Answer to Question #196746 in Real Analysis for Anuradhasingh jada

Question #196746

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
2021-05-24T19:10:22-0400

We know that directional derivative


Duf=D.D=f.uD_uf=D.D= \nabla f.\vec{u}

     = (fxi+fyj).u(f_x i+f_y j).\vec{u}


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


i.e. (fxi+fyj)((21)i+(22)j)=2f_x i+f_y j)((2-1)i+(2-2)j)=2


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


fx=2\Rightarrow f_x=2


Also Directional derivative towards (1,1)-


(fxi+fyj)((11)i+(12)j)=2(f_xi+f_yj)((1-1)i+(1-2)j)=-2

(fxi+fyj)(0ij)=2(f_xi+f_yj)(0i-j)=-2


fy=2-f_y=-2

fy=2\Rightarrow f_y=2


Gradient Δf=fxi+fyj\Delta f=f_x i+f_yj

           =2i+2j=2i+2j


Directional Derivative towards (4,6) is-

     =(2i+2j)((41)i+(62)j)=(2i+2j)(3i+4j)=66+8=14=(2i+2j)((4-1)i+(6-2)j)\\=(2i+2j)(3i+4j) \\ =66+8=14

    


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