/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 Sketch the graph of \(f\). $$ ... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 38

Sketch the graph of \(f\). $$ f(x, y)=4-x^{2}-y^{2} $$

Short Answer

Expert verified
The graph of the function is a downward-opening paraboloid with vertex at (0, 0, 4).

Step by step solution

01

Identify the Type of Surface

The function given is a quadratic function in the form of \[ f(x, y) = 4 - x^2 - y^2 \].This is the equation of a paraboloid, which is similar to the function of a circle extended into three dimensions.
02

Determine the Axis of Symmetry

For the function \[ f(x, y) = 4 - x^2 - y^2 \], the surface is centered at the origin and symmetric around the z-axis, since the x and y terms are squared and subtracted from a constant.
03

Determine the Vertex

The vertex of the paraboloid occurs at the maximum value of\[ f(x, y) = 4 - x^2 - y^2 \] which occurs when x = 0 and y = 0. Therefore, the vertex is at(0, 0, 4).
04

Identify the Intercepts

Substitute\[ f(x, y) = 0 \] to find where the surface intersects the xy-plane (i.e., solve\[ 4 - x^2 - y^2 = 0 \]). This gives the equation \[ x^2 + y^2 = 4 \], which represents a circle with radius 2.
05

Sketch the Graph

Draw the three-dimensional space with the z-axis vertically. Sketch the paraboloid opening downward from the vertex at (0, 0, 4). The base intersects the xy-plane along a circle with radius 2 centered at the origin.

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Key Concepts

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

Paraboloid
A paraboloid is a three-dimensional geometric shape that is similar in concept to a parabola. Think of a parabola, which is a U-shaped curve in two dimensions, being extended into the three-dimensional world. This creates a surface known as a paraboloid. Specifically, for the function given in the original exercise, the equation \( f(x, y) = 4 - x^2 - y^2 \) describes an elliptic paraboloid.
  • The term "elliptic" indicates the cross-section parallel to the \(xy\)-plane is an ellipse鈥攈ere, it simplifies to a circle.
  • This paraboloid opens downward because the \(x^2\) and \(y^2\) terms are subtracted from 4, indicating a reduction along the \(z\)-axis as \(x\) and \(y\) increase.
The top or peak of this paraboloid is called its vertex. In this scenario, calculating gives a vertex at the point (0, 0, 4). This emphasizes where the maximum point or the top of the surface stands in three-dimensional space.
Graph of a Function
Graphs are fundamental tools in mathematics to visualize functions. A graph of a function provides a picture of all the possible outputs \(z\) based on different inputs \((x, y)\). For multivariable functions like \( f(x, y) = 4 - x^2 - y^2 \), the graph is a three-dimensional surface.
  • In this case, graphing involves plotting points where each point corresponds to an ordered pair \((x, y, z)\).
  • These points indicate where the function has particular values, and collectively they form a surface over the defined domain.
  • The task of sketching this graph involves identifying important characteristics like the symmetry, vertex, and intercepts.
Understanding features like where it intersects the \(xy\)-plane help conceptualize its structure. For \( f(x, y) = 0 \), replacing \(z\) with 0 yields the circle \( x^2 + y^2 = 4 \), marking the boundary of the paraboloid's base.
Three-dimensional Space
Three-dimensional space is where we visualize objects with length, width, and height. It adds complexity to graphing, as we're not just considering a line or curve, but a spatial surface. Here, the coordinates \((x, y, z)\) describe locations within this space.
  • The "z-axis" is added to the traditional \(x\) and \(y\) axes, giving a sense of depth or height.
  • Functions like \(f(x, y) = 4 - x^2 - y^2\) utilize this third dimension, enabling us to see how outputs vary with two inputs (\(x\) and \(y\)).
  • When sketching in three-dimensional space, the \(xy\)-plane is often visualized as a horizontal plane, with the \(z\)-axis pointing upwards.
Grasping three-dimensional space is crucial for interpreting and drawing the shapes and surfaces associated with multivariable calculus, like paraboloids. This visualization aids in understanding geometric properties and spatial relationships in multiple dimensions.

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