Answer to Question #197603 in Real Analysis for Anuradhasingh jada

Question #197603

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).


1
Expert's answer
2021-05-24T15:34:36-0400

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 "\\nabla 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)\n\\\\\n =66+8=14"

   


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