Chapter 0. Preliminaries. 0.1 Things that you should know Derivatives

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1 Chapter Preliminaries These notes cover the course MATH232(Calculus& Applications) and are intended to supplement the lectures. The course does not follow any particular text and you do not need to buy any text books. A comprehensive summary of the theory is contained within Schaum s Outline of Differential Equations by R. Bronson & G. Costa and most of the material is in chapters 9, 3 and 7 of Calculus: Early Transcendentals by J. Stewart. An important component of the course is the use of computers and the software MATLAB to solve mathematical problems. MATLAB is the official mathematical computing package in the undergraduate degree programme and is available on most university PCs. A complete discussion of MATLAB would take many lecture courses and we shall use only a (small) subset of its capabilities. A quick introduction to the key commands is given in section.2 and examples will be given throughout the course. Just like learning mathematics, the only way to learn computing is by active participation. You should try to devote a reasonable proportion of your self-study time to learning and playing with MATLAB. Mastery of MATLAB is a great asset asset which will help you solve applied mathematics problems well beyound this course. This course follows from MATH3 and assumes that you are familiar with the basic concepts of ordinary and partial differentiation; (Riemann) integration; the fundamental theorem of calculus; Taylorseriesandhaveaworkingknowledgeofvectors. Ifyouareunsureaboutanyoftheseconcepts, summarised in the next section, you should carefully revise your first-semester lecture notes and your A-level texts.. Things that you should know.. Derivatives The (ordinary) derivative of a function of a single (real) variable y = f(x) is denoted by dy and dx represents the slope, or gradient, of the function, see Figure. The derivative is formally defined by the limit dy (x) = lim dx δx f(x+δx) f(x). (.) δx If the limit (.) exists then the function f(x) is said to be differentiable. If the limit (.) does not exist the function f(x) is non-differentiable. Functions that are non-differentiable can sometimes be made differentiable by restricting their domains. 2

2 f(x+δx) f(x) y = f(x) δx δy = f(x+δx) f(x) df dx δx, from equation (.) The slope of the tangent line is δy/δx dy dx x x+δx Figure : Geometric interpretation of the derivative, tangent to the curve y = f(x). dy, as the slope of the straight line that is dx Example.. A function with a single non-differentiable point The function y x, x >,.5 y = x x, x <,, x =, x y = x is differentiable over the domains x > and for x <, but is not differentiable at the point x =, where there is a discontinuity in the gradient. The partial derivative of a function of several variables f(x,y) with respect to one of its variables, x say, is denoted by f and is defined by the limit x f (x,y) = lim x δx f(x+δx,y) f(x,y), (.2) δx if such a limit exists. The partial derivative is equivalent to taking the ordinary derivative in one variable while simultaneously keeping all other variables fixed. It is the gradient in the direction of the chosen variable...2 (Riemann) Integration The integral of a function f(x) represents the area between the line y = f(x) and the x-axis, see Figure 2. Let f(x) be a continuous function defined over a region x [a,b], divided into n subintervals of equal width x = (b a)/n. The i-th subinterval is defined by x [x i,x i ], where x i = i x and we let x i be any sample point in the i-th subinterval. The definite (Riemann) integral of f from a to b is defined by b a f(x)dx = lim n n i= f(x i ) x. (.3) The continuity of f guarantees that the limit (.3) always exists and is independent of the choice of points x i.

3 y = f(x) f(x 3 ) Shaded area is b a f(x)dx a = x x x 2 x 3 x 4 = b Figure2: Geometric interpretation of theintegral b f(x)dxas the area between the curve y = f(x) a and the x-axis. The indefinite integral of a function f is written as a definite integral without limits and is defined to be an antiderivative of f if f(x)dx = F(x) then F (x) = f(x). x 3..3 Fundamental Theorem of Calculus The fundamental theorem of calculus establishes the relationship between integration and differentiation, namely that they are inverse to each other, as suggested by the definition of the indefinite integral. Theorem.. Fundamental Theorem of Calculus If a function f is continuous over the closed interval [a,b], and F is an antiderivative of f, a function such that F = f, then:. The function g(x) = x a f(ξ)dξ, a x b, is also continuous on the closed interval [a,b], differentiable on the open interval (a,b) and dg dx = f(x). 2. The definite integral b a f(x)dx = F(b) F(a).

