Short Answer

Gradients: Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.
f(x, y ,z) = ln(x + y + z), P = (e, 0, −1) .

$The gradient is \nabla f (x, y, z) = <\frac{1}{x + y + z}, \frac{1}{x + y + z}, \frac{1}{x + y + z}> and the direction in which increases most rapidly at p = (e, 0, - 1) is \nabla f (e, 0, - 1) = <\frac{1}{e - 1}, \frac{1}{e - 1}, \frac{1}{e - 1}>$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given

f(x, y ,z) = ln(x + y + z)

Step 2. Finding gradient of f(x, y ,z) = ln(x + y + z)

$The gradient of a function f (x, y, z) is defined by \nabla f (x, y, z) = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j + \frac{\partial f}{\partial z} k . here, f (x, y, z) = \ln (x + y + z) . So, its gradient is given by, \nabla f (x, y, z) = \frac{\partial (\ln (x + y + z))}{\partial x} i + \frac{\partial (\ln (x + y + z))}{\partial y} j + \frac{\partial (\ln (x + y + z))}{\partial z} k = \frac{1}{x + y + z} i + \frac{1}{x + y + z} j + \frac{1}{x + y + z} k . Hence, the gradient is \nabla f (x, y, z) = <\frac{1}{x + y + z}, \frac{1}{x + y + z}, \frac{1}{x + y + z}> .$

Step 3. Finding points in the direction in which f increases .

$As the gradient of a function at a point p, points in the direction in which f increases most rapidly Here p = (e, 0, - 1), so \nabla f (e, 0, - 1) = \frac{1}{e - 1} i + \frac{1}{e - 1} j + \frac{1}{e - 1} k Hence the direction in which increases most rapidly at p = (e, 0, - 1) is \nabla f (e, 0, - 1) = <\frac{1}{e - 1}, \frac{1}{e - 1}, \frac{1}{e - 1}>$

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Calculus

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Short Answer

Gradients: Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.
f(x, y ,z) = ln(x + y + z), P = (e, 0, −1) .

Step by Step Solution

Step 1. Given

Step 2. Finding gradient of f(x, y ,z) = ln(x + y + z)

Step 3. Finding points in the direction in which f increases .

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Calculus

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Short Answer

Gradients: Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P. f(x, y ,z) = ln(x + y + z), P = (e, 0, −1) .

Step by Step Solution

Step 1. Given

Step 2. Finding gradient of f(x, y ,z) = ln(x + y + z)

Step 3. Finding points in the direction in which f increases .

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Gradients: Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.
f(x, y ,z) = ln(x + y + z), P = (e, 0, −1) .