Chapter 12: Q. 32 (page 989)

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

Short Answer

Expert verified

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

Step by step solution

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,

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

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

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solution

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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)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Pure Maths

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Statistics

Study anywhere. Anytime. Across all devices.

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