4 ..4 Taylor Series The Taylor series of a smooth (infinitely differentiable) function f about the point a is a powerseries expansion f (n) (a) f(x) = (x a) n n! n= = f(a)+f (a)(x a)+ f (a) 2 (x a) 2 + f (a) (x a) 3 + (.4) 6 The form of the Taylor series ensures that every derivative of the Taylor series agrees with the corresponding derivative of the function f at the point a. The Taylor series is often used to write an approximation for a function at a small distance h from a given point x. In which case we can write f(x+h) = f(x)+hf (x)+ 2 h2 f (x)+o(h 3 ), (.5) where the notation O(h 3 ) means a unspecified function that when divided by h 3 is bounded above. Usually, but not always, the function has the form Nh 3 + where N is a constant...5 Vectors A vector in Ê 3 is a collection of three real numbers that represent a distance and direction in space. Vectors will be denoted by bold face, a Ê 3 and the components of the vectors by subscripts a = (a,a 2,a 3 ). The component a i represents the distance travelled in the x i -th direction. Dot (scalar) product The dot, or scalar, product between two vectors is the sum of the products of corresponding components of the two vectors a b = (a,a 2,a 3 ) (b,b 2,b 3 ) = a b +a 2 b 2 +a 3 b 3. The dot product is always a scalar, a single real number. The dot product can also be written in the form a b = a b cosθ, where a = a a is the magnitude, or length, of a vector and θ is the angle between the two vectors a and b. Cross (vector) product The cross, or vector, product between two vectors is a vector given by a a b = a 2 a 3 =. b b 2 b 3 a 2 b 3 b 2 a 3 a 3 b b 3 a a b 2 b a 2 The resulting vector is perpendicular to both a and b and has magnitude a b = a b sinθ, where θ is the angle between the vectors a and b.

5 ..6 Complex Numbers A complex number, z, is a number of the form z = a+ib, where a,b Ê are real numbers and i 2 =, by definition. The value a is called the real part of z, a = Re{z} and the value b is called the imaginary part of z, b = Im{z}. We will need to use complex numbers to represent all possible solutions of quadratic equations with real coefficients. Complex numbers can be visualised graphically on an Argand diagram, in which the real part is plotted against the imaginary part. Hence, the Cartesian coordinates of the complex number z are given by (a,b), see Figure 3. Im{z} 2.5 z = +i r.5 θ Re{z} Figure 3: An Argand diagram. The complex number z = + i has the Cartesian coordinates (,) when plotted on the complex, Re{z} Im{z}, plane; z is located at the point (r,θ) in polar coordinates, where r = 2 and θ = π/4. Complex numbers can also be represented by polar coordinates (r, θ), in the complex plane, where r = a 2 +b 2 is called the modulus of the complex number, z = r, and θ = tan(b/a) is called the argument of the complex number, argz = θ. Thus, a complex number can be written in many equivalent forms z = a+ib = rcosθ +irsinθ, Euler s formula states that and, hence, z = Re{z}+iIm{z} = z cosargz +i z sinargz. e iθ = cosθ+isinθ, z = rcosθ+irsinθ = r(cosθ+isinθ) = re iθ = z e iargz. We can use Euler s formula to represent trigonometric functions as the real or imaginary parts of complex exponentials, which can actually make calculations easier, particularly if there is lots of differentiation to be done. For example, sinx = Im{e ix } = Re{ ie ix } = Re{ie ix }, 2cos(2x)+7sin5x = Re{2e i2x 7ie i5x }.

