Q. 34 Find the gradient of the given f... [FREE SOLUTION]

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

Short Answer

Expert verified

$鈭� f (鈭� 2, 2) = (鈭� \frac{1}{4}, 鈭� \frac{1}{4}>$

Step by step solution

Step 1. Given Information

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

Step 2. Solution

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

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solution

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91影视

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鈭�1鈦�yx,P=(鈭�2,2)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solution

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Discrete Mathematics

Decision Maths

Probability and Statistics

Mechanics Maths

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