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Chapter 12: Q. 12 (page 943)

Consider the function f defined as in exercise 11

a) use the definition of the partial derivatives to show that fx(x0,0)every value of x0but fy(x0,0)exists only when x0=0

b)use the definition of the partial derivatives to show thatf0(0,y0)every value ofy0butfx(0,y0)exists only wheny0=0.

Short Answer

Expert verified

We proved both the parts and showed the results

Step by step solution

01

Given information

We are given a functionf(x,y)=0ifxy=01ifxy鈮�0

02

Part (a) Step 2. The explanation for part (a).

Now consider the partial derivative

we have,

limh鈫�0f(x+h,y)-f(x,y)hlimh鈫�0(x+h)y-xyhlimh鈫�0hyhlimh鈫�0y

This limit is zero at point (x0,0)

Hence fx(x0,0)exists for all values of x0

Now consider,

localid="1653498286992" limk鈫�0f(x,y+k)-f(x,y)klimk鈫�0x(y+k)-xyklimk鈫�0xkklimk鈫�0x

At this point, it varies as the value of x0changes Hence the limit exists only when x0=0

03

Part (b) Step 1. The explanation for part (b).

Consider the partial derivatives

limk鈫�0f(x,y+k)-f(x,y)klimk鈫�0x(y+k)-xyklimk鈫�0x

At point (0,y0)this value will be 0.

Hence limit exists for any y0

Now consider,

localid="1653498432339" limh鈫�0f(x+h,y)-f(x,y)hlimh鈫�0(x+h)y-xyhlimh鈫�0y

At point, this limit varies hence it will only exist at point y0=0

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Most popular questions from this chapter

Let f(x,y)be a differentiable function such that 鈭�f(x,y)鈮�0for every point in the domain of f, and let Rbe a closed, bounded subset of role="math" localid="1649887954022" 鈩�2.Explain why the maximum and minimum of f restricted to Roccur on the boundary ofrole="math" localid="1649888770915" R.

In Exercises 37-42, use the partial derivatives of role="math" localid="1650186824938" fx,y=xyand the point e,3specified to

afind the equation of the line tangent to the surface defined by the function in the xdirection,

bfind the equation of the line tangent to the surface defined by the function in the ydirection, and

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

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.

f(x,y,z)=xy2z3,P=(0,0,0),v=鉄�1,鈭�2,鈭�1鉄�

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.

In Exercises 37-42, use the partial derivatives of gx,y=xcosyand the point 1,蟺specified to

afind the equation of the line tangent to the surface defined by the function in the xdirection,

bfind the equation of the line tangent to the surface defined by the function in the ydirection, and

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