91Ó°ÊÓ

A unit vector is a vector with a magnitude of 1. It is often used to indicate direction without magnitude. In our problem, the unit vector \( \mathbf{u} \) is given as \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \). This means it points diagonally in a direction that splits equally between the x and y axes. To ensure that a vector is indeed a unit vector, it should satisfy the condition:

• \( |\mathbf{u}| = \sqrt{u_x^2 + u_y^2} = 1 \)

For \( \mathbf{u} = \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \):

• \( |\mathbf{u}| = \sqrt{ \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 } = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1 \)

This confirms our vector is a unit vector. Unit vectors are essential in directional derivatives to specify the exact direction in which the derivative is taken.

The concept of a limit is fundamental in calculus and describes the value that a function approaches as the input approaches a certain value. Limits can be incredibly important for understanding behavior near specific points, especially when these points cannot be evaluated directly. In the context of the directional derivative, the limit \( \lim_{h \rightarrow 0} \frac{f(x + hu_x, y + hu_y) - f(x, y)}{h} \) describes how the function \( f(x, y) \) changes as we take infinitesimally small steps in a given direction given by the unit vector \( \mathbf{u} \). By calculating this limit, we find the instantaneous rate of change of the function in the specified direction. The concept of limits ensures that derivatives are defined accurately and precisely. They allow us to rigorously deal with points in functions where direct substitution might not work or where a physical approximation is useful.

Function differentiation involves finding the derivative of a function - which tells us how the function changes. In general, the derivative measures the rate of change of a quantity.Differentiation can occur in various forms, including partial differentiation when managing functions of multiple variables.For the function \( f(x, y) = 3x - y^2 \), partial derivatives are:

• With respect to \( x \): \( \frac{\partial f}{\partial x} = 3 \)
• With respect to \( y \): \( \frac{\partial f}{\partial y} = -2y \)

In functions of two variables, partial derivatives describe how the function changes when one variable changes while the other remains constant. This approach, when combined with the direction defined by the unit vector, helps to compute the directional derivative. Differentiation, in this way, allows us to understand not merely where a function is but how it is moving or changing at any given point.

The directional derivative is a way to measure how a function changes as you move in the direction of a given vector. The formula used for the directional derivative of a function \( f \) at a point \((x, y)\) in the direction of a unit vector \( \mathbf{u} = (u_x, u_y) \) is:

• \( D_{\mathbf{u}} f(x, y) = \lim_{h \rightarrow 0} \frac{f(x + hu_x, y + hu_y) - f(x, y)}{h} \)

In essence, this formula allows us to compute the rate at which the function \( f \) changes at a point \((x, y)\) in the specified direction of \( \mathbf{u} \).The directional derivative is beneficial in fields like physics and engineering for finding gradients or changes along specific paths or directions, which is crucial for optimization problems and analyzing vector fields. Applying this formula with careful substitution and simplifications enables us to effectively capture directional changes, as seen in the given exercise.