Let f(x,y)=xex2−y and P=(1,1). (a) Calculate ‖∇fp‖. (b) Find the rate of change of f in the direction ∇fp (c) Find the rate of change of f in the direction of a vector making an angle of 45∘ with ∇fp

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Stuart Rountree · Accepted Answer

Step 1  As per given by the question,  There are given that the f(x,y)=xex2−y,P(1,1)  Now,  (a). Calculate ‖∇fp‖.  ∇f=&amp;lt;fx(x,y),fy(x,y)&amp;gt;  ∇f=&amp;lt;x(2xex2−y)+ex2−y,−xex2−y&amp;gt;  ∇f=&amp;lt;(2x2ex2−y)+ex2−y,−xex2−y&amp;gt;  Now,  Evaluate ∇f at the point P(1,1).  ∇f(1,1)=&amp;lt;2(1)e(1)−1+e(1)−1,−(1)e1−1&amp;gt;  ∇f(1,1)=&amp;lt;3,−1&amp;gt;  Step 2  Therefore,  ‖∇fp‖=(3)2+(−1)2=10  Hence, the value of ‖∇fp‖  is 10.

Ronnie Schechter · Accepted Answer

(b).  v=&amp;lt;1,1&amp;gt;  now,  From formula,  &amp;lt;&amp;gt;Duf(p)=1‖v‖∇fp⋅v  Step 3  Then,  Duf(p)=1‖v‖∇fp⋅v  Du(1,1)=1‖11‖&amp;lt;3,−1&amp;gt;⋅&amp;lt;1,1&amp;gt;  Du(1,1)=22

karton · Accepted Answer

(c) now, apply Duf(p)=(∇fp)cos⁡θ, so Duf(1,1)=10cos⁡45∘ Duf(1,1)=202 Hence, the rate of change of x is 202

Let f(x, y) = xe^{x^{2}-y} and P = (1, 1).

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