Exam 12: Functions of Several Variables

Does the point (-5, -1, -5)lie on any of the level surfaces of $f ( x , y , z ) = x ^ { 2 } + 4 y ^ { 2 } + z ^ { 2 }$ ? If so, what is the equation of that level surface?

Level surfaces of g(x,y,z)corresponding to different levels cannot intersect.

The points A = (1, 1, 1), B = (2, 4, 2), and C = (3, 2, 2)are the vertices of a triangle in space. Which of the vertices is closest to the origin?

If all of the contours of a function f(x, y)are parallel lines, then the function must be linear.

The level surfaces of the functions $g _ { 1 } ( x , y , z )$ $g _ { 2 } ( x , y , z )$ and $g _ { 3 } ( x , y , z )$ are shown in below respectively. Which function is is decreasing in the positive z-direction?

Let f(x, y, z)= ax + by + cz + d be a linear function of three variables, for constants a, b, c and d. Given that some of the cross sections of f are f(x, 1, 1)= 0 + 4x, f(0, y, -3)= 4 + 4y, and f(1, 1, z)= 6 - 2z, find a formula for f.

Match the graph with the function.

A certain piece of electronic surveying equipment is designed to operate in temperatures ranging from 0° C to 30° C.Its performance index, p(t, h), measured on a scale from 0 to 1, depends on both the temperature t and the humidity h of its surrounding environment.Values of the function p = f(t, h)are given in the following table.(The higher the value of p, the better the performance.) What is the value of p(0, 25)?

Sketch the level set of $f ( x , y ) = x ^ { 2 } + z ^ { 2 }$ corresponding to f = 16 and describe it in words.

Yummy Potato Chip Company has manufacturing plants in N.Y.and N.J.The cost of manufacturing depends on the quantities (in thousand of bags), q₁ and q₂, produced in the N.Y.and N.J.factories respectively.Suppose the cost function is given by $C \left( q _ { 1 } , q _ { 2 } \right) = 2 q _ { 1 } ^ { 2 } + q _ { 1 } q _ { 2 } + q _ { 2 } ^ { 2 } + 420$ (a)Find $C ( 10,25 )$ (b)By comparing the terms $2 q _ { 1 } ^ { 2 }$ and $q _ { 2 } ^ { 2 }$ in the above expression, the manager concluded that it is more expensive to produce in the N.Y.factory.Will shifting all the production to the N.J.factory minimize the production cost?

The picture below is the contour diagram of f(x, y).The areas between contours have been shaded.Lighter shades represent higher levels, while darker shades represent lower levels. Sketch the cross section of f(x, y)with x fixed at x = 0.5.

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

A certain piece of electronic surveying equipment is designed to operate in temperatures ranging from 0° C to 30° C.Its performance index, p(t, h), measured on a scale from 0 to 1, depends on both the temperature t and the humidity h of its surrounding environment.Values of the function p = f(t, h)are given in the following table.(The higher the value of p, the better the performance.) Describe the function p(10, h)and explain its meaning.

Given the table of some values of a linear function determine a formula for the function.

Match the following function with the graphs below. The function z = f(x, y)giving happiness as a function of health y and money x according to the statement of a fortune cookie: 'Whoever said money cannot buy happiness does not know where to shop.'

Suppose f is a function of two variables. The cross-sections of f with one variable fixed must also be functions.

Match the appropriate function $f ( x , y , z )$ with the geometric description of its level surface f = 0.

Find a linear function with f(-5, 5)= 1, whose graph is parallel to the level surfaces of g(x, y, z)= 10x + 15y + 5z.

The points A = (4, 1, 2), B = (3, -2, 3), and C = (-2, 3, -4)are the vertices of a triangle in space. Which of the vertices is closest to the yz-plane?

Consider the plane that passes through the points (1, 3, 10), (-1, -1, 0), and (2, 1, -1). If you were walking on this plane with no change in altitude, what would be the slope of your path in the xy-plane?