6 .2 A very brief introduction to MATLAB MATLAB is an environment designed to facilitate scientific computing and visualisation, which is a fancy way of saying solving maths problems and looking at the answers Once MATLAB has been started 2, you should be presented with a command window that contains a command prompt: >> at which you can enter MATLAB commands. Once you have typed in your command pressing the return key (also known as enter) will run that command..2. MATLAB command-editing syntax You should keep a collection of your MATLAB work in a separate folder somewhere. You can create such a folder and make it your working directory from within MATLAB by the commands >> mkdir p:\matlab >> cd p:\matlab The above commands create the folder on your network P drive, but you might prefer to save your work on a USB stick. You can record all the commands that you type into MATLAB by issuing the command >> diary MyFile.txt which creates a text file called MyFile.txt in your working directory. You can then edit the file with your favourite text editor once you have finished the MATLAB session. Typing a semicolon (;) after a MATLAB command will suppress any output. It is a good idea not to use the semicolon too much when you first start working with MATLAB so that you can check that the commands are doing what you expect. If you want to type the same command again (or a very similar command) you can use the up arrow key to scroll through previous commands. When you have got to the command you want you can edit it by using the left and right arrow keys and the delete key. The command will not run until you press enter..2.2 Using MATLAB as a calculator MATLAB can do everything that your desktop calculator can do. The standard mathematical operations use the symbols + Add Subtract Multiply / Divide ^ Exponentiate (Raise to the power of) A lot of information about MATLAB is available online and in books; a good place to start is 2 The mechanism for starting MATLAB varies between operating systems, but on the university PCs you can find it from the Start menu via All Programs -> Site Licensed Programs -> Mathematical Applications.

7 For example >> 2*(3 + 7)/5-3 MATLAB also has the standard scientific functions (trigonometric, hyperbolic, exponential) and some more fancy functions (Bessel, Gamma, elliptic, etc). A list of elementary functions can be obtained by typing help elfun. >> sin(.5*pi) The above example shows that pi is a built-in constant and that trigonometric functions work in radians by default. Complete lists of functions can be found in the on-line help files. MATLAB can also work with complex numbers and i is another built-in constant. >> i*i - >> sqrt(i) i.2.3 Scalar variables (named storage) Computers work by storing and manipulating data (internally represented using binary numbers). If you wish to store your own data, for later use, you must tell MATLAB where to store it by giving it a name (just like using memory on a calculator or storing numbers in a phone by labelling them with a name). >> x = x = The above command creates a variable 3 x and assigns it the value. Once a variable has been created it can be used in subsequent commands and its value may change. >> x = 2*x + 3 x = 5 3 so-called from analogy with a mathematical variable

8 Note that the above statement is not a mathematical equation, if it were then the solution would be x= 3. InMATLAB, andalmost allother programming languages, theoperator=isanassignment operator. MATLAB evaluates the right-hand-side for the current value of x (= 2 +3 = 5) and then sets x to this new value. We can also create new variables from existing variables using any of the built-in mathematical functions >> y = 7*x y = 35 >> z = exp(x) z = IMPORTANT NOTE: The values of the variables y and z are determined from the value of x at the moment the return key is pressed. If, later in the session, the value of x changes, the values of y and z will not change unless you explicitly update them yourself. In other words, y and z are not functions of x. It is possible to define functions in MATLAB, see Vector and matrix variables MATLAB variables can be vectors or matrices rather than simple scalars. Matrix or vector data are defined by using square brackets [] and row ( n) and column (n ) vectors are specified differently. Entries in a row are separated by spaces (or commas), whereas entries in a column are separated by semi-colons. >> a = [ 2] a = 2 >> b = [; ; 2] b = 2 Access to a single component of a vector is via round brackets >> b(2)

9 If you try to access a component that doesn t exist you will get an error >> a(5)??? Index exceeds matrix dimensions. If you assign a value to a component that doesn t exist, that component will be created and all components in between will also be created and set to zero >> b(5) = b = 2 It is very easy to accidentally create new entries in vectors rather than overwriting existing entries so be careful. The same syntax is used to define matrices 4 >> A = [ ; 2 ; 3] A = 2 3 Basic arithmetic operations on matrices and vectors are easy to perform >> b + [3 ; 6; 9] 3 7 >> A*b 4 If you don t know what a matrix is, you should learn about them this semester in MATH22. For now, think of them as 2D vectors : a grid of numbers that is represented by two indices.

