Short Answer

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.
$f (x, y) = \frac{x}{y^{'}} P = (4, 3)$

$<\frac{1}{3}, - \frac{4}{9}>$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

$The gradient of a function f (x, y) is the vector function defined by \nabla f (x, y) = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j, Here f (x, y) = \frac{x}{y} .$

Step 2. Solution

So its gradient is given by,

$\begin{array}{l} \nabla f (x, y) = \frac{\partial}{\partial x} (\frac{x}{y}) i + \frac{\partial}{\partial y} (\frac{x}{y}) j \\ = \frac{1}{y} i - \frac{x}{y^{2}} j \\ Therefore \nabla f (x, y) = <\frac{1}{y}, - \frac{x}{y^{2}}> \end{array}$

$As the gradient of a function f at a point p, points in the direction in which f increases most rapidly. At p = (4, 3), \nabla f (4, 3) = \frac{1}{3} i - \frac{4}{9} j Hence the direction in which f increases most rapidly at p = (4, 3) is \nabla f (4, 3) = < \frac{1}{3}, - \frac{4}{9} >$

Find Study Materials for

Create Study Materials

Select your language

Calculus

Answers without the blur.

Short Answer

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.
$f (x, y) = \frac{x}{y^{'}} P = (4, 3)$

Step by Step Solution

Step 1. Given Information

Step 2. Solution

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Calculus

Answers without the blur.

Short Answer

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)=xy′P=(4,3)

Step by Step Solution

Step 1. Given Information

Step 2. Solution

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Calculus

Pure Maths

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.
$f (x, y) = \frac{x}{y^{'}} P = (4, 3)$