Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find all points \((x, y)\) at which the tangent plane to the graph of \(z=x^{2}-6 x+2 y^{2}-10 y+2 x y\) is horizontal.

Step-by-Step Solution

Answer

The point is (1, 2).

Step 1: Understand the problem

We are asked to find points \(x, y\) on the surface \(z = x^2 - 6x + 2y^2 - 10y + 2xy\) where the tangent plane is horizontal. A plane is horizontal when its normal vector is vertical, implying the gradient of the function is zero.

Step 2: Calculate the partial derivatives

To find where the gradient is zero, calculate the partial derivatives of \(z = x^2 - 6x + 2y^2 - 10y + 2xy\). The partial derivatives are: \(\frac{\partial z}{\partial x} = 2x + 2y - 6\) and \(\frac{\partial z}{\partial y} = 4y + 2x - 10\).

Step 3: Set the gradients to zero

Set the gradients to zero for horizontal tangents: 1. \(2x + 2y - 6 = 0\) 2. \(4y + 2x - 10 = 0\). Solve these equations simultaneously for \(x\) and \(y\).

Step 4: Solve simultaneous equations

From equations \(2x + 2y - 6 = 0\) and \(4y + 2x - 10 = 0\), solve for \(x\) and \(y\). Subtract the first equation from the second: \( (4y + 2x - 10) - (2x + 2y - 6) = 0\) simplifies to \(2y - 4 = 0\), leading to \(y = 2\). Substitute \(y = 2\) into \(2x + 2y - 6 = 0\) to find \(x: 2x + 4 - 6 = 0\), thus \(x = 1\).

Step 5: Verify the solution

Verify \(x = 1, y = 2\) satisfies both equations: 1. \(2(1) + 2(2) - 6 = 0\) (True)2. \(4(2) + 2(1) - 10 = 0\) (True).Both equations being satisfied confirms the solution is correct.

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, especially when dealing with functions of multiple variables. They represent the rate of change of a function concerning one variable while keeping other variables constant.
For a surface given by a function like \( z = f(x, y) \), the partial derivative \( \frac{\partial f}{\partial x} \) tells us how \( z \) changes as \( x \) changes, with \( y \) remaining unchanged. Similarly, \( \frac{\partial f}{\partial y} \) describes the change in \( z \) with respect to changes in \( y \).

• To find a partial derivative, treat other variables as constants and differentiate with respect to the chosen variable.
• This helps determine slopes in the direction of each axis, crucial for understanding surface behavior.

In our problem, the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \) are critical for determining where the tangent plane is horizontal.

Gradient

The gradient is a vector that points in the direction of the greatest rate of increase of a function. It is denoted by \( abla f \) and comprises the partial derivatives of a function. For the function \( f(x, y) = z \), the gradient is \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \).

• The gradient tells us the slope of the tangent plane. When the gradient equals zero, the tangent plane is horizontal because there is no slope in any direction.
• A zero gradient means that at that point, the plane is perfectly flat, aligning with the concept of horizontal tangent planes.

In this exercise, finding where \( abla f = (0, 0) \) leads us to the points where the tangent plane to the surface is horizontal.

Simultaneous Equations

Simultaneous equations are sets of equations with multiple variables that are solved together, meaning the solutions must satisfy all equations in the set.
Solving these equations is vital when dealing with problems involving multiple variables. When we calculate partial derivatives and set them to zero to find horizontal tangent planes, we end up with simultaneous equations.

• Each equation represents a condition for one of the variables.
• Methods to solve them include substitution, elimination, or matrix approaches.

For our problem, finding the solutions of \( 2x + 2y - 6 = 0 \) and \( 4y + 2x - 10 = 0 \) simultaneously allows us to find the values of \( x \) and \( y \) where the tangent plane becomes horizontal.

Surface Equation

A surface equation describes a three-dimensional surface in a mathematical way. For example, the equation \( z = f(x, y) \) gives a surface where \( z \) is determined based on \( x \) and \( y \).
This problem's surface equation, \( z = x^2 - 6x + 2y^2 - 10y + 2xy \), provides a polynomial representation of a surface in three-dimensional space.

• It's essential for visualizing how changes in \( x \) and \( y \) affect \( z \), hence computing the nature of the tangent plane.
• The surface's shape and orientation guide us to where horizontal tangent planes can occur.

By understanding the surface equation, we can apply derivatives and gradients efficiently, aiming to find where the tangent behaves as required - in this case, a horizontal orientation.