Maximum Rate of Change

Calculus 3 · Axiom Academy

Standing on a hill, which way is steepest? One vector — the gradient — points that way, and tells you exactly how steep it is. Imagine a hill whose height above the point (x,y) is f(x,y)=100-x^2-y^2 — a smooth dome with its summit right over the origin. From wherever you stand, you could walk in any direction, and each one climbs at a different rate. There is exactly one direction that climbs fastest, and calculus hands it to you as a single vector: the gradient . Watch the gradient arrow grow at the marked point. Notice two things it does at once: it points straight toward the summit (the steepest way up), and it sits at a right angle to the ring it starts on. Its length is the maximum rate of climb, . The gradient is the arrow of steepest ascent — and it always crosses the level rings at a right angle. Every direction has its own rate of climb Fix the point and spin the direction. The blue arrow is the way you choose to walk (a unit step ); the teal arrow is the fixed gradient. The rate you climb is their dot product, — biggest when you point the same way as the gradient, zero when you walk along a ring, negative downhill. At its peak the rate equals — the maximum rate of change, reached only along the gradient. Always at a right angle to the rings

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