Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15 and 16, find the equation \(w=T(x, y, z)\) of the tangent "hyperplane" at \(\mathbf{p}\). $$ f(x, y, z)=x y z+x^{2}, \mathbf{p}=(2,0,-3) $$

Step-by-Step Solution

Answer

The equation of the tangent hyperplane is \(w = 4x - 6y - 4\).

Step 1: Determine the Gradient of f

The gradient of the function, denoted as \(abla f(x, y, z)\), is found by computing the partial derivatives. For the function \(f(x, y, z) = x y z + x^2\), we have:\[\frac{\partial f}{\partial x} = yz + 2x\]\[\frac{\partial f}{\partial y} = xz \]\[\frac{\partial f}{\partial z} = xy \]

Step 2: Evaluate the Gradient at the Point \(\mathbf{p}\)

Substitute the point \((x, y, z) = (2, 0, -3)\) into each of the partial derivatives:\[\frac{\partial f}{\partial x} \bigg|_{(2, 0, -3)} = (0)(-3) + 2(2) = 4\]\[\frac{\partial f}{\partial y} \bigg|_{(2, 0, -3)} = 2(-3) = -6 \]\[\frac{\partial f}{\partial z} \bigg|_{(2, 0, -3)} = (2)(0) = 0 \]

Step 3: Find the Function Value at \(\mathbf{p}\)

Calculate \(f(2, 0, -3)\) by substituting into the function:\[f(2, 0, -3) = (2)(0)(-3) + (2)^2 = 0 + 4 = 4\]

Step 4: Write the Equation of the Tangent Hyperplane

The equation of the tangent hyperplane at the point \(\mathbf{p} = (2, 0, -3)\) is given by:\[w = f(2, 0, -3) + abla f \cdot ((x, y, z) - (2, 0, -3))\]Substitute the evaluated gradient and function values:\[w = 4 + 4(x - 2) - 6(y - 0) + 0(z + 3)\]Simplify:\[w = 4 + 4x - 8 - 6y = 4x - 6y - 4\]

Step 5: Finalize the Tangent Hyperplane Equation

The simplified equation of the tangent hyperplane is:\[w = 4x - 6y - 4\]

Key Concepts

Gradient

The gradient is a crucial concept in calculus, particularly in understanding how a multivariable function changes at a given point. It is often represented by the symbol \(abla f\), and it consists of all the partial derivatives of a function. For a function like \(f(x, y, z) = xyz + x^2\), the gradient tells us how the function changes in the direction of each variable (i.e., \(x\), \(y\), and \(z\)).
The gradient vector, \(abla f(x, y, z)\), is a collection of these partial derivatives:
\[abla f(x, y, z) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\]
This vector points in the direction of the steepest ascent of the function. At the point \(\mathbf{p} = (2, 0, -3)\), the gradient was evaluated as \((4, -6, 0)\). This tells us how sharply \(f\) changes in each direction around this particular point.

Partial Derivatives

Partial derivatives are a foundational tool in multivariable calculus. They measure how a function changes as one of its variables is varied, keeping the others constant. For the function \(f(x, y, z) = xyz + x^2\), we find the partial derivatives with respect to each variable:
\[\frac{\partial f}{\partial x} = yz + 2x\]
\[\frac{\partial f}{\partial y} = xz\]
\[\frac{\partial f}{\partial z} = xy\]
This approach helps in examining the effect of changing one particular variable on the entire function. By calculating these at a specific point, such as \((2, 0, -3)\), we gain insights into the behavior of the function just around that immediate vicinity. The partial derivatives evaluated at this point become components of the gradient.

Multivariable Calculus

Multivariable calculus extends calculus concepts to functions with more than one variable. It involves the exploration of functions like \(f(x, y, z)\), offering a richer field of study since there's room to explore changes in multiple directions. Techniques in multivariable calculus include working with partial derivatives, optimizing functions, and exploring geometric interpretations (such as tangent planes).
These approaches allow us to:

• Understand local behavior of functions in their domain.
• Analyze how changes in input affect the output.
• Model 3D surfaces and curves in space.

Considering the function \(f(x, y, z) = xyz + x^2\), multivariable calculus tools help us to visualize this in three dimensions, calculate derivatives in multiple directions, and explore the nature of surfaces it might describe.

Equation of Tangent Plane

The equation of a tangent plane provides a linear approximation of a surface at a given point. For a multivariable function \(f(x, y, z)\), the tangent plane equation at point \(\mathbf{p}\) can be expressed with the help of its gradient.
Given function value \(w\) at some point, and the gradient \(abla f\) at that point, the formula for the equation of the tangent plane becomes:
\[w = f(x_0, y_0, z_0) + abla f \cdot ((x, y, z) - (x_0, y_0, z_0))\]
In our exercise, at the point \(\mathbf{p} = (2, 0, -3)\), this equation simplifies to:
\[w = 4 + 4(x - 2) - 6(y - 0) + 0(z + 3)\]
After simplification the formula resolves to:
\[w = 4x - 6y - 4\]
This linear equation accurately describes a plane that "touches" the surface of \(f(x, y, z)\) at \(\mathbf{p}\) without cutting through it, providing a localized flat approximation of the surface.