91Ó°ÊÓ

A partial derivative represents how a function changes as one of the variables changes while others are kept constant. In multivariable calculus, when you have a function like \( z = f(x, y) \), the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \), tells us how \( z \) changes as \( x \) changes, keeping \( y \) fixed.

To calculate the partial derivative \( f_x(x, y) \) of our function \( f(x, y) = \sin(x + y^2) \), we treat \( y^2 \) as a constant and differentiate \( \sin(x + y^2) \) with respect to \( x \). This process gives us \( f_x(x, y) = \cos(x + y^2) \).

Similarly, the partial derivative concerning \( y \), \( f_y(x, y) \), involves differentiating the function while treating \( x \) as a constant. For \( f(x, y) = \sin(x + y^2) \), it results in \( f_y(x, y) = 2y \cos(x + y^2) \). Calculating these derivatives helps understand how the function behaves when making small changes in \( x \) or \( y \).

To visualize a function of two variables, such as \( z = \sin(x + y^2) \), we graph it as a surface in 3D space. Here, \( x \) and \( y \) serve as the horizontal axes, while \( z \) is the vertical dimension.

Graphing helps us understand the structure of the function better, as it reveals features like peaks, valleys, and slopes. In our specific example, the surface resembles a wavy pattern due to the sine function, with variations dependent on both \( x \) and \( y^2 \).

Once we incorporate the tangent plane \( z = \frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}\pi}{8} \), we can observe where the plane touches the surface at a single point. This intersection gives insight into the local linear approximation of the surface at that point and helps verify the correctness of our calculations.

The equation of a plane in 3D space typically takes the form \( ax + by + cz = d \), where \( a \), \( b \), \( c \), and \( d \) are constants. For tangent planes to a surface, we use a specific form derived from the partial derivatives:

Substitute calculated partial derivatives and the point of tangency into the formula for a tangent plane:
\[z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0).\]

In the problem, this formula gave us the equation \( z = \frac{\sqrt{2}}{2}x + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}\pi}{8} \).

This equation represents the plane touching the surface \( z = \sin(x + y^2) \) at the point \( (\frac{\pi}{4}, 0, \frac{\sqrt{2}}{2}) \). The tangent plane can be regarded as a flat approximation of the surface at a given point and is particularly useful in optimization and linear approximation problems.