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César Pérez López - MATLAB Differential and Integral Calculus

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Book:
MATLAB Differential and Integral Calculus
Author:
Csar Prez Lpez
Publisher:
Apress
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Year:
2014
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MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java.

MATLAB Differential and Integral Calculus introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. In addition to giving a short introduction to the MATLAB environment and MATLAB programming, this book provides all the material needed to work with ease in differential and integral calculus in one and several variables. Among other core topics of calculus, you will use MATLAB to investigate convergence, find limits of sequences and series and, for the purpose of exploring continuity, limits of functions. Various kinds of local approximations of functions are introduced, including Taylor and Laurent series. Symbolic and numerical techniques of differentiation and integration are covered with numerous examples, including applications to finding maxima and minima, areas, arc lengths, surface areas and volumes. You will also see how MATLAB can be used to solve problems in vector calculus and how to solve differential and difference equations.

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Csar Prez Lpez 2014

Csar Prez Lpez MATLAB Differential and Integral Calculus 10.1007/978-1-4842-0304-0_1

1. Introduction and the MATLAB Environment

Csar Prez Lpez 1

(1)

Madrib, Spain

1.1 Numerical Computation with MATLAB

You can use MATLAB as a powerful numerical computer. While most calculators handle numbers only to a preset degree of precision, MATLAB performs exact calculations to any desired degree of precision. In addition, unlike calculators, we can perform operations not only with individual numbers, but also with objects such as arrays.

Most of the topics of classical numerical analysis are treated by this software. It supports matrix calculus, statistics, interpolation, least squares fitting, numerical integration, minimization of functions, linear programming, numerical and algebraic solutions of differential equations and a long list of further methods that well meet as this book progresses.

Here are some examples of numerical calculations with MATLAB. (To obtain the results simply press Enter once the desired command has been entered after the prompt >> .)

We calculate 4 + 3 to obtain the result 7. To do this, just type 4 + 3, and then Enter .

>> 4 + 3

ans =

We find the value of 3 to the power of 100, without having previously set the precision. To do this we simply enter 3 ^ 100.

>> 3 ^ 100

ans =

5. 1538e + 047

We can use the command format long e to obtain results to 15 digits (floating-point).

>> format long e

>> 3^100

ans =

5.153775207320115e + 047

We can also work with complex numbers. We find the result of the operation raising (2 + 3i) to the power 10 by typing the expression (2 + 3i) ^ 10.

>> (2 + 3i) ^ 10

ans =

-1 415249999999998e + 005 - 1. 456680000000000e + 005i

The previous result is also available in short format, using the format short command.

>> format short

>> (2 + 3i)^10

ans =

-1.4152e + 005- 1.4567e + 005i

We can calculate the value of the Bessel function J 0 at 11.5. To do this we type besselj(0,11.5).

>> besselj(0,11.5)

ans =

-0.0677

We can also perform numerical integration. To calculate the integral of sin(sin( x )) between 0 and we type int(sin((sin(x)), 0, pi).

>> int ('sin(sin(x))', 0, pi)

ans =

1235191162052677/2251799813685248 * pi

These ideas will be treated more thoroughly later in the book.

1.2 Symbolic Computation with MATLAB

MATLAB perfectly handles symbolic mathematical computations, manipulating and performing operations on formulae and algebraic expressions with ease. You can expand, factor and simplify polynomials and rational and trigonometric expressions, find algebraic solutions of polynomial equations and systems of equations, evaluate derivatives and integrals symbolically, find solutions of differential equations, manipulate powers, and investigate limits and many other features of algebraic series.

To perform these tasks, MATLAB first requires all the variables (or algebraic expressions) to be written between single quotes. When MATLAB receives a variable or expression in quotes, it is interpreted as symbolic.

Here are some examples of symbolic computations with MATLAB.

