Chapter 12: Q.19 (page 953)

$U \sin g D e f i n i t i o n 12.30 a s a m o d e l, p r o v i d e a d e f i n i t i o n o f d i f f e r e n t i a b i l i t y f o r a f u n c t i o n o f n v a r i a b l e s .$

Short Answer

Expert verified

$Differentiability for a function of n variables.$ $\nabla f (P) \cdot u = 0$

Step by step solution

Introduction

We have given $f (x, y) = x y + 2 x - y$ at specified point $P = (1, - 2)$ with unit vector $u = (α, β) = (1, 1)$

Step 2 Directional derivative of function at point P with directional unit vector u is given by,

$\nabla f (P) \cdot u = \nabla f (1, - 2) \cdot u = ({\frac{d f}{d x}|}_{(1, - 2)} i + {\frac{d f}{d y}|}_{(1, - 2)} j) \cdot (\frac{i + j}{\sqrt{i^{2} + j^{2}}})$

$= [(y + 2)_{(1, - 2)} i + (x - 1)_{(1, - 2)} j] \cdot (\frac{i + j}{\sqrt{2}})$

$= [(- 2 + 2) i + (1 - 1) j] \cdot (\frac{i + j}{\sqrt{2}})$

$= [0 i + 0 j] \cdot (\frac{i + j}{\sqrt{2}})$

$\nabla f (P) \cdot u = 0$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$U \sin g D e f i n i t i o n 12.30 a s a m o d e l, p r o v i d e a d e f i n i t i o n o f d i f f e r e n t i a b i l i t y f o r a f u n c t i o n o f n v a r i a b l e s .$

Short Answer

Step by step solution

Introduction

Step 2 Directional derivative of function at point P with directional unit vector u is given by,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Probability and Statistics

Applied Mathematics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

UsingDefinition12.30asamodel,provideadefinitionofdifferentiabilityforafunctionofnvariables.

Short Answer

Step by step solution

Introduction

Step 2 Directional derivative of function at point P with directional unit vector u is given by,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Probability and Statistics

Applied Mathematics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

$U \sin g D e f i n i t i o n 12.30 a s a m o d e l, p r o v i d e a d e f i n i t i o n o f d i f f e r e n t i a b i l i t y f o r a f u n c t i o n o f n v a r i a b l e s .$