Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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**Question 4: Level Curves and Plane Graphs** Suppose that the level curves of a function \( z = f(x, y) \) consist of straight lines. Must the graph of \( f \) be a plane? In this question, we are considering a function \( f \) of two variables \( x \) and \( y \), where the level curves (i.e., the curves along which the function \( f \) has a constant value) are straight lines. The question asks whether this implies that the graph of the function \( f \) itself must be a plane. To address this, recall that if the level curves of \( z = f(x, y) \) are straight lines, this suggests a specific property about the partial derivatives and the overall form of the function \( f \). If the function \( f \) is linear, the graph of \( f \) would indeed be a plane. However, it's necessary to analyze whether every case of straight-line level curves necessarily leads to a planar graph. This question provides an opportunity to explore the relationship between level curves and the graph of a function in multivariable calculus.
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Transcribed Image Text:**Question 4: Level Curves and Plane Graphs** Suppose that the level curves of a function \( z = f(x, y) \) consist of straight lines. Must the graph of \( f \) be a plane? In this question, we are considering a function \( f \) of two variables \( x \) and \( y \), where the level curves (i.e., the curves along which the function \( f \) has a constant value) are straight lines. The question asks whether this implies that the graph of the function \( f \) itself must be a plane. To address this, recall that if the level curves of \( z = f(x, y) \) are straight lines, this suggests a specific property about the partial derivatives and the overall form of the function \( f \). If the function \( f \) is linear, the graph of \( f \) would indeed be a plane. However, it's necessary to analyze whether every case of straight-line level curves necessarily leads to a planar graph. This question provides an opportunity to explore the relationship between level curves and the graph of a function in multivariable calculus.
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