Setup: a surface and one cut through it. Let them predict first.
Reveal: treat y as a number and differentiate in x: slope 2.456 at x = 1.2.
Freeze y, then use the product rule on x squared times ln x.
Setup: four slices at different y. Let them predict first.
Reveal: 2 ln x + 1 = 0, so x = e^(-1/2) = 0.6065 for every y (y not 0).
Freeze x instead and the slice is a straight line in y.
Setup: the log-density as a surface in x and beta, with three slices. Let them predict first.
Reveal: peaks at beta = 1, 1.5, 2 for x = 2, 3, 4: the ridge beta = x/2.
Seen from above, each x has its own best beta: the ridge is the MLE as a function of the data.
Five points. Then go and do the drills by hand.