Exam 12: Functions of Several Variables

Given the table of some values of a linear function complete the table:

In an exam, a student wrote the following answer. "The level curve of $f ( x , y )$ at the point (3,-4)is $x ^ { 2 } - 2 x y = 34$ ." How can you tell that the answer is wrong without even knowing the formula for f?

Consider the (partial)contour diagram below for a linear function Find an equation z = f(x, y)for the function.

Find a formula for a function f(x, y), given that its contour at level 9 has equation $x ^ { 2 } - 10 x y = - 1$

Find a formula for the linear function whose contours are shown below.

The diagram below shows the contour map for a circular island.Sketch the vertical cross-section of the island through the center.Your sketch should show concavity clearly.

Some level surfaces of the functions $g _ { 1 } ( x , y , z )$ $g _ { 2 } ( x , y , z )$ and $g _ { 3 } ( x , y , z )$ are shown below, respectively. Which function takes the value 0 at (0, 0, 0)?

Determine the equation of the plane which passes through the point (1, 3, -1), has slope 5 in the x-direction and slope -3 in the y-direction.

Describe in words, write equations, and give a sketch for the following set of points. $\text { Consider the set of points } ( x , y , z ) \text { whose distance from the } y z \text {-plane is } 3 \text {. }$

For what values of the constant k is the intersection between the set of points $y = x$ and the graph of $f ( x , y ) = 4 x ^ { 2 } - k y ^ { 2 }$ a straight line?

Match the function with the graph below.

If all the level surfaces of a function $f ( x , y , z )$ are parallel planes, then f must be a linear function.

Consider the function $f ( x , y ) = \frac { a x ^ { 2 } - b x } { - 7 + y }$ where a, b are positive numbers.For what values of (x, y)will the function f not be defined?