Q. 64 聽f(x,y)=ln鈦y2,P=(1,鈭�3)... [FREE SOLUTION]

91影视

Chapter 12: Q. 64 (page 954)

$f (x, y) = \ln 鈦� (x y^{2}), P = (1, 鈭� 3)$

Short Answer

Expert verified

The solution's response $f (x, y) = \ln 鈦� (x y^{2}), P = (1, 鈭� 3)$ is $3 x 鈭� 2 y 鈭� 3 z = 2.4084$

Step by step solution

Given data

The equation of line of segment is

$f_{x} (x_{0}, y_{0}) (x 鈭� x_{0}) + f_{y} (x_{0}, y_{0}) (y 鈭� y_{0}) = z 鈭� f (x_{0}, y_{0})$

$f_{x} (1, 鈭� 3) (x 鈭� 1) + f_{y} (1, 鈭� 3) (y + 3) = z 鈭� f (1, 鈭� 3)$ $(1)$

Find fx

$f_{x} (x_{0}, y_{0}) = {\frac{d}{d x} f (x, y)|}_{(x_{0}, y_{0})} = {\frac{d}{d x} \ln 鈦� (x y^{2})|}_{(x_{0}, y_{0})}$

$f_{x} (x_{0}, y_{0}) = {\frac{d}{d x} (\ln 鈦� (x y^{2})) \frac{d}{d x} (x y^{2})|}_{(x_{0}, y_{0})}$

$f_{x} (x_{0}, y_{0}) = {\frac{1}{x y^{2}} 脳 y^{2}|}_{(x_{0}, y_{0})}$

$f_{x} (x_{0}, y_{0}) = {\frac{1}{x}|}_{(x_{0}, y_{0})}$

$f_{x} (1, 鈭� 3) = {\frac{1}{1}|}_{(1, 鈭� 3)}$

$f_{x} (1, 鈭� 3) = 1$ $(2)$

Find fy

$f_{y} (x_{0}, y_{0}) = {\frac{d}{d y} f (x, y)|}_{(x_{0}, y_{0})} = {\frac{d}{d y} \ln 鈦� (x y^{2})|}_{(x_{0}, y_{0})}$

$f_{y} (x_{0}, y_{0}) = {\frac{d}{d y} (\ln 鈦� (x y^{2})) \frac{d}{d y} (x y^{2})|}_{(x_{0}, y_{0})}$

$f_{y} (x_{0}, y_{0}) = {\frac{1}{x y^{2}} 脳 2 x y|}_{(x_{0}, y_{0})}$

$f_{y} (1, 鈭� 3) = {\frac{2}{y}|}_{(1, 鈭� 3)}$

$f_{y} (1, 鈭� 3) = 鈭� \frac{2}{3}$ $(3)$

Find f(1,-3)

$f (x_{0}, y_{0}) = f (1, 鈭� 3) = \ln 鈦� (1 脳 (鈭� 3)^{2})$

$f (1, 鈭� 3) = \ln 鈦� (9)$

$f (1, 鈭� 3) = 2.1972$ $(4)$

Step 5:Solution

Subsututing Equation $f_{x} (1, 鈭� 3) = 1$ , $f_{y} (1, 鈭� 3) = 鈭� \frac{2}{3}$ and $f (1, 鈭� 3) = 2.1972$ in $f_{x} (1, 鈭� 3) (x 鈭� 1) + f_{y} (1, 鈭� 3) (y + 3) = z 鈭� f (1, 鈭� 3)$ get

$1 (x 鈭� 1) 鈭� \frac{2}{3} (y + 3) = z 鈭� 2.1972$

$x 鈭� 1 鈭� \frac{2}{3} y 鈭� 2 鈭� z = 鈭� 2.1972$

$x 鈭� \frac{2}{3} y 鈭� z = 鈭� 2.1972 + 1 + 2$

$x 鈭� \frac{2}{3} y 鈭� z = 0.8028$

multiple $3$ on both side ,get

$3 x 鈭� 2 y 鈭� 3 z = 2.4084$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

$f (x, y) = \ln 鈦� (x y^{2}), P = (1, 鈭� 3)$

Short Answer

Step by step solution

Given data

Find fx

Find fy

Find f(1,-3)

Step 5:Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Decision Maths

Probability and Statistics

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

91影视

f(x,y)=ln鈦�xy2,P=(1,鈭�3)

Short Answer

Step by step solution

Given data

Find fx

Find fy

Find f(1,-3)

Step 5:Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Decision Maths

Probability and Statistics

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

$f (x, y) = \ln 鈦� (x y^{2}), P = (1, 鈭� 3)$