Directional Derivative

The directional derivative of a function at a given point measures the rate at which the function changes along a specific direction. It provides information about how the function varies with respect to that direction.

Given Function

The given function is f(x, y) = xey. This function represents a surface in the Cartesian coordinate system. The value of the function at any point (x, y) on this surface is obtained by multiplying the x-coordinate by the exponential of the y-coordinate.

Direction of the Straight Line Segment

The problem asks for the directional derivative along the direction of the straight line segment connecting the points (2, 0) and (1/2, 2). To find this direction, we need to calculate the direction vector of the line segment.

The direction vector of a line segment between two points (x1, y1) and (x2, y2) is given by (x2 - x1, y2 - y1). In this case, the direction vector is ((1/2) - 2, 2 - 0) = (-3/2, 2).

Calculating the Directional Derivative

To calculate the directional derivative of f(x, y) along the direction vector (-3/2, 2) at the point (2, 0), we need to use the gradient of the function.

The gradient of a function f(x, y) is given by (∂f/∂x, ∂f/∂y), where ∂ denotes partial differentiation.

For the given function f(x, y) = xey, the partial derivatives are:

∂f/∂x = ey
∂f/∂y = xey

At the point (2, 0), these partial derivatives become:

∂f/∂x = e^0 = 1
∂f/∂y = 2e^0 = 2

Now, we can calculate the directional derivative using the formula:

D_v(f) = (∂f/∂x, ∂f/∂y) · (v/|v|)

where · denotes the dot product and v/|v| represents the unit vector of the direction vector v.

In our case, v = (-3/2, 2). To find the unit vector of v, we divide v by its magnitude:

|v| = sqrt((-3/2)^2 + 2^2) = sqrt(9/4 + 4) = sqrt(25/4) = 5/2

v/|v| = (-3/2, 2) / (5/2) = (-3/5, 4/5)

Plugging in the values, the directional derivative becomes:

D_v(f) = (1, 2) · (-3/5, 4/5) = (1)(-3/5) + (2)(4/5) = -3/5 + 8/5 = 5/5 = 1

Therefore, the directional derivative of the function f(x, y) = xey at the point (2, 0) along the direction of the straight line segment from (2, 0) to (1/2, 2) is