Question

Use the function $$f(x, y)=3-\frac{x}{3}-\frac{y}{2}$$ Find the maximum value of the directional derivative at (3,2) .

Short answer

The maximum value of the directional derivative at point (3, 2) is \(||\nabla f||\).

Step-by-step solution

Find the Gradient

The gradient of a function can be computed by taking the derivatives regarding to the variables x and y. For the given function \(f(x, y) = 3 - \frac{x}{3} - \frac{y}{2}\), we obtain the derivatives: \[f_x = -\frac{1}{3}\] and \[f_y = -\frac{1}{2}\] Hence, the gradient vector of the function is \[\nabla f = (f_x, f_y) = (-\frac{1}{3}, -\frac{1}{2})\]

Compute the Unit Vector

The unit vector or the direction vector needs to be calculated to find the direction in which the function increases most rapidly. It can be calculated using the formula: \[\hat{u} = \frac{\nabla f}{||\nabla f||}\] Here, ||\nabla f|| is the magnitude of the gradient vector, so \[||\nabla f|| = \sqrt{sum([(-\frac{1}{3})^2 + (-\frac{1}{2})^2])} = \sqrt{\frac{1}{9} + \frac{1}{4}}\] Now, calculate the unit vector: \[\hat{u} = \frac{(-\frac{1}{3}, -\frac{1}{2})}{||\nabla f||}\]

Find the Directional Derivative

The directional derivative in the direction of the unit vector \(\hat{u}\) is defined as the dot product of the gradient of the function and the unit vector. This is given by: \[D_u f = \nabla f \cdot \hat{u}\] and it represents the rate of change of the function in the direction of the vector \(\hat{u}\). Substitute the values we previously obtained for \(\nabla f\) and \(\hat{u}\) into the formula and calculate the dot product.

Find the Maximum Value

The maximum value of the directional derivative is obtained when the direction vector and the gradient vector go in the same direction. The dot product of both vectors gives the directional derivative, but in order to maximize this value, the direction vector must be equal to the unit gradient vector. Hence, the maximum value of the directional derivative is equal to the magnitude of the gradient vector, or \(||\nabla f||\). Simply calculate this value and we have the maximum value of the directional derivative.