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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State whether this is true or false. If true, give an explanation. If false, give a counterexample.
![# Gradient and Directional Derivatives
## Maximizing the Dot Product of the Gradient and a Unit Vector
Let \( f \) be a differentiable function. Then, the quantity \( \nabla f \cdot \hat{u} \), where \( \hat{u} \) is a unit vector, is maximized if:
\[
\hat{u} = \frac{\nabla f}{|\nabla f|}
\]
In other words, the directional derivative of \( f \) in the direction of \( \hat{u} \) is maximized when \( \hat{u} \) points in the direction of the gradient \( \nabla f \). Here, \( \nabla f \) represents the gradient of \( f \), which is a vector of partial derivatives, and \( |\nabla f| \) is the magnitude of the gradient vector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b9a2144-dfc4-40f3-97fb-de24320ca2ba%2Fc85899e5-bfbe-4248-b8df-48f0e6269274%2Fmgodx3_processed.png&w=3840&q=75)
Transcribed Image Text:# Gradient and Directional Derivatives
## Maximizing the Dot Product of the Gradient and a Unit Vector
Let \( f \) be a differentiable function. Then, the quantity \( \nabla f \cdot \hat{u} \), where \( \hat{u} \) is a unit vector, is maximized if:
\[
\hat{u} = \frac{\nabla f}{|\nabla f|}
\]
In other words, the directional derivative of \( f \) in the direction of \( \hat{u} \) is maximized when \( \hat{u} \) points in the direction of the gradient \( \nabla f \). Here, \( \nabla f \) represents the gradient of \( f \), which is a vector of partial derivatives, and \( |\nabla f| \) is the magnitude of the gradient vector.
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