Tangent Planes to Surfaces

Calculus 3 · Axiom Academy

LESSON Tangent Planes to Surfaces Understanding how to construct the tangent plane at a point on a surface using partial derivatives Consider a surface defined by z = f(x, y) and a point P_0 = (x_0, y_0, z_0) on that surface. The tangent plane at P_0 is the plane that "just touches" the surface there: it passes through P_0 , and in every direction you can leave P_0 , it leaves with exactly the surface's slope. 2. Building from Partial Derivatives The partial derivatives give us the slopes of the tangent plane in two key directions: Hold y = y_0 constant and walk in the x -direction. The surface cuts out a curve, and f_x(x_0,y_0) is that curve's slope at P_0 . Hold x = x_0 constant and walk in the y -direction. The slope of that curve at P_0 is f_y(x_0,y_0) . These two slopes determine the orientation of the tangent plane. Think of them as the "tilt" in each direction: dial the x -tilt until it matches f_x(x_0,y_0) , dial the y -tilt until it matches f_y(x_0,y_0) , and the plane through P_0 is completely pinned down — there is exactly one. Using these slopes, we can write the equation of the tangent plane. Start at the height z_0 = f(x_0,y_0) and step out to a nearby point. Moving in the x -direction changes the height by ; then moving changes it by . Stack the two contributions on the starting height and you land exactly on the plane. 4. Connection to Linear Approximation

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