91Ó°ÊÓ

Partial derivatives are a fundamental concept in calculus, essential for understanding how functions behave when more than one variable is involved. They represent how the function changes in relation to one variable while keeping other variables constant.
In the given function, \(f(x, y) = 3x - 5y\), we can find partial derivatives with respect to \(x\) and \(y\), denoted as \(f_x\) and \(f_y\). For \(f_x\), differentiate \(f\) treating \(y\) as a constant:

• \(f_x = \frac{\partial}{\partial x}(3x - 5y) = 3\)

Similarly, for \(f_y\), treat \(x\) as a constant:

• \(f_y = \frac{\partial}{\partial y}(3x - 5y) = -5\)

By evaluating these derivatives at the point \(P(4,2)\), you obtain the rates of change of the function in both the \(x\) and \(y\) directions at that point:

• \(f_x(4,2) = 3\)
• \(f_y(4,2) = -5\)

Partial derivatives are crucial for constructing parametric equations for tangent lines to surfaces, which help in understanding the geometry of the function.

Parametric equations are used to describe a path of points in a coordinate system. Instead of relying on a single equation in terms of x and y, parametric equations express each coordinate as a separate function of an independent variable, usually represented by \(t\).
For directional tangent lines to the function \(f(x, y)\) at point \(P(4,2)\), we construct the parametric equations \(\ell_x(t)\) and \(\ell_y(t)\), which move horizontally and vertically from \(P\), respectively.

• For \(\ell_x(t)\): Move along the x-axis while keeping the y-coordinate constant. \[\ell_x(t) = (4 + t, 2, f(4,2) + 3t) = (4 + t, 2, 2 + 3t)\]
• For \(\ell_y(t)\): Move along the y-axis while keeping the x-coordinate constant. \[\ell_y(t) = (4, 2 + t, f(4,2) - 5t) = (4, 2 + t, 2 - 5t)\]

These parametric forms trace directions of change, showing how the function behaves right from the point \(P\) along specified axes.

A unit vector is a vector with a magnitude (or length) of one. It indicates direction and is often used in directional derivatives, where the change in a function is measured in a specific direction.
To find the unit vector \(\vec{u}\) in the direction of a given vector \(\vec{v}\), we first calculate the magnitude of \(\vec{v}\). For \(\vec{v} = \langle 1, 1 \rangle\), the magnitude is calculated as:

• \[\|\vec{v}\| = \sqrt{1^2 + 1^2} = \sqrt{2}\]

The unit vector \(\vec{u}\) is obtained by dividing each component of \(\vec{v}\) by its magnitude:

• \[\vec{u} = \left\langle \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right\rangle\]

Using this unit vector, we can build the parametric equation \(\ell_{\vec{u}}(t)\), which describes the line moving through point \(P\) in the direction of \(\vec{v}\). This helps understand how the function changes in that specific vector direction.