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Chapter 12: Q. 39 (page 954)

In Exercises 39-42, show that the directional derivative of the given function at the specified point pis zero for every unit vector u.

f(x,y)=xy+2x鈭�yatP=(1,鈭�2)

Short Answer

Expert verified

The given function's directional derivative is鈭�f(P).u=0.

Step by step solution

01

Unit vector.

We've f(x,y)=xy+2y-y placed a unit vector P(1,-2) at the supplied location u=(伪,尾)=(1,1).

02

Derivation of Function.

With directional unit vector u, the directional derivative of a function at point pis given by,

localid="1650642877881" 鈭�f(P)鈰�u=鈭�f(1,鈭�2)脳u=dfdx(1,鈭�2)i+dfdy(1,鈭�2)j脳i+ji2+j2

localid="1650642908587" =(y+2)(1,鈭�2)2i+(x鈭�1)(1,鈭�2)j脳i+j2

localid="1650642922908" =[(鈭�2+2)i+(1鈭�1)j]脳i+j2

localid="1650642941206" =[0i+0j]脳i+j2

localid="1650458792936" 鈭�f(P).u=0

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Most popular questions from this chapter

In Exercises, find the maximum and minimum of the function f subject to the given constraint. In each case explain why the maximum and minimum must both exist.

f(x,y)=x+ywhenx2+4y2=16

Construct examples of the thing(s) described in

the following.

Try to find examples that are different than

any in the reading.

(a) A function z = f(x, y) for which 鈭噁(0, 0) = 0 but f is

not differentiable at (0, 0).

(b) A function z = f(x, y) for which 鈭噁(0, 0) = 0 for every

point in R2.

(c) A function z = f(x, y) and a unit vector u such that

Du f(0, 0) = 鈭噁(0, 0) 路 u.

Solve the exact differential equations in Exercises 63鈥�66.ycos(xy)鈭�3+(xcos(xy)+2)dydx=0.

Gradients: Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.

f(x, y ,z) = ln(x + y + z), P = (e, 0, 鈭�1) .

Given a function of n variables, z=fx1,x2,鈥�,xn,and a constraint equation, gx1,x2,鈥�,xn=0,how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?
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