Suppose that z is a linear function of x and y with slope2 in the x-direction and slope

Exercises 12 each contain a partial table of values for a linear function. Fill in the blanks. 1

Exercises 12 each contain a partial table of values for a linear function. Fill in the blanks. 2

In Exercises 36, could the tables of values represent a linear function? 3

In Exercises 36, could the tables of values represent a linear function? 4

In Exercises 36, could the tables of values represent a linear function? 5

In Exercises 36, could the tables of values represent a linear function? 6

In Exercises 78, could the contour diagram represent a linear function? 7

In Exercises 78, could the contour diagram represent a linear function? 8

Find the equation of the linear function z = c+mx+ny whose graph contains the points (0, 0, 0), (0, 2, 1), and (3, 0, 4).

Find the linear function whose graph is the plane through the points (4, 0, 0),(0, 3, 0) and (0, 0, 2).

Find an equation for the plane containing the line in the xy-plane where y = 1, and the line in the xz-plane where z = 2.

Find the equation of the linear function z = c+mx+ny whose graph intersects the xz-plane in the line z = 3x + 4 and intersects the yz-plane in the line z = y + 4.

Suppose that z is a linear function of x and y with slope 2 in the x-direction and slope 3 in the y-direction. (a) A change of 0.5 in x and 0.2 in y produces what change in z? (b) If z = 2 when x = 5 and y = 7, what is the value of z when x = 4.9 and y = 7.2?

(a) Find a formula for the linear function whose graph is a plane passing through point (4, 3, 2) with slope 5 in the x-direction and slope 3 in the y-direction. (b) Sketch the contour diagram for this function.

A store sells CDs at one price and DVDs at another price. Figure 12.67 shows the revenue (in dollars) of the music store as a function of the number, c, of CDs and the number, d, of DVDs that it sells. What is the price of a CD? What is the price of a DVD?

The charge, C, in dollars, to use an Internet service is a function of m, the number of months of use, and t, the total number of minutes on-line: C = f(m, t) = 35 + 15m + 0.05t. (a) Is f a linear function? (b) Give units for the coefficients of m and t, and interpret them as charges. (c) Interpret the intercept 35 as a charge. (d) Find f(3, 800) and interpret your answer.

A manufacturer makes two products out of two raw materials. Let q1, q2 be the quantities sold of the two products, p1, p2 their prices, and m1, m2 the quantities purchased of the two raw materials. Which of the following functions do you expect to be linear, and why? In each case, assume that all variables except the ones mentioned are held fixed. (a) Expenditure on raw materials as a function of m1 and m2. (b) Revenue as a function of q1 and q2. (c) Revenue as a function of p1 and q1.

Problems 1820 concern Table 12.10, which gives the number of calories burned per minute for someone roller-blading, as a function of the persons weight and speed. Does the data in Table 12.10 look approximately linear? Give a formula for B, the number of calories burned per minute in terms of the weight, w, and the speed, s. Does the formula make sense for all weights or speeds?

Problems 1820 concern Table 12.10, which gives the number of calories burned per minute for someone roller-blading, as a function of the persons weight and speed. Who burns more total calories to go 10 miles: A 120 lb person going 10 mph or a 180 lb person going 8 mph? Which of these two people burns more calories per pound for the 10-mile trip?

Problems 1820 concern Table 12.10, which gives the number of calories burned per minute for someone roller-blading, as a function of the persons weight and speed. Use Problem 18 to give a formula for P, the number of calories burned per pound, in terms of w and s, for a person weighing w lbs roller-blading 10 miles at s mph.

For Problems 2122, find possible equations for linear functions with the given contour diagrams. 21

For Problems 2122, find possible equations for linear functions with the given contour diagrams. 22

For Problems 2324, find an equation for the linear function with the given values. 23

For Problems 2324, find an equation for the linear function with the given values. 24

It is difficult to graph a linear function by hand. One method that works if the x, y, and z-intercepts are positive is to plot the intercepts and join them by a triangle as shown in Figure 12.68; this shows the part of the plane in the octant where x 0, y 0, z 0. If the intercepts are not all positive, the same method works if the x, y, and z-axes are drawn from a different perspective. Use this method to graph the linear functions in Problems 2528. z = 2 2x + y

It is difficult to graph a linear function by hand. One method that works if the x, y, and z-intercepts are positive is to plot the intercepts and join them by a triangle as shown in Figure 12.68; this shows the part of the plane in the octant where x 0, y 0, z 0. If the intercepts are not all positive, the same method works if the x, y, and z-axes are drawn from a different perspective. Use this method to graph the linear functions in Problems 2528. z = 2 x 2y

