91Ó°ÊÓ

When you encounter a multivariable function, like the one in our exercise, understanding how each variable affects the overall function requires calculating partial derivatives. Partial derivatives are like the regular derivatives from single-variable calculus, but they focus on one variable at a time, while keeping the others constant.

For the function given in the exercise, \( f(x, y) = x^2y - xy^2 \), we first need to find the partial derivatives with respect to \( x \) and \( y \).

• Partial derivative with respect to \( x \): This measures how the function changes as \( x \) changes, while \( y \) remains fixed. Here, it is calculated as \( \frac{\partial f}{\partial x} = 2xy - y^2 \).
• Partial derivative with respect to \( y \): This tells us how the function changes with respect to \( y \), with \( x \) constant. For this function, it is \( \frac{\partial f}{\partial y} = x^2 - 2xy \).

These partial derivatives are crucial as they form the components of the gradient vector, which points in the direction of the steepest increase of the function.

The concept of a tangent plane arises in multivariable calculus when you need to approximate a surface at a certain point with a flat plane. It essentially touches the surface at that point and mirrors its slope and incline as closely as possible.

In our problem, once the gradient vector \( abla f(-2, 3) = (-21, 16) \) is evaluated, we use it to construct the equation of the tangent plane. The formula for the tangent plane to a surface \( z = f(x, y) \) at point \( (x_0, y_0) \) is:
\[ z - f(x_0, y_0) = \frac{\partial f}{\partial x}(x - x_0) + \frac{\partial f}{\partial y}(y - y_0) \]

By substituting the given values and simplifying, we derived the tangent plane's equation as \( z = -21x + 16y - 96 \). It tells us how the plane touches and approximates the surface of the function at point \((-2, 3)\). This plane can be used for linear approximations and predictions close to that point.

Multivariable calculus extends the principles of calculus to functions of several variables. This opens a path to study how different variables collectively affect a system, allowing for a richer understanding of changes and behaviors in more complex settings.

In multivariable calculus, key tools like the gradient vector and tangent planes become essential:

• The gradient vector combines partial derivatives, offering insights into how a function changes across multiple dimensions. It's a vector pointing towards the direction of greatest increase in function value, symbolized by \( abla f \).
• Tangent planes serve a similar role to tangent lines in single-variable calculus. They approximate surfaces at a point, providing a useful tool for predicting values and understanding local behavior.

This branch of calculus allows us to solve real-world problems in areas like physics and engineering, where multiple forces and variables interact. In our exercise, computing these elements enabled us to grasp the underlying surfaces and directions of increase, offering a layer of understanding that supports solving more intricate equations and models.