Chapter 12: Q. 36 (page 989)

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) .

Short Answer

Expert verified

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

Step by step solution

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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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) .

Short Answer

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) .

Short Answer

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 .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Calculus

Study anywhere. Anytime. Across all devices.

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) .