Importance of Gradients: Maximizing the Directional Derivative
Suppose we have a function f of two or three variables and we
consider all possible directional derivatives of f at a given point.
These give the rates of change of f in all possible directions. We can
then ask the questions: in which of these directions does f change
fastest and what is the maximum rate of change? The answers are provided
by the following theorem.
Theorem Suppose f is a differentiable function of
two or three variables. The maximum value of the directional derivative
\(D_{\mathbf{u}}f(\mathbf{x})\) is
\(|\nabla f(\mathbf{x})|\) and it
occurs when u has the same direction as the gradient
vector \(\nabla f(\mathbf{x})\).
where \(\theta\) is the angle
between \(\nabla f\) and
u. The maximum value of \(\cos\theta\) is 1 and this occurs when
\(\theta = 0\). Therefore the maximum
value of \(D_{\mathbf{u}}f\) is \(|\nabla f|\) and it occurs when \(\theta = 0\), that is, when
u has the same direction as \(\nabla f\).
EXAMPLE (a) If \(f(x, y) =
xe^y\), find the rate of change of f at the point P(2, 0) in the
direction from P to Q(1/2, 2). (b) In what direction does f have the
maximum rate of change? What is this maximum rate of change?
SOLUTION (a) We first compute the gradient
vector:
The unit vector in the direction of \(\vec{PQ} = \langle -3/2, 2 \rangle\) is
\(\mathbf{u} = \langle -3/5, 4/5
\rangle\), so the rate of change of f in the direction from P to
Q is
According to Theorem 15, f increases fastest in the direction of the
gradient vector \(\nabla f(2, 0) = \langle 1,
2 \rangle\). The maximum rate of change is
EXAMPLE 7 Suppose that the temperature at a point
(x, y, z) in space is given by \(T(x, y, z) =
80/(1 + x^2 + 2y^2 + 3z^2)\), where T is measured in degrees
Celsius and x, y, z in meters. In which direction does the temperature
increase fastest at the point (1, 1, -2)? What is the maximum rate of
increase?
By Theorem 15 the temperature increases fastest in the direction of
the gradient vector \(\nabla T(1, 1, -2) =
\frac{5}{8}(-\mathbf{i} - 2\mathbf{j} + 6\mathbf{k})\) or,
equivalently, in the direction of \(-\mathbf{i} - 2\mathbf{j} + 6\mathbf{k}\)
or the unit vector \((-\mathbf{i} -
2\mathbf{j} + 6\mathbf{k})/\sqrt{41}\). The maximum rate of
increase is the length of the gradient vector: