The following simulation in Fig.6.1 is a plotter for arbitrary functions $y=f\left(x\right)$. From a selection menu you can choose preset functions, that can be changed. You can also enter totally new analytic functions.

The original function itself is shown in red. With option switches you can call several functions to calculate and display them: inverse function, first derivative, second derivative and integral.

The derivatives are calculated in secant approximation , the integrals in the parabolic approximation.

inverse function : x = g(y).

This involves the problem of finding for a given function $y=f\left(x\right)$ for a certain image variable $y$ the preimage $x$. Graphically this corresponds to a reflection of $y=f\left(x\right)$ on the angle bisector $y=x$ or swapping of $x$ and $y$. This line is shown in the corresponding plot. In the following picture this is shown for a polynomial of fifth order with three zeros. The angle bisector is shown in gray, the inverse function in light brown. The plot of this function is an example for the situation where the function $y=f\left(x\right)$ is unique, each ${x}_{i}$ is mapped to exactly one ${y}_{i}$, but the inverse function is not unique, i.e. there are many ${y}_{i}$ for which three ${x}_{j}$ exist.

first derivative: $y=\frac{d}{dx}f\left(x\right)$ is shown in magenta.

second derivative: ${y}^{\u2033}=\frac{{d}^{2}}{d{x}^{2}}f\left(x\right)=\frac{d}{dx}{y}^{\prime}$ is shown in green.

The integral $\underset{{x}_{min}}{\overset{x}{\int}}f\left(x\right)dx$ with adjustable initial value ${I}_{0}$ for ${x}_{min}$, is shown in blue.

When calculating the integral you have to remember, that the calculation starts at ${x}_{0}$ with and initial value ${I}_{0}$. The variable region and the initial value must be chosen in such a way, that the integral curve stays in the window.

Fig.6.2 shows the Gaussian $y={e}^{-{x}^{2}}$ with inverse function, first and second derivative and integral.

The command panel allows for up to three parameters $a,b$ and $c$ a continuous variation and for a fourth parameter a choice of integers.

With the coloured option boxes the inverse function, 1. and 2. derivative and integral can be shown or suppressed.

The presentation is for abscissa and ordinate symmetric to the origin. In the first white field on the bottom you can adjust the variable range $-{x}_{max}<x<{x}_{max}$ by hand, in the second field the $y$-range and in the the third the initial value of the integral for $-{x}_{max}$. If the symmetric range of variation is not sufficient for your function, you can increase or decrease it via entering factors in the formulas.

As usual the window can be pulled to full screen size and after marking a point you may read off its coordinates on the lower boundary of the graphic.

The preset functions are

functions constant $p$-th power, $p>0$ and integer $b$-th power, (b rational; x $>$ 0) sine cosine sine with three parameters cosine with three parameters power of sine power of cosine tangent with three parameters exponential function exponential decay natural logarithm hyperbolic sine hyperbolic cosine hyperbolic tangent Gauss distribution with three parameters $\frac{sinx}{x}$ ${\left(\frac{sinx}{x}\right)}^{2}$ | formulas in Java-syntax $a$ a*xˆp a*xˆb sin(x) cos(x) a*sin(b*x + c) a*cos(b*x + c) sin(a*x)ˆp cos(a*x)ˆp a*tan(b*x + c) a*exp(x/b) a*exp(-x/b) ln(x/a) (exp(a*x) - exp(-a*x))/2 (exp(a*x) + exp(-a*x))/2 (exp(a*x) - exp(-a*x))/(exp(a*x) + exp(-a*x)) a*exp(-b*(x-c)ˆ2 sin(a*x)/(a*x) (sin(a*x)/(a*x))ˆ2 |

In the simulation you may change the preset functions or enter new formulas from scratch.