Doing Calculus with MATLAB in Visual Studio .NET Creation PDF-417 2d barcode in Visual Studio .NET Doing Calculus with MATLAB

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Doing Calculus with MATLAB using visual studio .net tobuild qr code with web,windows application About QR Code Doing Calculus with MATLAB MATLAB has comma nds for most of the computations of basic calculus in its Symbolic Math Toolbox. This toolbox includes part of a separate program called Maple , which processes the symbolic calculations..

Differentiation You can use diff to differentiate symbolic expressions, and also to approximate the derivative of a function given numerically (say by an M- le):. >> syms x; diff(x 3) ans = 3*x^2 Here MATLAB has gured out that the variable is x. (See Default Variables at the end of the chapter.) Alternatively,.

>> f = inline( x 3 , x ); diff(f(x)) ans = 3*x^2 The syntax for s econd derivatives is diff(f(x), 2), and for nth derivatives, diff(f(x), n). The command diff can also compute partial derivatives of expressions involving several variables, as in diff(x 2*y, y), but to do multiple partials with respect to mixed variables you must use diff repeatedly, as in diff(diff(sin(x*y/z), x), y). (Remember to declare y and z symbolic.

) There is one instance where differentiation must be represented by the letter D, namely when you need to specify a differential equation as input to a command. For example, to use the symbolic ODE solver on the differential equation xy + 1 = y, you enter. dsolve( x*Dy + 1 = y , x ). 4: Beyond the Basics Integration MATLAB can compu te de nite and inde nite integrals. Here is an inde nite integral:. >> int ( x 2 , x ) ans = 1/3*x^3 As with diff, yo u can declare x to be symbolic and dispense with the character string quotes. Note that MATLAB does not include a constant of integration; the output is a single antiderivative of the integrand. Now here is a de nite integral:.

>> syms x; qr codes for .NET int(asin(x), 0, 1) ans = 1/2*pi-1. You are undoubte dly aware that not every function that appears in calculus can be symbolically integrated, and so numerical integration is sometimes necessary. MATLAB has three commands for numerical integration of a function f (x): quad, quad8, and quadl (the latter is new in MATLAB 6). We recommend quadl, with quad8 as a second choice.

Here s an example:. >> syms x; QR for .NET int(exp(-x 4), 0, 1) Warning: Explicit integral could not be found. > In /data/matlabr12/toolbox/symbolic/@sym/int.

m at line 58 ans = int(exp(-x^4),x = 0 .. 1) >> quadl(vectorize(exp(-x 4)), 0, 1) ans = 0.

8448. The commands quad, quad8, and quadl will not accept Inf or -Inf as a limit of integ Visual Studio .NET qrcode ration (though int will). The best way to handle a numerical improper integral over an in nite interval is to evaluate it over a very large interval.

. Doing Calculus with MATLAB You have another .NET QR Code 2d barcode option. If you type double(int( )), then Maple s numerical integration routine will evaluate the integral even over an in nite range.

. MATLAB can also do multiple integrals. The following command computes the double integral 0 0 sin x (x 2 + y2 ) dy dx :. >> syms x QR Code for .NET y; int(int(x 2 + y 1, y, 0, sin(x)), 0, pi) ans = pi^2-32/9. Note that MATLAB presumes that the variable of integration in int is x unless you prescribe otherwise. Note also that the order of integration is as in calculus, from the inside out . Finally, we observe that there is a numerical double integral command dblquad, whose properties and use we will allow you to discover from the online help.

. Limits You can use limi t to compute right- and left-handed limits and limits at in nity. For example, here is lim sin(x)/x:. >> syms x; limit(sin(x)/x, x, 0) ans = 1 To compute one-s ided limits, use the right and left options. For example,. >> limit(a QR Code ISO/IEC18004 for .NET bs(x)/x, x, 0, left ) ans = -1. Limits at in nit y can be computed using the symbol Inf:. >> limit(( visual .net QR-Code x 4 + x 2 - 3)/(3*x 4 - log(x)), x, Inf) ans = 1/3. 4: Beyond the Basics Sums and Products Finite numerical sums and products can be computed easily using the vector capabilities of MATLAB and the commands sum and prod. For example,. >> X = 1:7; >> sum(X) ans = 28 >> prod(X) ans = 5040 You can do nite and in nite symbolic sums using the command symsum. To illustrate, here is the telescoping sum.
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