# Understanding 3D Graphs of Multivariable FunctionsFigure 16.35 presents computer-generated graphs of a multivariable function. The function is \( z = (x^2 - 3x)(1 + y^2) \).## Graph Descriptions### (a) Coordinate Grid Lifted to the Graph- **Description:** This graph uses a mesh grid to illustrate how the function \( z \) changes over the \( xy \)-plane. The surface varies in height, showing positive and negative regions.- **Axes:** The \( x \)-axis and \( y \)-axis form the base, while the \( z \)-axis represents the function's output.### (b) Level Curves Lifted to the Graph- **Description:** This graph shows contours, or level curves, indicating points where the function \( z \) maintains a constant value. The curves help visualize how the function’s value changes over different regions.- **Axes:** Similar to (a), with \( x \), \( y \), and \( z \) demonstrating the function's behavior in three dimensions.## Exercise InsightThe exercise refers to Figure 16.35 and asks to identify the maximum and minimum points of the function. Understanding the topography of these graphs aids in finding where the regions of highest and lowest values are located.### Analyzing the Graphs- **Peaks and Valleys:** The mesh and contour plots allow observation of peaks (maximum points) and valleys (minimum points).- **Critical Analysis:** By studying the surface undulations and contour proximity, one gains insight into the extrema.These visual representations are essential tools in calculus, especially when analyzing more complex multivariable functions.

1. Refer to Fig. 16.3.5, a computer-generated graph 3x-3x)/(1 + y^). Where are the maxi Figure 16.3.5. Computer- generated graph of := (x- 3x)/(1+ y3). (a) Coordinate grid lifted to the graph. X-AXIS X-RXI5 (b) Level curves lifted to the graph. mum and points?

1. Refer to Fig. 16.3.5, a computer-generated graph 3x-3x)/(1 + y^). Where are the maxi Figure 16.3.5. Computer- generated graph of := (x- 3x)/(1+ y3). (a) Coordinate grid lifted to the graph. X-AXIS X-RXI5 (b) Level curves lifted to the graph. mum and points?

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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