Problem 67 - Calculus Early Transcendentals

Question

Find an equation for the family of level surfaces pe corresponding to \(f .\) Describe the level surfaces.$$f(x, y, z)=x^{2}+y^{2}-z.$$

Step-by-Step Solution

Answer

Answer: The level sets of the function \(f(x,y,z)=x^2+y^2-z\) represent a family of paraboloids.

Step 1: Set the function equal to a constant

To find the level surfaces of the given function \(f(x,y,z)=x^2+y^2-z\), set it equal to a constant \(c\) to form the equation: $$x^2+y^2-z=c.$$

Step 2: Solve for z

Solve the equation for z to better visualize the form of the level surfaces: $$z=x^2+y^2-c.$$

Step 3: Describe the level surfaces

The equation \(z=x^2+y^2-c\) represents a paraboloid. The value of \(c\) determines the position of the paraboloid along the z-axis, and the level surfaces for different values of \(c\) will be a family of paraboloids. The paraboloids open upwards and their vertex is translated along the z-axis as \(c\) changes.

Key Concepts

Paraboloid

A paraboloid is a three-dimensional surface that resembles a parabola when sliced along specific planes. In the exercise, the equation \( z = x^2 + y^2 - c \) describes a paraboloid. This specific form of paraboloid is an elliptic paraboloid because the cross-sections parallel to the \( xy \)-plane are ellipses, though they appear circular when \( x^2 + y^2 \) has equal coefficients, as is the case here.

• The paraboloid opens upwards along the \( z \)-axis.
• The equation can be rearranged to visualize the situation: \( z = x^2 + y^2 - c \).
• The parameter \( c \) shifts the vertex of the paraboloid up or down the \( z \)-axis.

Understanding paraboloids is crucial in multivariable calculus, as they represent simple, yet vital, three-dimensional shapes.

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions of multiple variables, often involving 2, 3, or more interdependent variables. It is a powerful tool to analyze and describe changes within dimensions beyond the typical graphable functions of basic calculus.

• Functions like \( f(x, y, z) = x^2 + y^2 - z \) require methods from multivariable calculus for analysis.
• Such studies involve concepts like partial derivatives, gradients, and examining surfaces like paraboloids.
• Level surfaces, as seen in this exercise, are a particular feature of multivariable functions.

In this exercise, using multivariable calculus allows us to better understand and visualize the family of surfaces represented by paraboloids.

Equation of Surfaces

Equations of surfaces describe the mathematical relationship defining the points on a surface in space. Here, for \( f(x, y, z) = x^2 + y^2 - z \), setting \( f \) equal to a constant \( c \) offers an equation of a surface: \( x^2 + y^2 - z = c \).

• This particular equation is a form of equation highlighting level surfaces.
• Each surface for different \( c \) values describes varying positions of the same type of shape, a paraboloid.
• These surfaces assist in visualizing how certain functions delineate three-dimensional space.

Recognizing and manipulating the equations of surfaces is a foundational skill in multivariable calculus, allowing one to comprehend complex spatial interactions.