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Analysis

Analysis is the area of mathematics devoted to the study of functions.

 >>fast Differentiation Integration Integral calculus Definite integral Calculus of limits Limit Horizontal limit Taylor Series Series Differential equations vector fields integral curves integral curve

To differentiate use the icon, the differentiate command or the 'sign, corresponding to the apostrophe.

Clicking on the icon brings up the common expression for differentiation with respect to a variable, including two empty green boxes. In the upper box, enter the expression to differentiate, and in the lower box the variable with respect to which we wish to differentiate.

The differentiate command takes 2 arguments; the first corresponds to the expression to be differentiated and the second to the variable with respect to which we wish to differentiate. In the case of a function of a single variable, the second argument may be omitted.

The ' sign can be used after the expression to be derived, in accordance with normal mathematical notation. Note that here there is no need to state the variable with respect to which we are differentiating, because WIRIS will identify it automatically. If this operator is applied to an expression with more than one variable, an error is returned.

The ' sign can also be used to differentiate functions. In fact, if f=f(t) is a function of a single variable f' is the derivative of f with respect to t). Therefore, the derivative of f at point a is f'(a) in accordance with normal analysis notation. Let's look at some examples.

Integral calculus

To obtain the antiderivative or indefinite integral of a given function, we use the Icon or icons or the command integrate.

Clicking on the icon brings up the common expression for the indefinite integral with respect to a variable, including two empty green boxes. In the first box, enter in the expression to be integrated, and in the second enter the variable of integration. If f is the integrand, F is the indefinite integral and x is the variable of integration, it can be said that F is a primitive (or primitive function) of f and it can be verified that the derivative of F with respect to x is f.

Alternatively the integrate command can be used with two arguments, the first corresponding to the expression and the second to the variable.

If there is no doubt about the variable of integration, the indefinite integral of the function can also be obtained using the icon, . On clicking this icon, a symbol appears with an empty green box in which to enter the function to be integrated.

If the expression to be integrated has no variables, WIRIS will integrate it with respect to a made up variable. If there is only one variable, it will be integrated with respect to it, and if there is more than one, an error will be returned. In all cases, the result is a primitive function of the argument.

Alternatively, the integrate command can be used, with a single argument, instead of the icon; the command works exactly as described for the icon.

Definite integral

To calculate the definite integral between two limits, use the or icons or the command integrate. WIRIS attempts to calculate the integral of the function and apply Barrow's rule, which only requires evaluating the integral obtained at the limits of integration and then subtracting. If it cannot find the integral, it carries out the calculation using numerical methods and returns a warning message.

Upon clicking the , the standard definite integral symbol will appear with four green, empty boxes. Those which appear at the upper and lower extremes of the integral sign correspond to the upper and lower limits of integration, respectively. In the other two boxes, enter the expression to be integrated in the first and the variable of integration in the second.

Alternatively the integrate command can be used with four arguments. The first corresponds to the expression, the second to the variable and the third and fourth correspond to the lower and upper limits of integration, respectively.

Where there is no doubt about the variable of integration, the definite integrals of functions can be obtained using the icon . Upon clicking the icon, the standard definite integral symbol will appear with three empty boxes. Those which appear at the upper and lower extremes of the integral sign correspond to the upper and lower limits of integration, respectively. Enter the function or expression to be integrated in the third box. If the expression to be integrated has no variables, it will be integrated with respect to a made up variable. If there is only one variable, it will be integrated with respect to it, and if there is more than one, an error will be returned.

Alternatively the integrate command can be used with three arguments. The first corresponds to the function or expression to be integrated and the second and third correspond to the upper and lower limits of integration, respectively.

To calculate function limits, use the , or icons or the command limit.

Limit

Upon clicking the icon, the standard limit symbol will appear along with three green, empty boxes. In the upper box, to the right of lim enter the expression for which the limit should be calculated. In the lower boxes, enter the variable in the first box and the limit we wish the variable to approach in the second. If the limit command is used instead of the icon, the limit of function f when x approaches a can be entered using one of the following methods:

limit(f,x->a)
limit(f,x,a)
Notice that the icon allows us to create a symbol equivalent to: -> .

The value for a can be a real number, positive infinity (the icon), minus infinity (the icon), or unsigned infinity (the icon).

Horizontal limit

The and icons allow us to calculate the left and right limits, respectively. The parameters to be entered in the empty boxes are the same as for the icon .

To calculate right and left limits, use the command limit. To calculate the limit of function f when x approaches a from the right (or from the left), either of the two following expressions can be used:

limit(f,x->a,1) (from the left, limit(f,x->a,-1) )
limit(f,x,a,1) (from the left, limit(f,x,a,-1) )

WIRIS supports calculation of the Taylor Series of a real function at a given point.

To calculate the Taylor Series of a function at a given point, use the taylor_series command with three arguments. The first argument corresponds to the function, the second to the variable and the third corresponds to the value for which the Taylor Series should be calculated (remember that the Taylor Series permits us to approximate the value of any function at a given point). If you would like to view a specific number of terms of the series, which is infinite, specify this number in the fourth argument.

In order to obtain the Taylor polynomial of a given order for a particular function, use the taylor command, followed by the four arguments just described. Note, the fourth argument is necessary for this purpose.

WIRIS allows us to determine whether a series is convergent, as well as calculating the sum of a convergent series.

Use standard mathematical notation to express the series, as shown in the following examples. The response will give the value for the sum of the series if it is convergent (or if it is divergent, but WIRIS knows how to calculate the relevant infinite value), and the series itself in other cases.

To ask WIRIS about whether a series is convergent or divergent, use the convergent? command, and enter the series as the only argument.

See the solve command to find the exact solutions of a differential equation.

vector fields:  command vector_field

Vector fields in the plane can be used to analyze first degree differential equations. We can plot such a vector field by using the vector_field command.

integral curves:  command integral_curves

It draws a sample of solutions of a differential equation, which can be plotted over its vector field.

integral curve:  command integral_curve

Calculates a particular solution of a differential equation.