Problem 1 - Calculus Early Transcendentals

Question

I-6 Find an equation of the tangent plane to the given surface at the specified point. $$ z=4 x^{2}-y^{2}+2 y, \quad(-1,2,4) $$

Step-by-Step Solution

Answer

The equation of the tangent plane is \(z = -8x - 2y + 4\).

Step 1: Find the partial derivatives

To determine the equation of the tangent plane, calculate the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \). For the given surface \( z = 4x^2 - y^2 + 2y \), the partial derivatives are:\[ \frac{\partial z}{\partial x} = 8x \]\[ \frac{\partial z}{\partial y} = -2y + 2 \]

Step 2: Evaluate partial derivatives at point

Substitute \( x = -1 \) and \( y = 2 \) into the partial derivatives:\[ \frac{\partial z}{\partial x} = 8(-1) = -8 \]\[ \frac{\partial z}{\partial y} = -2(2) + 2 = -4 + 2 = -2 \]

Step 3: Use point-slope form for tangent plane

The equation for the tangent plane to a surface at point \((x_0, y_0, z_0)\) is given by:\[ z - z_0 = \frac{\partial z}{\partial x} (x - x_0) + \frac{\partial z}{\partial y} (y - y_0) \]Using \( x_0 = -1 \), \( y_0 = 2 \), and \( z_0 = 4 \), substitute the values from previous steps:\[ z - 4 = -8(x + 1) - 2(y - 2) \]

Step 4: Simplify the equation

Expand and simplify the equation from Step 3:\[ z - 4 = -8x - 8 - 2y + 4 \]\[ z = -8x - 2y - 4 + 4 + 4 \]Simplified further:\[ z = -8x - 2y + 4 \]

Key Concepts

Partial Derivatives

Partial derivatives are an essential concept in calculus, particularly when dealing with functions of multiple variables. They measure how one variable in a function changes while keeping the other variables constant. For instance, when finding the tangent plane to a surface, understanding partial derivatives is crucial.

Let's say you have a function, such as the surface given by:

• \( z = 4x^2 - y^2 + 2y \).

To find the rates of change about each variable, you take the partial derivative with respect to each independent variable. For our function, the partial derivatives are:

• With respect to \( x \):\[ \frac{\partial z}{\partial x} = 8x \]
• With respect to \( y \):\[ \frac{\partial z}{\partial y} = -2y + 2 \]

These derivatives help predict how the surface behaves around a point by measuring the slope in the direction of each axis.

Surface Topology

Surface topology refers to the features and properties of a surface in space, like smoothness, curves, or peaks. Understanding the topology is vital when working with tangent planes, as these characteristics tell us how a surface is contoured at any point. In topology, a surface might be bumpy or smooth, influencing how we construct tangent planes.

Consider a point on a surface, such as

• \((-1, 2, 4)\).

By examining the topology around this point, you can develop a clear picture of the local behavior of the surface. Smoothness and gradient are particularly important, giving insight into how fast the surface is rising or declining in different directions. This allows us to find a precise tangent plane by utilizing the localized linear approximation of the surface.

Point-Slope Form

The point-slope form is a common way of expressing linear equations, which is very helpful when determining the equation of a tangent plane. Essentially, it creates the equation of a line using a known point and a given slope. When applied to surfaces, you use a point and the partial derivatives (slopes in different directions) to form the tangent plane equation.

To illustrate, consider the point-slope form for a surface at

• \((x_0, y_0, z_0)\): \[ z - z_0 = \frac{\partial z}{\partial x} (x - x_0) + \frac{\partial z}{\partial y} (y - y_0) \].

This formula connects the slope information acquired from partial derivatives with the specific coordinates of the point, enabling us to derive the plane that just "touches" or "kisses" the surface at that point without "breaking" the surface.

Calculus

Calculus is the mathematical study of change, including operations such as differentiation and integration. It is particularly important in analyzing and understanding curves and surfaces. Finding derivatives and rates of change lies at the core of calculus and is directly applied in problems involving tangent planes to surfaces.

When dealing with multivariable functions like

• \( z = 4x^2 - y^2 + 2y \),

calculus allows for the calculation of partial derivatives, aiding in the determination of tangent planes. It's calculus that allows breaking down complex surfaces into understandable local linear approximations, like those found with tangent planes. Mastering calculus not only helps in solving these kinds of problems but also prepares one to navigate more advanced mathematical territories and real-world applications.