E1: Select the antiderivative for the normalized Gaussian. Change its width (a) and position (b) and observe that the end value of the antiderivative is always 1, as long as the initial value at -xmax is sufficiently close to zero. If the Gaussian is understood as a statistical distribution of some value, what is the relevance of its normalization, its center and its width?
E2: Choose the Gaussian with additive noise. Vary the noise level and watch the antiderivative. Explain what you observe in general, and with the antiderivative specifically.
E3: As E2, but observe the first derivative. You will have to enlarge the ordinate range to fully see it. Do the same for the second derivative.
E4: Explain why integration smoothes a fluctuating function, while differentiation exaggerates its fluctuations. Do you see technical consequences in processing signals?
E5: Study the modulation schemes. Increase the carrier frequency (b) to have a more realistic relation of carrier frequency / modulation frequency. Consider how you could retrieve the modulation frequency from the modulated signal.
E6: Study Planck´s formula. Vary the temperature (b) and determine the relation of the spectral maximum and of the integral on temperature (increase xmax and ymax to observe the whole range; adjust amplitude (a) accordingly).
E7: Vary Planck´s formula and study the physical consequences. Remember that Planck in a way guessed his formula experimentally, starting with the knowledge of the limit cases at very small and very large wavelengths.
E8: Input your own formulas and adjust scales to have physically relevant ranges.