Question

find the gradient vector of the given function at the given point \(\mathbf{p}\). Then find the equation of the tangent plane at \(\mathbf{p}\) \((\) see Example 1). $$ f(x, y)=x^{3} y+3 x y^{2}, \mathbf{p}=(2,-2) $$

Short answer

The gradient vector is \\( (-12, -16) \\\) and the tangent plane is \\( z = -12x - 16y - 8 \\\).

Step-by-step solution

Differentiate with respect to x

To find the gradient vector, first calculate the partial derivative of the function with respect to \( x \). Differentiate the function \( f(x, y) = x^3 y + 3xy^2 \) with respect to \( x \): \( \frac{\partial f}{\partial x} = 3x^2y + 3y^2 \).

Differentiate with respect to y

Next, calculate the partial derivative of the function with respect to \( y \). Differentiate \( f(x, y) = x^3 y + 3xy^2 \) with respect to \( y \): \( \frac{\partial f}{\partial y} = x^3 + 6xy \).

Evaluate at \( \mathbf{p} \)

Evaluate the partial derivatives at the point \( \mathbf{p} = (2, -2) \).For \( \frac{\partial f}{\partial x} \): \\( 3(2)^2(-2) + 3(-2)^2 = -24 + 12 = -12 \).For \( \frac{\partial f}{\partial y} \): \\( (2)^3 + 6(2)(-2) = 8 - 24 = -16 \).

Construct the gradient vector

The gradient vector, \( abla f \), at point \( \mathbf{p} \) is given by:\( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (-12, -16) \).

Set up the tangent plane equation

The formula for the equation of the tangent plane to the surface \( z = f(x, y) \) at a point \( (x_0, y_0, z_0) \) is:\( z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \).Given \( z_0 = f(2, -2) \), evaluate \( f(2, -2) = (2)^3(-2) + 3(2)(-2)^2 = -16 + 24 = 8 \).

Substitute into the tangent plane equation

Substitute \( (x_0, y_0, z_0) = (2, -2, 8) \) and \( (f_x, f_y) = (-12, -16) \) into the tangent plane equation:\( z - 8 = (-12)(x - 2) + (-16)(y + 2) \).Simplifying gives:\( z - 8 = -12x + 24 - 16y - 32 \rightarrow z = -12x - 16y - 8 \).

Key concepts

Partial Derivatives

Partial derivatives are like the building blocks of calculus for functions with more than one variable. Imagine you have a surface that looks like a rolling wave. To understand how it changes, you can look at one direction at a time, like north-south or east-west. That's what partial derivatives do.

When we take the partial derivative of a function with respect to one variable, we are looking at how the function changes as we move along that axis, while keeping all other variables constant. For example, in the function \( f(x, y) = x^3 y + 3xy^2 \), the partial derivative with respect to \( x \) is written as \( \frac{\partial f}{\partial x} \). This tells us how \( f \) changes as \( x \) changes, with \( y \) fixed.

• Partial with respect to \( x \): \( \frac{\partial f}{\partial x} = 3x^2y + 3y^2 \)
• Partial with respect to \( y \): \( \frac{\partial f}{\partial y} = x^3 + 6xy \)

By evaluating these partial derivatives at a specific point, like in our exercise at \( (2, -2) \), we gain insights into the slope or inclination of the surface at that point in each respective direction.

Tangent Plane

A tangent plane is like a flat surface that just *touches* a curved surface at one point, without cutting through it. Think of how a piece of paper might sit on the side of a round ball, making contact at one specific spot.

In the context of multivariable calculus, if you have a function that describes a surface, the tangent plane smoothly aligns with the surface at a given point, offering the best linear approximation of the surface near that point. We use the partial derivatives to tell us how to tilt our flat piece of paper.

• The formula for the tangent plane is: \( z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \)
• "\( z_0 \)" represents the height of the surface at our point of interest \((x_0, y_0)\).
• Substitute values from our exercise:
  • At \( (2, -2) \), \( z_0 = f(2, -2) = 8 \)
  • The gradient components are \( f_x = -12 \) and \( f_y = -16 \)

By plugging in these values, we get the equation of the tangent plane: \( z = -12x - 16y - 8 \). This plane best describes the area around our given point on the surface.

Multivariable Calculus

Multivariable calculus extends the ideas of single-variable calculus to functions with more than one input. Instead of a line, imagine finding the area of a hilly landscape or the volume of a shapely 3D object. This branch of math allows us to analyze how these complex shapes change across different directions.

Key concepts in multivariable calculus include:

• **Functions of several variables**: These are functions like \( f(x, y) \) which depend on two or more variables.
• **Gradient vectors**: These are collections of partial derivatives. They point in the direction of the greatest rate of increase of the function.
• **Tangent planes and linear approximations**: These help us approximate the behavior of a surface near a specific point by using flat surfaces.

Multivariable calculus is essential for fields like physics, engineering, and computer graphics since it deals with multiple dimensions of change. Our exercise, for example, used these concepts to find how a function of two variables varies and to describe its behavior near a given point using a gradient vector and a tangent plane.