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Chapter 14: Problem 41

Display the values of the functions in two ways: (a) by sketching the surface \(z=f(x, y)\) and (b) by drawing an assortment of level curves in the function's domain. Label each level curve with its function value. $$f(x, y)=x^{2}-y$$

Short Answer

Expert verified
Sketch 3D surface of \( z = x^2 - y \) and plot parabolas \( y = x^2 - c \) for different \( c \).

Step by step solution

01

Understanding the Function

The given function is a two-variable function, \( f(x, y) = x^2 - y \). Here, \( x^2 \) is the paraboloid that opens upwards in the \( xz \)-plane, and \( -y \) shifts it downward along the \( y \)-axis.
02

Sketching the Surface

To sketch the surface \( z = f(x, y) = x^2 - y \), imagine a 3D plot where \( z \) represents the height above each point \((x, y)\). For constant \( y \), the surface forms a parabola opening upwards with vertex at \( z = -y \). Similarly, for constant \( x \), \( z \) decreases linearly as \( y \) increases. The surface will look like a parabolic trough extending along the \( y \)-axis.
03

Drawing Level Curves

The level curves are defined by setting \( z = c \), where \( c \) is a constant, in the equation \( x^2 - y = c \). Solving for \( y \), we get \( y = x^2 - c \). This represents a series of parabolas each translated down \( c \) units along the \( y \)-axis. Plot several level curves for different values of \( c \) (e.g., \( c = 0, 1, -1, 2, -2 \)) and label them appropriately.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sketching Surfaces
In multivariable calculus, sketching surfaces involves visualizing a function of two variables, like \( z = f(x, y) = x^2 - y \), in a three-dimensional space. Imagine you have a graph where the x and y values form a plane, and the z-value represents the height above this plane for each \(x, y\) pair.
To sketch a surface:
  • You first identify the shape formed by holding one variable constant. For \( z = x^2 - y \), keep \( y \) constant and you get parabolas opening upwards in the xz-plane. Change the y-value, and the vertex of a parabola shifts along the z-axis.
  • Similarly, keep \( x \) constant to see that z linearly decreases with increasing y. The surface thus extends like a parabolic trough along the y-axis.
Visualizing such graphs helps understand how changes in variables affect the output, making abstract functions more tangible.
Level Curves
Level curves provide a slice through 3D graphs to help visualize changes in two-dimensional plots. They occur where a function of two variables, say \( z = x^2 - y \), equals a constant value \( c \). Replacing \( z \) with \( c \) simplifies this to an equation of y in terms of x: \( y = x^2 - c \).
To draw level curves:
  • Select various \( c \) values (e.g., 0, 1, -1, 2, -2) and solve for \( y \).
  • Each value of \( c \) corresponds to a quadratic equation, forming a parabola on the xy-plane.
  • These parabolas are parallel curves shifting vertically, giving a sense of elevation and depth on paper.
Labeling these curves with their respective \( c \) values offers a snapshot of the function's topographical nature, depicting how \( z \) levels fluctuate.
Functions of Two Variables
A function of two variables, such as \( f(x, y) = x^2 - y \), is crucial for processing changes across an area, not just along a line. Here, both x and y are independents affecting the output z, making the function multivariable.
Key points include:
  • A function of two variables maps from pairs of numbers onto heights or depths, often requiring interpretation in a three-dimensional space.
  • Contoured by level curves and sketched surfaces, these functions reveal multi-directional relationships between variables.
  • They are frequently used in fields such as physics and economics, where systems often depend on several interlinked factors.
Comprehending these multifaceted functions enhances grasp over real-world phenomena and mathematical modeling.

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