Question 1

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

Accepted Answer

The answer of A plane passes through the points (1,...

Question 2

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.

Accepted Answer

The answer of A linear function f(x,y)has cross-sections f(x,4)= 2x...

Question 3

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

Accepted Answer

The answer of Sketch the level set of \(f (...

Question 4

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?

Accepted Answer

The answer of Consider the plane that passes through the...

Question 5

A linear function f(x, y)has the values f(4, 2)= 10, f(1, 2)= 4 and f(4, 1)=7.Find an equation for f.

Accepted Answer

The answer of A linear function f(x, y)has the values...

Question 6

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

Accepted Answer

The answer of Given the table of some values of...

Question 7

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

Accepted Answer

The answer of Find a formula for the linear function...

Question 8

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

Accepted Answer

The answer of Consider the (partial)contour diagram below for a...

Question 9

If f is a linear function, then $f ( 4,2 ) - f ( 4,1 ) = f ( 0,2 ) - f ( 0,1 )$

Accepted Answer

For a linear function, the difference in function values depends only on the difference in the input values, not on their specific values.

Question 10

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.

Accepted Answer

The answer of Determine a formula for the linear function...

Question 11

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

Accepted Answer

A linear function of two variables, say f(x, y), generally takes the form ax + by + c. The property f(x, y + 1) = f(x + 1, y) would imply a specific relationship between a and b that does not hold for all linear functions, thus this statement is not universally true.

Question 12

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

Accepted Answer

The answer of Find a formula for a linear function...

Question 13

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

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The answer of Given the table of some values of...

Question 14

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?

Accepted Answer

Substituting the point (-5, -1, -5) into the function $f(x, y, z) = x^2 + 4y^2 + z^2$ gives $f(-5, -1, -5) = (-5)^2 + 4(-1)^2 + (-5)^2 = 25 + 4 + 25 = 54$. Therefore, the point lies on the level surface defined by $x^2 + 4y^2 + z^2 = 54$.

Question 15

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.

Accepted Answer

The answer of Determine the equation of the plane which...

Question 16

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?

Accepted Answer

The answer of Describe the set of points whose distance...

Question 17

Find an equation for the plane passing through (1, -5, -2)and containing the x-axis.

Accepted Answer

The answer of Find an equation for the plane passing...

Question 18

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.

Accepted Answer

The answer of What is the domain of \(g (...

Question 19

Below is a contour diagram depicting D, the average fox population density as a function of $x _ { E }$ , kilometers east of the western end of England, and $x _ { N }$ , kilometers north of the same point.  Is D increasing or decreasing at the point (120, 25)in the northern direction?

Accepted Answer

The answer of Below is a contour diagram depicting D,...

Question 20

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

Accepted Answer

If all contours of a function f(x, y) are parallel lines, it indicates that the function changes at a constant rate in a specific direction, but this does not necessarily mean the function is linear. Non-linear functions can also have contours that are parallel lines, such as f(x, y) = ax + by + c, where a, b, and c are constants, and the function represents a plane, or more complex functions where the rate of change in one direction is constant.

Quiz 12: Functions of Several Variables

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

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.

Sketch the level set of f(x,y)=x2+z2f ( x , y ) = x ^ { 2 } + z ^ { 2 }f(x,y)=x2+z2 corresponding to f = 16 and describe it in words.

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?