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Chapter 12: Q. 21 (page 953)

In Exercises21-28 , find the directional derivative of the given

function at the specified point Pand in the direction of the

given unit vector u.

f(x,y)=x2鈭�y2atP=(2,3),u=22,22

Short Answer

Expert verified

The directional derivative of the given

function is鈭�2

Step by step solution

01

Given data

f(x,y)=x2鈭�y2

P=x0,y0=(2,3)u=(伪,尾)=22,22

02

Solution

Consider directional derivative

Dwfx0,y0=Limh鈫�0鈦�fx0+伪h,y0+尾h鈭�fx0,y0h

Dwf(2,3)=Limh鈫�0鈦�f2+22h,3+22h鈭�f(2,3)h

Therefore

f2+22h,3+22h=2+22h3鈭�3+22h2

=4+22h+12h2鈭�9+32h+12h2

Equation 2

fx0+y0=22鈭�32=鈭�5 Equation 3

03

Substitute

Substituting equation 2and 3

Dvf(2,3)=Limh鈫�0鈦�鈭�5鈭�2h+5h

=Limb鈫�0鈭�2

Dvf(2,3)=鈭�2

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Most popular questions from this chapter

Achainruleforfunctionsoftwovariables:Letf(x,y)=x2y3,x=scost,andy=ssint.Find鈭�f鈭�sand鈭�f鈭�tusingtheprecedingresults.

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cfind the equation of the plane containing the lines you found in parts aand b.

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Evaluate the following limits, or explain why the limit does not exist.

lim(x,y)鈫�(1,2)x2+y2x2-y2

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.

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