91Ó°ÊÓ

Partial derivatives are a fundamental concept in calculus. They are used to find the rate at which a function changes with respect to one variable, while keeping others constant.
When dealing with functions of multiple variables, such as in our exercise, partial derivatives help us understand these changes.To find the partial derivatives for a function like \[ f(x, y, z) = x^3 + y^3 - 3xyz \]we calculate the derivative with respect to each variable individually, treating the other variables as constants at each step.
For our function, the partial derivatives are:

• \( f_x = 3x^2 - 3yz \)
• \( f_y = 3y^2 - 3xz \)
• \( f_z = -3xy \)

These derivatives are essential for finding the gradient vector, which leads us to the tangent plane.

The gradient vector plays a crucial role when determining planes tangent to surfaces.
It can be seen as a vector containing all partial derivatives of a function. Denoted by \( abla f \), it guides us in finding the normal vector for the tangent plane.For the given function, our previously calculated partial derivatives are combined into the gradient vector:\[abla f = (f_x, f_y, f_z) = (3x^2 - 3yz, 3y^2 - 3xz, -3xy)\]This vector points in the direction where the function increases most rapidly. When evaluating this gradient at a specific point, such as \( (1, 2, \frac{3}{2}) \), we derive the components of the normal vector of the tangent plane.
The result is the vector \((-6, \frac{15}{2}, -6)\). This vector is perpendicular to the surface at the given point.

The normal vector is an essential component when dealing with tangent planes.
It represents the direction that is perpendicular to the surface at a specific point. In our case, this normal vector was obtained by evaluating the gradient vector at point \( (1, 2, \frac{3}{2}) \).The calculation gave us the normal vector:\[(-6, \frac{15}{2}, -6)\]This vector tells us that any plane parallel to it is parallel to the given surface.
We use this vector in the equation of the tangent plane to ensure it is perpendicular to any vectors lying on the plane.
Ultimately, it helps us define the exact position and orientation of the tangent plane in space.

Simplifying an equation is an important step that makes it more understandable and usable.When we formed the equation of the tangent plane using the normal vector and the reference point, we initially got:\[-6(x - 1) + \frac{15}{2}(y - 2) - 6(z - \frac{3}{2}) = 0\]Simplification involves expanding and combining like terms, which reduces complexity and eliminates fractions.
By simplifying, we make it easier to handle problems where solving is involved.After simplification, our equation becomes:\[-6x + \frac{15}{2}y - 6z = 0\]Further simplification, such as multiplying the whole equation by 2, can eliminate fractions, resulting in:\[-12x + 15y - 12z = 0\]This form is often preferable, especially in subsequent calculations or interpretations.