^{2}) + y

^{→ }= i

^{→ }+ 2j

^{→ }at the point where x = √e, y = e.

^{→ }of maximum change at this point

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

For the surface:

g(x,y) = In(y + x^{2}) + 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.

--------------------------

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

Tutors, sign in to answer this question.

Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...

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 + x^{2}) + y this works out to

grad (g) = [ 2x/(y + x^{2}) ] i^{→} + [ 1/(y + x^{2}) + 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_component^{2} + y_component^{2}) 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) )

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