Understanding the Given Surface

Oct 13, 2025 by magicalceylontours.com 6 views

Introduction

In mathematics, the concept of tangent planes is a crucial aspect of multivariable calculus. It involves finding the equation of a plane that is tangent to a given surface at a specific point. In this article, we will explore how to find the equation of the tangent plane to the given surface $ z = \sqrt{xy} $ at the point $ (7, 7, 7) $.

Understanding the Given Surface

The given surface is defined by the equation $ z = \sqrt{xy} $. This is a type of surface known as a radical surface, which involves the square root of a function of $ x $ and $ y $. To find the equation of the tangent plane, we need to understand the behavior of this surface at the specified point.

Finding the Partial Derivatives

To find the equation of the tangent plane, we need to find the partial derivatives of the surface with respect to $ x $ and $ y $. The partial derivative of $ z $ with respect to $ x $ is denoted as $ \frac{\partial z}{\partial x} $, and the partial derivative of $ z $ with respect to $ y $ is denoted as $ \frac{\partial z}{\partial y} $.

Let's find the partial derivatives of the given surface:

\frac{\partial z}{\partial x} = \frac{\partial}{\partial x} \sqrt{xy} = \frac{1}{2} (xy)^{-\frac{1}{2}} y = \frac{y}{2\sqrt{xy}}

\frac{\partial z}{\partial y} = \frac{\partial}{\partial y} \sqrt{xy} = \frac{1}{2} (xy)^{-\frac{1}{2}} x = \frac{x}{2\sqrt{xy}}

Evaluating the Partial Derivatives at the Specified Point

Now that we have found the partial derivatives, we need to evaluate them at the specified point $ (7, 7, 7) $. Let's substitute $ x = 7 $ and $ y = 7 $ into the partial derivatives:

\frac{\partial z}{\partial x} (7, 7) = \frac{7}{2\sqrt{7 \cdot 7}} = \frac{7}{14} = \frac{1}{2}

\frac{\partial z}{\partial y} (7, 7) = \frac{7}{2\sqrt{7 \cdot 7}} = \frac{7}{14} = \frac{1}{2}

Finding the Equation of the Tangent Plane

Now that we have evaluated the partial derivatives at the specified point, we can find the equation of the tangent plane. The equation of the tangent plane 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)

where $ (x_0, y_0, z_0) $ is the specified point, and $ \frac{\partial z}{\partial x} (x_0, y_0) $ and $ \frac{\partial z}{\partial y} (x_0, y_0) $ are the partial derivatives evaluated at the specified point.

Substituting the values, we get:

z - 7 = \frac{1}{2} (x - 7) + \frac{1}{2} (y - 7)

Simplifying the Equation of the Tangent Plane

We can simplify the equation of the tangent plane by combining like terms:

z - 7 = \frac{1}{2} x - \frac{7}{2} + \frac{1}{2} y - \frac{7}{2}

z - 7 = \frac{1}{2} x + \frac{1}{2} y - 7

z = \frac{1}{2} x + \frac{1}{2} y

Conclusion

In this article, we have found the equation of the tangent plane to the given surface $ z = \sqrt{xy} $ at the point $ (7, 7, 7) $. We have used the partial derivatives of the surface to find the equation of the tangent plane, and then simplified the equation to obtain the final result.

Final Answer

The final answer is: $\boxed{z = \frac{1}{2} x + \frac{1}{2} y}$

Introduction

In our previous article, we explored how to find the equation of the tangent plane to the given surface $ z = \sqrt{xy} $ at the point $ (7, 7, 7) $. In this article, we will answer some frequently asked questions related to finding the equation of the tangent plane.

Q: What is the tangent plane?

A: The tangent plane is a plane that touches a given surface at a specific point. It is a crucial concept in multivariable calculus, and it is used to approximate the behavior of a surface near a given point.

Q: How do I find the equation of the tangent plane?

A: To find the equation of the tangent plane, you need to find the partial derivatives of the surface with respect to $ x $ and $ y $. Then, you need to evaluate these partial derivatives at the specified point. Finally, you can use the equation of the tangent plane formula to find the equation of the tangent plane.

Q: What are the partial derivatives?

A: The partial derivatives are the derivatives of the surface with respect to $ x $ and $ y $. They are denoted as $ \frac{\partial z}{\partial x} $ and $ \frac{\partial z}{\partial y} $, respectively.

Q: How do I evaluate the partial derivatives at the specified point?

A: To evaluate the partial derivatives at the specified point, you need to substitute the values of $ x $ and $ y $ into the partial derivatives. Then, you can simplify the expressions to obtain the values of the partial derivatives at the specified point.

Q: What is the equation of the tangent plane formula?

A: The equation of the tangent plane formula 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)

Q: Can I use the equation of the tangent plane to approximate the behavior of a surface near a given point?

A: Yes, you can use the equation of the tangent plane to approximate the behavior of a surface near a given point. The tangent plane provides a linear approximation of the surface near the given point.

Q: What are some common applications of the tangent plane?

A: The tangent plane has many applications in physics, engineering, and economics. Some common applications include:

Finding the equation of a tangent line to a curve
Finding the equation of a tangent plane to a surface
Approximating the behavior of a surface near a given point
Finding the maximum or minimum value of a function

Q: Can I use the tangent plane to find the maximum or minimum value of a function?

A: Yes, you can use the tangent plane to find the maximum or minimum value of a function. The tangent plane provides a linear approximation of the function near a given point, and you can use this approximation to find the maximum or minimum value of the function.

Conclusion

In this article, we have answered some frequently asked questions related to finding the equation of the tangent plane. We have covered topics such as the definition of the tangent plane, how to find the equation of the tangent plane, and some common applications of the tangent plane.

Final Answer

The final answer is: $\boxed{z = \frac{1}{2} x + \frac{1}{2} y}$