Problem 179 - Calculus Volume 3

Question

Find the equation for the tangent plane to the surface at the indicated point. \(z=\sin x+\sin y+\sin (x+y), P(0,0,0)\)

Step-by-Step Solution

Answer

The tangent plane at P(0,0,0) is \( z = 2x + 2y \).

Step 1: Identify the function and point

We are given the surface function as \( z = \sin x + \sin y + \sin (x+y) \), and the point \( P(0,0,0) \) where we need to find the equation of the tangent plane.

Step 2: Calculate the Partial Derivatives

To find the tangent plane, first calculate the partial derivatives of \( z \) with respect to \( x \) and \( y \). The partial derivative with respect to \( x \) is \( \frac{\partial z}{\partial x} = \cos x + \cos (x+y) \). The partial derivative with respect to \( y \) is \( \frac{\partial z}{\partial y} = \cos y + \cos (x+y) \).

Step 3: Evaluate the Partial Derivatives at Point P

Evaluate the partial derivatives at the point \( P(0,0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the partial derivatives: \( \frac{\partial z}{\partial x} (0,0) = \cos 0 + \cos 0 = 2 \) and \( \frac{\partial z}{\partial y} (0,0) = \cos 0 + \cos 0 = 2 \).

Step 4: Find the Tangent Plane Equation

The equation for the tangent plane at a point \((x_0, y_0, z_0)\) is given by: \[z - z_0 = \frac{\partial z}{\partial x} (x_0, y_0)(x - x_0) + \frac{\partial z}{\partial y} (x_0, y_0)(y - y_0)\]Substitute \( (x_0, y_0, z_0) = (0,0,0) \) and the evaluated partial derivatives: \[z = 2(x - 0) + 2(y - 0) = 2x + 2y\]Thus, the equation for the tangent plane is \( z = 2x + 2y \).

Key Concepts

Partial Derivatives

When working with functions of multiple variables, the concept of partial derivatives is crucial. A partial derivative measures the rate at which the function changes as one of the variables changes, while keeping the other variables constant. In simpler terms, it's a bit like looking at a slice of multi-dimensional data.
For the given surface function, you have:

• The partial derivative with respect to \( x \), denoted as \( \frac{\partial z}{\partial x} \), shows how \( z \) changes if you slightly vary \( x \) and keep \( y \) constant.
• Similarly, \( \frac{\partial z}{\partial y} \) represents the change in \( z \) as \( y \) varies, holding \( x \) fixed.

Calculating these is quite straightforward, especially with trigonometric functions. Here, the partial derivatives of \( z = \sin x + \sin y + \sin(x+y) \) were calculated as: \ \( \frac{\partial z}{\partial x} = \cos x + \cos(x+y) \) and \( \frac{\partial z}{\partial y} = \cos y + \cos(x+y) \). After computing the partial derivatives, they're used to find how a surface tilts at specific points.

Surface Function

A surface function describes a surface in a three-dimensional space using a variable equation, often in the form \( z = f(x,y) \). In our example, the surface is defined by \( z = \sin x + \sin y + \sin (x+y) \). This equation represents a wave-like surface, combining sinusoidal variations due to \( x \), \( y \), and the sum \( x+y \).
Understanding surface functions involves recognizing how changes in \( x \) and \( y \) affect \( z \). These relationships can be visualized as ripples on a water surface, where each point on the surface holds a specific \( z \) value directly related to its \( x \) and \( y \) coordinates.

• When \( x \) increases, certain parts of the surface rise or fall due to the \( \sin x \) component.
• The same happens with the \( \sin y \) and \( \sin(x+y) \). Each part adds a layer of complexity to the shape.

These elements make surface functions fascinating, showing intricate patterns and highlighting the significance of each variable in determining surface properties.

Tangent Plane Equation

The concept of a tangent plane is an extension of the tangent line idea from two-dimensional calculus to three dimensions. While a tangent line just touches a curve at a point, a tangent plane just "kisses" a surface at a given point.
To find the tangent plane equation, you use the partial derivatives you've calculated. These partials represent slopes in the \( x \) and \( y \) directions.
The general formula for the tangent plane equation is: \[ z - z_0 = \frac{\partial z}{\partial x} (x_0, y_0)(x - x_0) + \frac{\partial z}{\partial y} (x_0, y_0)(y - y_0) \] This equation tells you exactly how the plane is oriented at point \((x_0, y_0, z_0)\).

• The terms \( \frac{\partial z}{\partial x}(x - x_0) \) and \( \frac{\partial z}{\partial y}(y - y_0) \) define the tilt or slope of the plane.
• Plugging in our example's values at point \( (0,0,0) \), where both partials equal 2, the equation simplifies to \( z = 2x + 2y \). This shows a plane rising equally in the \( x \) and \( y \) direction.

Tangent planes are vital in approximations, as they provide a flat, easy-to-understand surface that mimics our complicated function closely at a specific point.