Q.19 - Calculus | StudyQuestionHub

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Q.19

Question

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

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Answer

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

1Step 1 : Introduction

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

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

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

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

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

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

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

Other exercises in this chapter

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Using Theorem 12.31 as a model, provide a conjecture you think would be sufficient to