Exam 12: Functions of Several Variables

Suppose that the temperature T of any point (x, y)is given by $T ( x , y ) = 100 - x ^ { 2 } - y ^ { 2 }$ . Sketch isothermal curves (i.e.contours)for T = 100, T = 75, T = 50 and T = 0.Be sure to label each contour. What does the graph of T(x, y)look like if it is sliced by the plane x = 4?

Classify the following surface as ellipsoid, elliptical paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, hyperboloid of two sheets, or cone. $- 9 x ^ { 2 } - 4 y ^ { 2 } + z ^ { 2 } = 9$

You are in a nicely heated cabin in the winter.Deciding that it's too warm, you open a small window.Let T be the temperature in the room, t minutes after the window was opened, x feet from the window.Is T an increasing or decreasing function of x?

Determine the values of a and b such that the following function is continuous everywhere. $f ( x , y ) = \left\{ \begin{array} { l l } 5 + \infty x + b y , & y \geq 2 \\4 + 5 x + 2 y , & y < 2\end{array} \right.$

Consider the function $z = f ( x , y ) = - 3 y - 2 x ^ { 2 }$ Suppose you are standing on the surface at the point where x = 2, y = -1.What is your altitude?

Represent (if possible)the following surfaces as graphs of functions f(x, y), and as level surfaces of the form g(x, y, z)= c.(There are many possible answers.) (a)The upper half of the sphere of radius 3, centered at (5, 5, 0). (b)The lower half of the cylinder of radius 4 around the x-axis. (c)The cone $z ^ { 2 } = 4 x ^ { 2 } + 4 y ^ { 2 }$

(a)Find an equation of the largest sphere that can fit inside the cubical space enclosed by the planes x = 1, x = 5, y = 2, y = 6, z = 2 and z = 6. (b)If we replace the plane z = 6 in part (a)with z = 7, what will be the new equation of the largest sphere?

Which of the following is a contour diagram for f(x, y)= sin x?

Your monthly payment, C(s, t), on a car loan depends on the amount, s, of the loan (in thousands of dollars), and the time, t, required to pay it back (in months).What is the meaning of C(7, 48)= 250?

Suppose that the graph of a linear function f passes through the point (3,1,6)and that the $z = 9$ level curve of f is the line $y = \frac { 1 } { 2 } x - 2$ .Find a formula for f.

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.

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?

Consider the sphere $( x + 1 ) ^ { 2 } + ( y - 0 ) ^ { 2 } + ( z - 1 ) ^ { 2 } = 4$ (a)What are the center and radius of this sphere? (b)Find an equation of the circle (if any)where the sphere intersects the plane x = -2.

The graph of the function f(x, y)is shown below. Draw graph of cross-sections with y fixed at y = 0, and y = 1.

Consider the figure shown below. Match the level curve with the function.

Your monthly payment, C(s, t), on a car loan depends on the amount, s, of the loan (in thousands of dollars), and the time, t, required to pay it back (in months).Is C an increasing or decreasing function of t?

Suppose that for all x and y the function f satisfies $| f ( x , y ) - 3 | > \sqrt { x ^ { 2 } + y ^ { 2 } } \text { and } | f ( x , y ) - 4 | \leq 5 \left( ( x - 4 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \right)$ . Determine, if possible, the values of the following limits.Explain your answer. Note that the limit may not exist nor be determined from the given information. (a) $\lim _ { ( x , y ) \rightarrow ( 0 \rho ) } f ( x , y )$ (b) $\lim _ { ( x , y ) \rightarrow ( 4,2 ) } f ( x , y )$

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

Let $f ( x , y ) = \frac { x - 5 y } { x + 5 y }$ .Describe the level curve of f at level 2.

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