Question

a differentiable real valued function f has at the point (1, 2), directional derivatives +2 in the direction toward (2, 2) and -2 in the direction toward (1, 1). determine the gradient vector at (1, 2) and compute the directional derivative in the direction toward (4, 6).

Answer 1

We know that directional derivative

$D_uf=D.D= \nabla f.\vec{u}$

     = $(f_x i+f_y j).\vec{u}$

Now directional derivative in the direction towards (2,2) is +2

i.e. ($f_x i+f_y j)((2-1)i+(2-2)j)=2$

$(f_xi+f_yj).(i+0j)=2$ f

$\Rightarrow f_x=2$

Also Directional derivative towards (1,1)-

$(f_xi+f_yj)((1-1)i+(1-2)j)=-2$

$(f_xi+f_yj)(0i-j)=-2$

$-f_y=-2$

$\Rightarrow f_y=2$

Gradient $\Delta f=f_x i+f_yj$

           $=2i+2j$

Directional Derivative towards (4,6) is-

     $=(2i+2j)((4-1)i+(6-2)j)\\=(2i+2j)(3i+4j) \\ =66+8=14$