Chapter 12: Q. 34 (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.
$f (x, y) = \tan^{- 1} (\frac{y}{x}), P = (- 2, 2)$

Short Answer

Expert verified

$\nabla f (- 2, 2) = (- \frac{1}{4}, - \frac{1}{4}>$

Step by step solution

Step 1. Given Information

$The gradient of a function f (x, y) is the vector function defined by \nabla f (x, y) = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j, Here f (x, y) = \tan^{- 1} \frac{y}{x} .$

Step 2. Solution

$G r a d i e n t i s g i v e n b y, \begin{array}{l} \nabla f (x, y) = \frac{\partial}{\partial x} (\tan^{- 1} \frac{y}{x}) i + \frac{\partial}{\partial y} (\tan^{- 1} \frac{y}{x}) j \\ = \frac{1}{1 + {(\frac{y}{x})}^{2}} (\frac{y}{x^{2}}) + \frac{1}{1 + {(\frac{y}{x})}^{2}} (\frac{1}{x}) j \\ = \frac{- y}{x^{2} + y^{2}} i + \frac{x}{x^{2} + y^{2}} j \\ H e n c e t h e g r a d i e n t i s \nabla f (x, y) = <\frac{- y}{x^{2} + y^{2}}, \frac{x}{x^{2} + y^{2}}> \\ As the gradient of a function f at a point p points in the direction in which f increases most rapidly andat \\ \begin{array}{l} \nabla f (- 2, 2) = \frac{- 2}{(- 2)^{2} + 2^{2}} i + \frac{- 2}{(- 2)^{2} + 2^{2}} j \\ = - \frac{1}{4} i - \frac{1}{4} j \\ S o, t h e d i r e c t i o n a t w h i c h f i n c r e a s e s m o s t r a p i d l y a t p = (- 2, 2) i s \nabla f (- 2, 2) = (- \frac{1}{4}, - \frac{1}{4}> \end{array} \end{array}$

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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.
$f (x, y) = \tan^{- 1} (\frac{y}{x}), P = (- 2, 2)$

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)=tan−1⁡yx,P=(−2,2)

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

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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.
$f (x, y) = \tan^{- 1} (\frac{y}{x}), P = (- 2, 2)$