Exam 12: Functions of Several Variables

Two contours of the function f(x, y)corresponding to different values of f cannot ever cross.

You are at (4, 2, 4)facing the yz-plane.You walk 3 units, turn right and walk for another 2 units.What are your coordinates now? Are you above or below the xy-plane?

Lawn King sells a push mower at price p₁ and a ride-on mower at price p₂.The demand function for each mower is a linear function of p₁ and p₂.Market research shows that if the price of the push mower is increased by $40, then the demand for it drops by 6400 units, while the demand for the ride-on mower increases by 1200 units.On the other hand, if the price of the ride-on mower is decreased by $60, then demand for it increases by 3000 units, while demand for the push mower drops by 9000 units.Currently, the company is selling 142,000 push mowers at a price of $300 and 19,000 ride-on mowers at a price of $1200.Find the two demand functions.

A linear function f(x,y)has cross-sections f(x,4)= 2x - 14 and f(2,y)= -4y + 6.Find an equation for f.

Consider the function $z = f ( x , y ) = 3 y - 4 x ^ { 3 }$ Suppose you are standing on the surface at the point where x = 3, y = 1.If you start to move on the surface parallel to the y-axis in the direction of increasing y, does your height increase or decrease?

Let $f ( x , y ) = y ^ { 2 } - 8 y x + 16 x ^ { 2 }$ Find the contour of f that passes through the point $( 0,2 )$ .

Let $g ( x , y , z ) = 5$ .Describe the level surfaces of g.

What value of c (if any)makes the following function continuous at (0, 0)? $f ( x , y ) = \left\{ \begin{array} { c c } \frac { x ^ { 4 } - y ^ { 4 } } { x - y } & \text { if } ( x , y ) \neq ( 0,0 ) \\c & \text { otherwise }\end{array} \right.$

A function f(x, y)is defined for (x, y) $\neq$ (0, 0).Does $\lim _ { ( x , y ) \rightarrow ( 0 \rho ) } f ( x , y )$ exist if f has the contour diagram below? Assume that different contours represent different values of the function.Explain your answer.

If f is any linear function of two variables, then f(x, y + 1)= f(x + 1, y).

Match the graph with the function.

The following figure contains the graphs of the cross sections z = f(a, y)for a = -2, -1, 0, 1, 2.Which of the graphs of z = f(x, y)in A and B best fits this information?

A spherical ball of radius four units is in a corner touching both walls and floor.What is the radius of the largest spherical ball that can be fit into the corner behind the given ball? (Hint: The smaller ball will not touch the corner point where the walls meet the floor.)

Using the fact that $\left| \frac { y ^ { 2 } ( x - 2 ) } { ( x - 2 ) ^ { 2 } + y ^ { 2 } } \right| \leq \sqrt { ( x - 2 ) ^ { 2 } + y ^ { 2 } }$ for all x and y except $( x , y ) = ( 2,0 )$ , show that $\lim _ { ( x , y ) \rightarrow ( 2 \rho ) } \frac { y ^ { 2 } ( x - 2 ) } { ( x - 2 ) ^ { 2 } + y ^ { 2 } } = 0$ .

Suppose that $g ( x , y ) = \cos ( A x + B y )$ .Find A and B if you know that the period of $g ( x , 0 )$ is $\frac { 2 \pi } { 11 }$ and the period of $g ( 0 , y )$ is $3$ .

Let $f ( x , y ) = \cos \sqrt { 9 - x ^ { 2 } - y ^ { 2 } }$ What is the domain of f? What are the possible values of f?

What is the domain of $g ( x , y , z ) = \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 3 } ?$ Describe the level surface g =5.

Consider the function $f ( x , y ) = a ^ { x } b ^ { y }$ , for certain constants a and b.Simplify the following expression. $\frac { f ( x + 4 , y + 5 ) } { f ( x , y ) }$

Suppose f is a function of two variables. Given a point (x,y)in the plane, there is a level curve of f passing through (x,y).

Find a formula for a linear function z = f(x, y)whose f = 0 contour is the line y = 2x + 2.