Question

In your own words, give a geometric description of the directional derivative of \(z=f(x, y)\).

Short answer

Geometrically, the directional derivative of a function \(z=f(x, y)\) at a point \((x, y)\) in the direction of a unit vector \(u\) provides the rate of change (slope) of the function in the direction of the vector \(u\).

Step-by-step solution

Define the Directional Derivative

The directional derivative of a function \(z=f(x, y)\) in the direction of a unit vector \(u = \) is a derivative that measures the rate at which the function changes at a point in the direction of that vector. Mathematically, it is defined as \(D_{u}f(x, y) = f_{x}(x, y)a + f_{y}(x, y)b\).

Describe the Geometric Interpretation

Geometrically, the directional derivative of \(z=f(x, y)\) at a point \((x, y)\), in the direction of a unit vector \(u\), provides the slope of the function at that point in that specific direction. More intuitively, it measures 'how fast' the function is increasing or decreasing at point \((x, y)\) if you start moving in the direction of the given vector \(u\). This is essentially the projection of the gradient vector \(\nabla f\) onto \(u\).