures within the above range; save the function into a function file and use it to obtain the viscosity at 22.5 °C.

12. The integral of a function f(x) can be calculated by the left Riemann sum equation:

    where n is the number of points in the integration interval [a,b], Δx = (b – a)/(n – 1) is the spacing of the points, and f_i is the value of the function at point i = 1, 2, …, n – 1. Calculate by this the bacterial population

    where time t is in hours, the proliferation rate v(t) = 0.6931.N₀.2^t, the initial bacterial amount N₀ = 10 cells, and the integration limits are a = 0 and b = 7 hours. Write and save the program as a function file with input parameters a, b, n, N₀ and output parameter N; run it with n = 1000. The resulting N should be given rounded to the nearest integer value.

13. Solve the integral in exercise 12 by the right Riemann sum equation:

    where Δx and f_i have the same sense as in exercise 12. Write and save the program as a function file; run it with n = 100, 500 and 1000. Use the function written in exercise 12 to display the results obtained for the left and right Riemann methods. Note: the input and output parameters of the function created should be suitable to solve the problem.

14. Solve the integral in exercise 12 with the quad command. Write and save the program as a function file. Use the functions written in exercises 12 and 13 to display the results obtained with the quad command and with the left and right Riemann methods (for n = 1000). Note: the input and output parameters of the function created should be suitable to solve the problem.

15. Bacterial population growth is expressed by the polynomial p = 10⁶ + 10⁴ t – 10³ t², where t is time in hours. Write the script that determines numerically the bacterial growth rate r = dp/dt when t = 1 hour. Take a time step equal to 0.01 hour and count the derivatives one step longer than the required t.

16. The decay rate of a certain radioisotope is described by the equation m = m⁰e^−kt where t is time, m and m₀ are the isotope amounts at the starting and current time, respectively, and k is the decay constant. Values of m of 500, 250, 125, 63 and 31 μCi were measured on days 0, 20, 40, 60 and 80, respectively. Write a function without parameters that fits these data by the above equation, displays the fitting equation with the found coefficients m₀ and k, and plots the fitting curve and original data.

2. (b) Input/output parameters, only written in the function definition line.

4. (b) ae^−kx – b = 0

6. Enter temperature as Fahrenheit 82

Temperature 82.00 degrees Fahrenheit is 27.78 degrees Celsius

8.

> > V = Ch4_8(6,323,2,5.43,0.0304)

V =

3.4495

11.

> > etai = Ch4_11(22.5)

etai =

1.6224

14.

> > [N_right,N_left,N_Simpson] = Ch4_14(0,7,1000,10)

N_right =

1272

N_left =

1266

N_Simpson =

1269

16.

> > Ch4_16

The fitted equation is:

m = 500.80386 e^{^}-0.1499 t