Help in calculus problem!!

Question

Hi, can anyone teach me how can I solve this problem? 

 

For the surface: 

g(x,y) = In(y + x²) + y

 

-- Find the directional derivative of ƒ(x,y) in the direction  u^→ = i^→ + 2j^→    at the point where x = √e, y = e.

 

-- Find the direction u^→ of maximum change at this point

 

-- Find the maximum change possible at this point. 

 

 

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Thank you so much and I will really appreciate your help. 

Answer 1

The directional derivative is usually called the gradient.   The gradient operator (grad ) operates on a scalar function to produce a vector function.    In two dimensions we have

  

   grad( g(x,y)) =    d g(x,y) / dx  i^→  +  d g(x,y) / dy j^→      For g = ln(y + x²) + y   this works out to

 

   grad (g) =       [ 2x/(y + x²) ] i^→    + [ 1/(y + x²)  + 1 ] j^→    for x = √e   and y = e   this becomes

 

   grad (g )   = [ 1/√e ] i^→    +   [ 1/2e +1 ] j^→  

 

   The maximum change is the magnitude of this vector which is given by the usual formula for the

magnitude of a vector quantity i.e.   sqrt ( x_component² + y_component²)   so

 

  maximum change =  sqrt ( 1/e + (1/2e +1)² )

 

The direction of this max change is arctan( y_component / x_component)   so

 

 direction ( angle with respect to x axis ) =  arctan( ( 1/2e +1 ) /(1/√e) )