10 2 6 >> a*a 2 6 Note that the non-commutativity 5 of a matrix-vector product is implemented in MATLAB. Note also that the dimensions of the matrices and vectors must be consistent or there will be errors. For example, the commands A*a and b*a do not work. The division operator acting on matrices is rather special because the result requires the solution (or minimisation) of a linear system, see MATH22. The operation a/a returns the solution to the linear system xa = a and the operation A\b returns the solution to the linear system Ax = b. >> a/a >> ans*a >> A\b >> A*ans 2 5 For matrix-vector products Ax xa, in general, even after we turn the column vector x into a row vector by taking the transpose, x T.

11 It is also possible to perform element-by-element arithmetic by using the special dot operators.* and./ >> B = 2*ones(3) B = >> A./ B The built-in command ones(n) creates an n n matrix in which every entry has the value. A single entry of a matrix can be extracted using round brackets, just like vectors, but using two indices separated by a comma. >> A(,).5 >> A(3,2) = A(2,2) + A(3,3); >> A A = A range of entries of a matrix (or vector) can be extracted using round brackets combined with the colon : operator. On its own the colon represents the entire range of the index. Alternatively, lower and upper limits of the index can be specified on the left and right of the colon, respectively. >> A(:,).5

12 >> A(,:).5 >> A(:2,:2).5. Finally, the size command gives the dimensions of the matrix. >> size(a) Plotting All plotting packages use essentially the same idea: sample the function to be plotted at a discrete set of points and join the points by straight lines. MATLAB is no exception and the basic plot command requires two vectors that specify the x and y coordinates of the points to be plotted. The easiest way to generate the x coordinates is to create a set of equally-spaced points in a given range, which can be quickly achieved using the colon (:) notation. >> x = :.: x = Columns through Columns 9 through.8.9. The vector x contains points between and with a spacing of., generated by the command :.:. If you want to plot the function y = x 2, say, then you can create the y coordinates that correspond to each x value by the following command >> y = x.^2 y = Columns through 8

13 Columns 9 through The use of the special.^ command is used to perform element-by-element exponentiation 6. We now create the plot by >> plot(x,y) which should bring up the desired plot in a separate Figure window, see Figure4(a) (a) (b) Figure 4: MATLAB Figure window (a) after issuing the command plot(x,y) with x and y defined as in the text; (b) after issuing the subsequent commands hold on; plot(x,z, g ); with z defined as in the text. You can print out your figure by using the print command >> print but be careful not to waste paper by excessive printing. If you want to create an image file you can use the command >> print -djpeg fig.jpg which will create a JPEG image of your plot called fig.jpg in your working directly. For a complete list of possible graphics output formats see help print. You can also save your figures, create labels and customise your graphs by using the drop-down menus in the Figure window. If you want to add another line to your figure you can type 6 The MATLAB default for the ^ operator acting on vectors and matrices is to apply the matrix power function.

14 >> hold on which means that the Figure window will not be cleared. We can add the function e x to our figure >> z = exp(x); >> plot(x,z, g ); and the argument g causes the new line to be drawn in green, see Figure 4(b)..2.6 Programming The whole point of using computers is to avoid doing tedious and repetitive tasks by hand. The idea of programming is to give the computer a set of instructions so that it can do the tedious tasks for you. A programming language refers to such a set of easy(ish) to remember instructions. Loops Loops are the building blocks of (nearly) all computer programs. A loop causes a set of commands to be issued several times, but without our having to type all the commands. We can make a vector that contains the first ten square numbers by using a for loop. >> for i=: % Press enter now sq(i) = i*i; % Set the i-th entry of the vector sq to be i*i end >>sq sq = Anything after a percent (%) sign is a comment and is ignored by MATLAB, but is very useful to remind us what is going on in the program. The loop has an index (in this case we ve chosen to label it i) that ranges from to (In fact i takes the values contained in a vector consisting of the integers to specified using the colon notation.). You should press the enter button after the for command and then type the instruction(s) that you would like to take place within the loop. Once you have typed all the instructions, the loop is finished by typing the command end. If you like, you can see what is going on in each iteration of the loop if you remove the semicolon (;) from the command sq(i) = i*i. Conditional evaluation Another important component of programming is the use of conditional evaluation, or if else constructs. The idea is that you only want to perform certain tasks if a condition is true. >> a = -; >> if(a > ) % Test if a is positive fprintf( %g is positive\n,a); % Print the value of a else % Otherwise a is not positive fprintf( %g is zero or negative\n,a); % Print the value of a end % End of the if-else construct - is zero or negative >>