We can expand the following algebraic expression: (( x + 1)( x + 2) - ( x + 2) ^ 2)^3. This is done by typing: expand(((x + 1)(x + 2) - (x + 2) ^ 2) ^ 3). The result will be another algebraic expression:

>> syms x; expand(((x + 1) *(x + 2)-(x + 2) ^ 2) ^ 3)

ans =

-x ^ 3-6 * x ^ 2-12 * x-8

We can factor the result of the calculation in the above example by typing: factor(((x + 1) *(x + 2) - (x + 2) ^ 2) ^ 3)

>> syms x; factor(((x + 1)*(x + 2)-(x + 2)^2)^3)

ans =

-(x+2)^3

We can find the indefinite integral of the function ( x ^ 2) sin( x ) ^ 2 by typing: int(x ^ 2 * sin(x) ^ 2, x)

>> int('x^2*sin(x)^2', 'x')

ans =

x ^ 2 *(-1/2 * cos(x) * sin(x) + 1/2 * x)-1/2 * x * cos(x) ^ 2 + 1/4 * cos(x) * sin(x) + 1/4 * 1/x-3 * x ^ 3

We can simplify the previous result:

>> syms x; simplify(int(x^2*sin(x)^2, x))

ans =

sin(2*x)/8 -(x*cos(2*x))/4 -(x^2*sin(2*x))/4 + x^3/6

We can present the previous result using a more elegant mathematical notation:

>> syms x; pretty(simplify(int(x^2*sin(x)^2, x)))

ans =

2 3

sin(2 x) x cos(2 x) x sin(2 x) x

-------- - ---------- - ----------- + --

8 4 4 6

We can find the series expansion up to order 12 of the function x ^ 2 * sin (x) ^ 2, presenting the result in elegant form:

>> pretty(taylor('x^2*sin(x)^2',12))

4 6 8 10 12

x - 1/3 x + 2/45 x - 1/315 x + o(x)

We can solve the equation 3 ax - 7 x ^ 2 + x ^ 3 = 0 (where a is a parameter):

>> solve('3*a*x-7*x^2 + x^3 = 0', 'x')

ans =

[ 0]

[7/2 + 1/2 *(49-12*a) ^(1/2)]

[7/2-1/2 *(49-12*a) ^(1/2)]

We can find the five solutions of the equation x ^ 5 + 2 x + 1 = 0:

>> solve('x^5+2*x+1','x')

ans =

RootOf(_Z^5+2*_Z+1)

As the result does not explicitly give five solutions, we apply the allvalues command:

>> allvalues(solve('x^5+2*x+1','x'))

ans =

[-.7018735688558619-. 8796971979298240 * i]

[-. 7018735688558619 +. 8796971979298240 * i]

[-. 4863890359345430]

[.9450680868231334-. 8545175144390459 * i]

[. 9450680868231334 +. 8545175144390459 * i]

On the other hand, MATLAB can use the Maple program libraries to work with symbolic math, and can thus extend its field of action. In this way, MATLAB can be used to work on such topics as differential forms, Euclidean geometry, projective geometry, statistics, etc.

At the same time, Maple can also benefit from MATLABs powers of numerical calculation, which might be used, for example, in combination with the Maple libraries (combinatorics, optimization, number theory, etc.)

1.3 MATLAB and Maple

Provided the Extended Symbolic Math Toolbox is installed then MATLAB can extend its symbolic calculation abilities by making use of the Maple libraries. To use a Maple command from MATLAB, use the command maple followed by the corresponding Maple syntax.

To use a Maple command from Matlab, the syntax is as follows:

maple ('Maple_command_syntax')

or alternatively:

maple 'Maple_command_syntax'

To use a Maple command with N arguments from Matlab, the syntax is as follows:

maple('Maple_command_syntax', argument1,

argument2,..., argumentN)

Here are some examples.

We can calculate the limit of the function (x ^ 3-1) / (x-1) as x tends to 1:

>> maple('limit((x^3-1)/(x-1),x=1)')

ans =

We could also have used the following syntax:

>> maple limit('(x^3-1)/(x-1),x=1)';

ans =

We can calculate the greatest common divisor of 10,000 and 5,000:

>> maple('gcd', 10000, 5000)

ans =

5000

1.4 Graphics with MATLAB

MATLAB can generate two- and three-dimensional graphs, as well as contour and density plots. You can graphically represent data lists, controlling colors, shading and other graphics features. Animated graphics are also supported. Graphics produced by MATLAB are portable to other programs.

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