Tangent Planes and Linear Approximation
Replace a surface near a point by the plane its partials define, and use that plane to estimate values and propagate small errors.
What a learner can do afterwards
- Write the equation of the tangent plane at a point on a surface
- Estimate a function value near a known point with the linear approximation
- Give a function that has both partials at a point but no tangent plane
1 · Read
The tangent plane is the flat surface that best hugs a smooth graph near a point. Its equation is z = f + fx(x minus a) + fy(y minus b). For z = x squared + y squared at (1, 2): height 5, fx = 2, fy = 4, so z = 5 + 2(x minus 1) + 4(y minus 2).
Dropping the z gives the linear approximation L, a quick estimator for nearby values. With L above: L(1.1, 1.9) = 5 + 0.2 minus 0.4 = 4.8, and L(0.9, 2.1) = 5 minus 0.2 + 0.4 = 5.2. The rhythm is always evaluate, differentiate, assemble.
Each partial tilts the plane in its own direction, and (fx, fy, minus 1) points straight out of it. Here that normal is (2, 4, minus 1). If you can read the normal, you can check your plane in one glance.
A subtle point: partials alone do not guarantee the plane fits. Wild functions can have both slopes on the axis slices but no true tangent. Only trust the plane on a smooth surface near its base point.
Build the plane from height plus two slopes, use it for nearby estimates, and trust it only when the surface is smooth.
More to read
4.4 Tangent Planes and Linear Approximations
The tangent plane is the flat surface that best hugs a smooth graph near a point, built from the two partials there. Its equation is z = f(a, b) + fx(a, b)(x minus a) + fy(a, b)(y minus b), which is worth memorizing cold. Dropping the z gives the linear approximation L, a quick estimator for nearby values. A subtle point: partials alone do not guarantee the plane fits, since wild functions can have slopes but no true tangent. Work one full example by hand, then reuse its L for estimates.
Tangent planes
This article walks the plane formula with pictures of the surface and its flat approximation touching at one point. Use it to see why each partial controls tilt in its own direction. The examples show the same evaluate, differentiate, assemble rhythm every time. Compare the approximated values with the true ones to feel the error shrink near the base point. Keep this page open while doing the estimation items.
Quadratic approximation
Quadratic approximation adds the second order terms on top of the tangent plane for a closer fit. Read it to see where the linear estimate sits in the bigger picture: plane first, bowl correction second. The extra terms use the second partials you already know. You will not need quadratics for the items, but the contrast shows exactly what the plane leaves out. Finish by naming which term dominates very close to the base point.
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2 · Watch
Take it off screen
Where it sits
Learn first
Optional connections
This opens up
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.