15 You should press the enter key after typing the if command. The instructions that you type next will be carried out if the condition is true. When you have specified all the instructions to be performed if the test is true then type else on a separate line. The set of commands that you type next will be carried out if the condition is not true and the entire structure is closed by the end command. In the above example, a is which is negative, so the else block is the block that is executed. The fprintf function is used to write instructions to the screen the special character %g is replaced by the value of the variable a and the special character \n prints a newline. The standard comparison operators that can be used in conditions are given by Functions > Greater than < Less than == Equal to >= Greater than or equal to <= Less than or equal to = Not equal to The combination of loops and conditional evaluation is the basis of all programming and allows you to do almost anything, but the program must still by typed in each time you want to use it. The final important element of programming is the use of functions in which we store a set of commands for future use. The easiest way to create your own functions is to type the command >> edit positive where positive is the name of the function that we are going to write. After pressing return you should obtain an editor window for a file called positive.m. All MATLAB functions must be saved in such M-files. It is extremely important that the name of the file is the same as the name of the function. Type the commands %Test whether a is positive if(a > ) fprintf( %g is positive\n,a); else fprintf( %g is zero or negative\n,a); end into the editor and save the file (Save in the File drop-down menu). We can now replay these commands by typing the command positive at the command prompt. >> a = ; >> positive is positive >> a = -5; >> positive -5 is zero or negative >>

16 Thefilepositive.m iscalledascript M-fileandisvery useful forrepeating longsetsofcommands, but what if we want to check whether a different variable (not a) is positive. We could copy that value into a and then type positive, but that is not a very elegant solution. Instead, we can turn our file into a full-blown MATLAB function. A function, like a mathematical function, takes input (arguments) and can return output. In the present context, the input to the function is the number that we wish to test for positivity and the output will be displayed on the screen. The M-file should be modified as follows function positive(a) %positive(a) %Determine whether the input variable a is positive or not if(a > ) fprintf( %g is positive\n,a); else fprintf( %g is zero or negative\n,a); end; The first statement in the file specifies that you are defining a function called positive that has a single input variable called a. The input variable is defined in round brackets () immediately after the function name. It is very important to realise that the variable a is now a local variable, which means that it is only defined within the function itself. Its value has nothing to do with the variable a that was defined in your main MATLAB command window. The next two lines are comments that explain what the function does. You can see these comments when you type help positive at the command prompt, so it s a good idea to make them informative so that other people can see how to use your function. Finally, the new function can be used by passing the input value between round brackets. >>a =5; >>positive() is positive >>positive(-) - is zero or negative >>positive(a) 5 is positive In the first use of function although the global variable a has the value 5, it is the value that is passed to the function. In the second use, the global variable a is passed to the function. Functions can return output data, specified before the name of the function with an equals sign. function y=positive(a) %positive(a) %Determine whether the input variable a is positive or not if(a > ) fprintf( %g is positive\n,a); y=; else fprintf( %g is zero or negative\n,a); y=-; end;

17 The function positive now sets a return value of if the input is positive or if the input is negative. >>positive() is positive The use of functions, loops and conditional statements is, essentially, all that is required to build programs to do almost anything. More information about programming is provided in the on-line MATLAB help files, in books and on the Internet. The best way to learn to program is by regular practice (and having a problem that you really want to solve). Computer problems will be provided each week on the example sheets. Please find the time to have a go at them.