<jupyter_start><jupyter_text>Title of investigation Mathematical model Want to understand $f(t) = \exp(-\alpha t) \cos(\omega t)$ Code / Data<jupyter_code>def f(t, alpha, omega):
    """Computes and returns exp(-alpha*t) * cos(omega*t)"""
    return exp(-alpha * t) * cos(omega * t)<jupyter_output><empty_output><jupyter_text>Interactive explorationWe can execute the function for valuel of $\alpha$ and $\omega$:<jupyter_code>f(t=0.1, alpha=1, omega=10)
f(t=0.0, alpha=1, omega=10)<jupyter_output><empty_output><jupyter_text>Although sometimes a plot is more instructive:<jupyter_code>def plot_f(alpha, omega):
    ts = linspace(0, 5, 4000)    # many points in the interval [0, 5]
    ys = f(ts, alpha, omega)
    pylab.plot(ts, ys, '-')
plot_f(alpha=0.1, omega=20)
plot_f(alpha=0.5, omega=100)<jupyter_output><empty_output><jupyter_text>A wide range of convenience tools, for example graphical interaction elements called *Widgets*:<jupyter_code>plot_f(alpha=0.8, omega=17)
interact(plot_f, alpha=(0, 2, 0.1), omega=(0, 50, 0.5))<jupyter_output><empty_output>
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