Exam 12: Functions of Several Variables

The contours of the function f(x, y)= 8x + 4y are all parallel lines with slope 2.

A plane passes through the points (1, 3, 7), (-1, 0, 6), and (2, 1, -3).Determine the equation of the plane.

The following table gives the number f(x, y)of grape vines, in thousands, of age x in year y. In 1986 a successful advertising campaign led to a dramatic increase in demand for premium wines.The growers followed by adding many more plants.Suppose a vine (the plant)produces the first harvestable grapes at age five, and is removed after sixteen years.How many (thousand)grape vines that bear fruit were there in the year 1986 and how many will be there in the year 1992 (assuming that no current vines die before 1992)? Enter your answers separated by a semi-colon.

Determine a formula for the linear function f(x, y), such that its cross-section with y fixed at y = 1 has equation z = 4x - 1, and its contour at level 0 is the line y = (4 + 4x)/5.

Describe the set of points whose distance from the y-axis equals the distance from the xz-plane. Write an equation for this set in the form G(x, y, z)= 0. Can the set be described as the graph of a function of two variables?

The following table gives the number f(x, y)of grape vines, in thousands, of age x in year y. In one year a fungal disease killed most of the older grapevines, and in the following year a long freeze killed most of the young vines.Which are these years?

Suppose f and g are different functions of three variables.Is it possible for the level surface f = 1 and the level surface g = 1 to be the same surface?

A soft drink company is interested in seeing how the demand for its products is affected by price.The company believes that the quantity, q, of soft drinks sold depends on $p _ { 1 }$ , the average price of the company's soft drinks, and $p _ { 2 }$ , the average price of competing soft drinks.Which of the graphs below is most likely to represent q as a function of $p _ { 1 }$ and $p _ { 2 }$ ?

What is the slope of the contour lines of the function f(x, y)= $- 3 + 9 x + 10 y ?$

Suppose f is a function of two variables.The contours of f must also be functions.

Find a possible formula for a function f(x, y)with the given values.

Draw a possible contour diagram for the function whose graph is shown below.Label your contours with reasonable z-values.

Let $f ( x , y ) = 3$ .Describe the cross-sections, the contours, and the graph of f.

Consider $f ( x , y ) = \frac { ( x - 2 ) ^ { 2 } } { ( x - 2 ) ^ { 2 } + y - 5 }$ .Use the cross-sections $f ( 2 , y )$ and $f ( x , 5 )$ to explain why $\lim _ { ( x , y ) \rightarrow ( 2,5 ) } f ( x , y )$ does not exist.

Draw the level curves for z = 2ln(x)- ln(y).

The points A = (-4, 5, -3), B = (-1, -3, -4), and C = (-2, 4, -4)are the vertices of a triangle in space. What is the length of the longest side of the triangle?

Find a function f(x, y, z)whose level surface at level 4 is the graph of the function $h ( x , y ) = 5 x ^ { 2 } y + 5 y - 1$ .

The contour diagram below shows the level curves of the difference between July and January mean temperatures in ° F. Does this graph support or contradict the claim that the largest annual temperature variations are found on the coasts of continents?

Describe in words the intersection of the surfaces $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and $z = 7 - 6 \left( x ^ { 2 } + y ^ { 2 } \right)$

Given that the equation of the level surface of f(x, y, z)at level -3 is $3 x ^ { 2 } + x y + z = 15$ , determine f(2, 3, -3)if possible.