The Given Surface and Point

Oct 13, 2025 by magicalceylontours.com 81 views

Introduction

In mathematics, the concept of tangent planes is a crucial aspect of multivariable calculus, particularly in the study of surfaces and their properties. A tangent plane to a surface at a given point is a plane that just touches the surface at that point and has the same slope as the surface at that point. In this article, we will explore how to find an equation of the tangent plane to a given surface at a specified point.

The Given Surface and Point

The given surface is defined by the equation:

${ z = 4x^2 - y^2 + 5y }$

The specified point is given as:

${ (-2, 4, 20) }$

Finding the Tangent Plane

To find the equation of the tangent plane, we need to find the partial derivatives of the surface equation with respect to x and y. The partial derivative of z with respect to x is denoted as ∂z/∂x, and the partial derivative of z with respect to y is denoted as ∂z/∂y.

Partial Derivative of z with Respect to x

To find the partial derivative of z with respect to x, we will differentiate the surface equation with respect to x while treating y as a constant.

${ \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} (4x^2 - y^2 + 5y) }$

Using the power rule of differentiation, we get:

${ \frac{\partial z}{\partial x} = 8x }$

Partial Derivative of z with Respect to y

To find the partial derivative of z with respect to y, we will differentiate the surface equation with respect to y while treating x as a constant.

${ \frac{\partial z}{\partial y} = \frac{\partial}{\partial y} (4x^2 - y^2 + 5y) }$

Using the power rule of differentiation, we get:

${ \frac{\partial z}{\partial y} = -2y + 5 }$

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 (-2, 4, 20).

Evaluating the Partial Derivative of z with Respect to x at the Specified Point

Substituting x = -2 into the partial derivative of z with respect to x, we get:

${ \frac{\partial z}{\partial x} (-2) = 8(-2) = -16 }$

Evaluating the Partial Derivative of z with Respect to y at the Specified Point

Substituting y = 4 into the partial derivative of z with respect to y, we get:

${ \frac{\partial z}{\partial y} (4) = -2(4) + 5 = -8 + 5 = -3 }$

Finding the Equation of the Tangent Plane

The equation of the tangent plane to the surface at the specified point can be written in the form:

${ z - z_0 = \frac{\partial z}{\partial x} (x_0) (x - x_0) + \frac{\partial z}{\partial y} (y_0) (y - y_0) }$

where (x0, y0, z0) is the specified point.

Substituting the values of the partial derivatives and the specified point, we get:

${ z - 20 = -16 (x + 2) - 3 (y - 4) }$

Simplifying the equation, we get:

${ z - 20 = -16x - 32 - 3y + 12 }$

Combining like terms, we get:

${ z - 20 = -16x - 3y - 20 }$

Adding 20 to both sides, we get:

${ z = -16x - 3y }$

Conclusion

In this article, we have found the equation of the tangent plane to the given surface at the specified point. We have used the partial derivatives of the surface equation with respect to x and y to find the equation of the tangent plane. The equation of the tangent plane is given by:

${ z = -16x - 3y }$

This equation represents the tangent plane to the surface at the point (-2, 4, 20).

Introduction

In our previous article, we explored how to find an equation of the tangent plane to a given surface at a specified point. We used the partial derivatives of the surface equation with respect to x and y to find the equation of the tangent plane. 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 to a surface?

A: The tangent plane to a surface at a given point is a plane that just touches the surface at that point and has the same slope as the surface at that 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 equation with respect to x and y. Then, you can use these partial derivatives to find the equation of the tangent plane.

Q: What are the partial derivatives of the surface equation?

A: The partial derivatives of the surface equation with respect to x and y are denoted as ∂z/∂x and ∂z/∂y, respectively. To find these partial derivatives, you need to differentiate the surface equation with respect to x and y while treating the other variable as a constant.

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.

Q: What is the equation of the tangent plane in terms of x, y, and z?

A: The equation of the tangent plane to the surface at the specified point can be written in the form:

${ z - z_0 = \frac{\partial z}{\partial x} (x_0) (x - x_0) + \frac{\partial z}{\partial y} (y_0) (y - y_0) }$

where (x0, y0, z0) is the specified point.

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

A: To simplify the equation of the tangent plane, you need to combine like terms and add or subtract constants.

Q: What is the significance of the tangent plane in mathematics?

A: The tangent plane is a crucial concept in mathematics, particularly in the study of surfaces and their properties. It is used to describe the behavior of a surface at a given point and is an essential tool in many areas of mathematics and science.

Q: Can I find the equation of the tangent plane to a surface at any point?

A: Yes, you can find the equation of the tangent plane to a surface at any point. However, you need to have the partial derivatives of the surface equation with respect to x and y.

Q: How do I find the equation of the tangent plane to a surface at a point where the partial derivatives are not defined?

A: If the partial derivatives are not defined at a point, it means that the surface is not differentiable at that point. In this case, you cannot find the equation of the tangent plane.

Q: Can I use the equation of the tangent plane to find the slope of the surface at a given point?

A: Yes, you can use the equation of the tangent plane to find the slope of the surface at a given point. The slope of the surface at a given point is equal to the slope of the tangent plane at that point.

Q: How do I use the equation of the tangent plane to find the slope of the surface at a given point?

A: To find the slope of the surface at a given point, you need to substitute the values of x and y into the equation of the tangent plane and solve for the slope.

Conclusion

In this article, we have answered some frequently asked questions related to finding the equation of the tangent plane to a given surface at a specified point. We have covered topics such as the definition of the tangent plane, how to find the equation of the tangent plane, and how to use the equation of the tangent plane to find the slope of the surface at a given point.