Question

Find the slope of the surface at the given point in (a) the \(x\) -direction and (b) the \(y\) -direction. $$ \begin{aligned} &z=x y \\ &(1,2,2) \end{aligned} $$

Short answer

The slope of the surface at the point (1,2,2) in the x-direction is 2 and in the y-direction is 1.

Step-by-step solution

Compute the Partial Derivatives

For a surface described by the equation \( z = f(x, y) \) where \( f(x, y) = xy \), calculate the partial derivatives: \n\n\( \frac{\partial z}{\partial x} = y \) \n\n\( \frac{\partial z}{\partial y} = x \)

Substitute the Values

Substitute the given point (1,2,2) into the partial derivatives. In the x-direction: substitute \(x = 1\), \(y = 2\) into \( \frac{\partial z}{\partial x} = y \) to get the slope in x-direction.\n\nIn the y-direction: substitute \(x = 1\), \(y = 2\) into \( \frac{\partial z}{\partial y} = x \) to get the slope in y-direction.

Final Result

So, the slope in the x-direction at the point (1,2,2) is \(2\), and the slope in the y-direction at the point (1,2,2) is \(1\).

Key concepts

Calculus

Calculus is a branch of mathematics that focuses on how things change. It's all about understanding movement and change, much like tracking a curve down a winding road. Calculus helps us dive into functions and their intricate behaviors, providing profound insights into shapes, areas, slopes, and curves. The two main branches of calculus are differentiation and integration. Differentiation focuses on finding the rate of change or the slope of a curve at any point, while integration deals with accumulating quantities, such as areas under a curve.

Using calculus, we can solve problems involving motion, growth, decay, and many natural phenomena. Its tools and methods, like derivatives and integrals, are fundamental for sciences, engineering, economics, and many other fields. For instance, derivatives, a core part of differential calculus, measure how a function's output value changes as its input changes, giving us insights into the behavior and rate of change in diverse situations.

Slope of Surface

The slope of a surface is a concept that tells us how steep a surface is at a particular point. In mathematics, especially when dealing with functions of two variables, the slope can be understood through partial derivatives. Partial derivatives help measure how the surface changes as we move in specific directions, such as the x-direction or the y-direction.

For instance, consider a function like \(z = xy\). To find how the surface defined by this function changes, we calculate its partial derivatives:

• \(\frac{\partial z}{\partial x} = y\): This tells us the slope of the surface when moving along the x-direction, keeping y constant.
• \(\frac{\partial z}{\partial y} = x\): This informs about the slope when moving in the y-direction, keeping x fixed.

The partial derivatives provide an analytical way to understand the surface's slope at any point, making them essential for studying multivariable functions.

Multivariable Functions

Multivariable functions involve two or more variables and model how these variables interact with one another within one function. These functions extend single-variable functions into higher dimensions, adding complexity and richness to their behavior.

Consider a function \(z = f(x, y)\) that incorporates variables \(x\) and \(y\). Here, \(z\) represents the output while \(x\) and \(y\) represent the input variables. Such a function defines a surface in three-dimensional space. Multivariable functions are pivotal in understanding phenomena where variables are interdependent such as economics for modeling market behaviors, physics for simulating forces and fields, or any setting where outcomes depend on multiple conditions.

These functions are analyzed using partial derivatives to comprehend how change in one variable affects the function while keeping other variables constant. This technique provides a detailed picture of how the function behaves in a multi-dimensional setting, crucial for fields that rely heavily on optimization and modeling.