Q.20 - Calculus

Question

$U\sin g Theorem 12.31 as a model, provide a conjecture you think would be sufficient to guarantee that a function of n variables is differentiable at a point in its domain.$

Step-by-Step Solution

Answer

  The function variables is differntlable at a point in its domain  

Step 1 : Introduction

$We have given f(x, y)=3 x^\left\{2\right\}-4 x y-2 y^\left\{2\right\} at specified point P=(0,0) with unit vector u=(1,1)$

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

              $\nabla f\left(P\right)·u=\nabla f(0,0)·u=\left({\left.\frac{df}{dx}\right|}_{(0,0)}i+{\left.\frac{df}{dy}\right|}_{(0,0)}j\right)·\left(\frac{i+j}{\sqrt{i^2+j^2}}\right)$

                                                       $=\left[(6x-4y{)}_{(0,0)}i+(-4x-4y{)}_{(0,0)}j\right]·\left(\frac{i+j}{\sqrt{2}}\right)$