It is difficult to graph a linear function by hand. One method that works if the x, y, and z-intercepts are positive is to plot the intercepts and join them by a triangle as shown in Figure 12.68; this shows the part of the plane in the octant where x 0, y 0, z 0. If the intercepts are not all positive, the same method works if the x, y, and z-axes are drawn from a different perspective. Use this method to graph the linear functions in Problems 2528. z =4+ x 2y

It is difficult to graph a linear function by hand. One method that works if the x, y, and z-intercepts are positive is to plot the intercepts and join them by a triangle as shown in Figure 12.68; this shows the part of the plane in the octant where x 0, y 0, z 0. If the intercepts are not all positive, the same method works if the x, y, and z-axes are drawn from a different perspective. Use this method to graph the linear functions in Problems 2528. z = 6 2x 3y

Let f be the linear function f(x, y) = c + mx + ny, where c, m, n are constants and n = 0. (a) Show that all the contours of f are lines of slope m/n. (b) For all x and y, show f(x + n, y m) = f(x, y). (c) Explain the relation between parts (a) and (b).

Problems 3031 refer to the linear function z = f(x, y) whose values are in Table 12.11. Each column of Table 12.11 is linear with the same slope, m = z/x = 4/5. Each row is linear with the same slope, n = z/y = 3/2. We now investigate the slope obtained by moving through the table along lines that are neither rows nor columns. (a) Move down the diagonal of the table from the upper left corner (z = 3) to the lower right corner (z = 31). What do you notice about the changes in z? Now move diagonally from z = 6 to z = 27. What do you notice about the changes in z now? (b) Move in the table along a line right one step, up two steps from z = 19 to z = 9. Then move in the same direction from z = 22 to z = 12. What do you notice about the changes in z? (c) Show that z = mx + ny. Use this to explain what you observed in parts (a) and (b).

Problems 3031 refer to the linear function z = f(x, y) whose values are in Table 12.11. If we hold y fixed, that is we keep y = 0, and step in the positive x-direction, we get the x-slope, m. If instead we keep x = 0 and step in the positive y-direction, we get the y-slope, n. Fix a step in which neither x = 0 nor y = 0. The slope in the x, y direction is Slope = Rise Run = z Length of step = z (x)2 + (y)2 . (a) Compute the slopes for the linear function in Table 12.11 in the direction of x = 5, y = 2. (b) Compute the slopes for the linear function in Table 12.11 in the direction of x = 10, y = 2.

For the contour diagrams (I)(IV) on 2 x, y 2, pick the corresponding function. f(x, y)=2x + 3y + 10 k(x, y) = 2x + 3y + 12 g(x, y)=2x + 3y + 60 m(x, y) = 2x + 3y + 60 h(x, y)=2x 3y + 12 n(x, y) = 2x 3y + 14 j(x, y)=2x 3y + 60 p(x, y) = 2x 3y + 60

In Problems 3334, explain what is wrong with the statement. If the contours of f are all parallel lines, then f is linear.

In Problems 3334, explain what is wrong with the statement. A function f(x, y) with linear cross-sections for x fixed and linear cross-sections for y fixed is a linear function.

In Problems 3536, give an example of: A table of values, with three rows and three columns, for a nonlinear function that is linear in each row and in each column.

In Problems 3536, give an example of: A linear function whose contours are lines with slope 2.

Are the statements in Problems 3748 true or false? Give reasons for your answer. The planes z = 3+2x + 4y and z = 5+2x + 4y intersect.

Are the statements in Problems 3748 true or false? Give reasons for your answer. The function represented in Table 12.12 is linear.

Are the statements in Problems 3748 true or false? Give reasons for your answer. Contours of f(x, y)=3x + 2y are lines with slope 3.

Are the statements in Problems 3748 true or false? Give reasons for your answer. If f is linear, then the contours of f are parallel lines.

Are the statements in Problems 3748 true or false? Give reasons for your answer. If f(0, 0) = 1, f(0, 1) = 4, f(0, 3) = 5, then f cannot be linear.

Are the statements in Problems 3748 true or false? Give reasons for your answer. The graph of a linear function is always a plane.

Are the statements in Problems 3748 true or false? Give reasons for your answer. The cross-section x = c of a linear function f(x, y) is always a line.

Are the statements in Problems 3748 true or false? Give reasons for your answer. There is no linear function f(x, y) with a graph parallel to the xy-plane.

Are the statements in Problems 3748 true or false? Give reasons for your answer. There is no linear function f(x, y) with a graph parallel to the xz-plane.

Are the statements in Problems 3748 true or false? Give reasons for your answer. A linear function f(x, y)=2x+ 3y5, has exactly one point (a, b) satisfying f(a, b)=0.

Are the statements in Problems 3748 true or false? Give reasons for your answer. In a table of values of a linear function, the columns have the same slope as the rows.

Are the statements in Problems 3748 true or false? Give reasons for your answer. There is exactly one linear function f(x, y) whose f = 0 contour is y = 2x + 1.