Total estimated height change=estimated height change from Δx +estimated height change from Δy

You may have come across the following formula and wondered what it means geometrically:

Estimated height change = fx(a, b)Δx + fy(a, b)Δy

You’re in the right place. This page visualizes an example in which the two terms cancel each other, so the estimated height change is 0. In general, each term can independently be positive, negative, or zero.

If you’re interested and know how to program or vibe code, you can download the Python script and explore different cases yourself by changing the starting point and the increments Δx and Δy. The script requires Python with NumPy, Matplotlib, and Tkinter.

Visualization (Drag to rotate)

Surface: z = x² + y² · Start = (0.5, 0.5) · End = (0.8, 0.2) · Δx = 0.3, Δy = −0.3
Step 1: (0.5, 0.5) → (0.8, 0.5) · Step 2: (0.8, 0.5) → (0.8, 0.2)
The surface reaches height 0.68 at the endpoint; the tangent plane predicts 0.5. Each coordinate step contributes 0.09 to the total error of 0.18.

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Conclusion: height change = 0

The orange and green vectors have equal heights (0.3) and opposite directions. Their height contributions cancel, so the tangent plane predicts no height increase. The actual surface rises by 0.18.