Finding Peaks and Valleys

Calculus 3 · Axiom Academy

Every surface has high points, low points, and one sneaky in-between. Here is how calculus spots all three. The moment a surface goes flat Picture a landscape whose height above every point is a function z = f(x, y) — hills, basins, ridges. Its summits and its lowest hollows share a hidden signature: right at the very top of a hill, or the very bottom of a valley, the surface is momentarily flat . Find that flatness and you have found the peak. Watch the tangent plane ride up the hill. Out on the slope it tilts along with the terrain — but as it reaches the summit it settles perfectly level. That horizontal tangent plane is the exact signal calculus hunts for. A level tangent plane means both slopes vanish at once: and . A contour map is that same surface seen from directly above — each ring a line of constant height, tightly packed where the ground is steep. Drag the marker around. The arrow is the gradient : it points straight uphill, and its length is how steep the climb is. Go hunting for the spots where it shrinks away to nothing. A critical point is exactly where the gradient vanishes: — every peak and every valley lives there. Not every flat spot is a peak or a valley. A mountain pass is level too — the highest point along the trail, but the lowest point across it. Here is the cleanest one, z = x^2 - y^2 . Sweep the direction of the slice and watch its cross-section flip from a valley to a ridge and back.

This is the written version of the interactive lesson above. See the full Calculus 3 course.