The cornea is the front surface of the eye. Corneal specialists use a TMS, or

Figure 8.20 shows contours for the function z = f(x, y). Is z an increasing or a decreasing function of x? Is z an increasing or a decreasing function of y? 1 2 3 4 5 6 1 2 3 4 5 6 10 9 8 7 6 5 4 3 2 1 x y Figure 8.20

Figure 8.21 is a contour diagram for the sales of a product as a function of the price of the product and the amount spent on advertising. Which axis corresponds to the amount spent on advertising? Explain. 1 2 3 4 5 6 1 2 3 4 5 6 Q = 5000 Q = 4000 Q = 3000 Q = 2000 x y Figure 8.21

Figure 8.22 shows the contours of the temperature H in a room near a recently opened window. Label the three contours with reasonable values of H if the house is in the following locations. (a) Minnesota in winter (where winters are harsh). (b) San Francisco in winter (where winters are mild). (c) Houston in summer (where summers are hot). (d) Oregon in summer (where summers are mild). Window Room Figure 8.22

A topographic map is given in Figure 8.23. How many hills are there? Estimate the x- and y-coordinates of the tops of the hills. Which hill is the highest? A river runs through the valley; in which direction is it flowing?

In Problems 512, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. f(x, y) = x + y

In Problems 512, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. f(x, y) = 3x + 3y

In Problems 512, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. f(x, y) = x + y + 1

In Problems 512, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. f(x, y) = 2x y

In Problems 512, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. f(x, y) = x y

In Problems 512, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. f(x, y) = y x2

In Problems 512, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. f(x, y) = x2 + y2

In Problems 512, sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. f(x, y) = xy

Draw a contour diagram for C(d,m) = 40d + 0.15m. Include contours for C = 50, 100, 150, 200.

Maple syrup production is highest when the nights are cold and the days are warm. Make a possible contour diagram for maple syrup production as a function of the high (daytime) temperature and the low (nighttime) temperature. Label the contours with 10, 20, 30, and 40 (in liters of maple syrup).

Hiking on a level trail going due east, you decide to leave the trail and climb toward the mountain on your left. The farther you go along the trail before turning off, the gentler the climb. Sketch a possible topographical map showing the elevation contours.

Give the range of daily high temperatures for: (a) Pennsylvania (b) NorthDakota (c) California

Sketch a possible graph of the predicted high temperature T on a line north-south through Topeka.

Sketch possible graphs of the predicted high temperature on a north-south line and an east-west line through Boise.

A manufacturer sells two products, one at a price of $4000 a unit and the other at a price of $13,000 a unit. A quantity q1 of the first product and q2 of the second product are sold at a total cost of $(4000 + q1 + q2) to the manufacturer. (a) Express the manufacturers profit, , as a function of q1 and q2. (b) Sketch contours of for = 10,000, = 20,000, and = 30,000 and the break-even curve = 0.

The contour diagram in Figure 8.24 shows your happiness as a function of love and money. (a) Describe in words your happiness as a function of: (i) Money, with love fixed. (ii) Love, with money fixed. (b) Graph two different cross-sections with love fixed and two different cross-sections with money fixed. 4 6 8 10 love money Figure 8.24

Sketch a contour diagram for z = y sin x. Include at least four labeled contours. Describe the contours in words and how they are spaced.

Each of the contour diagrams in Figure 8.25 shows population density in a certain region. Choose the contour diagram that best corresponds to each of the following situations. Many different matchings are possible. Pick any reasonable one and justify your choice. (a) The center of the diagram is a city. (b) The center of the diagram is a lake. (c) The center of the diagram is a power plant. 300 400 500 (I) 100 200 300 (II) 110 105 100 (III) Figure 8.25

Figure 8.26 shows contours of the function giving the species density of breeding birds at each point in the US, Canada, and Mexico.3 Are the following statements true or false? Explain your answers. (a) Moving from south to north across Canada, the species density increases. (b) In general, peninsulas (for example, Florida, Baja California, the Yucatan) have lower species densities than the areas around them. (c) The species density around Miami is over 100. (d) The greatest rate of change in species density with distance is in Mexico. If this is true, mark the point and direction giving the maximum rate. 40 140 80 80 100 120 140 160 160 180 120 180 140 100 200 160 200 140 100 180 160 280 240 320360 440 480 240 180 Miami Yucatan Baja California Figure 8.26

The concentration, C, of a drug in the blood is given by C = f(x, t) = tet(5x), where x is the amount of drug injected (in mg) and t is the number of hours since the injection. The contour diagram of f(x, t) is given in Figure 8.27. Explain the diagram by varying one variable at a time: describe f as a function of x if t is held fixed, and then describe f as a function of t if x is held fixed. 1 2 3 4 5 1 2 3 4 5 x t 0.1 0.2 0.3 0.4 0.5 Figure 8.27

The wind chill tells you how cold it feels as a function of the air temperature and wind speed. Figure 8.28 is a contour diagram of wind chill (F). (a) If the wind speed is 15 mph, what temperature feels like 20F? (b) Estimate the wind chill if the temperature is 0F and the wind speed is 10 mph. (c) Humans are at extreme risk when the wind chill is below 50F. If the temperature is 20F, estimate the wind speed at which extreme risk begins. (d) If the wind speed is 15 mph and the temperature drops by 20F, approximately how much colder do you feel? 60 40 20 0 20 40 10 20 30 80 60 40 20 0 20 air temp, F wind speed, mph Figure 8.28

In a small printing business, P = 2N0.6V 0.4, where N is the number of workers, V is the value of the equipment, and P is production, in thousands of pages per day. (a) If this company has a labor force of 300 workers and 200 units worth of equipment, what is production? (b) If the labor force is doubled (to 600 workers), how does production change? (c) If the company purchases enough equipment to double the value of its equipment (to 400 units), how does production change? (d) If both N and V are doubled from the values given in part (a), how does production change?

Figure 8.29 shows a contour map of a hill with two paths, A and B. (a) On which path, A or B, will you have to climbmore steeply? (b) On which path, A or B, will you probably have a better view of the surrounding countryside? (Assume trees do not block your view.) (c) Alongside which path is there more likely to be a stream?

Figure 8.30 shows cardiac output (in liters per minute) in patients suffering from shock as a function of blood pressure in the central veins (in mm Hg) and the time in hours since the onset of shock.4 1 2 3 4 5 2 4 6 8 10 12 2 4 8 6 10 12 time (hours) pressure (mm Hg) Figure 8.30 (a) In a patient with blood pressure of 4 mm Hg, what is cardiac output when the patient first goes into shock? Estimate cardiac output three hours later. How much time has passed when cardiac output is reduced to 50% of the initial value? (b) In patients suffering from shock, is cardiac output an increasing or decreasing function of blood pressure? (c) Is cardiac output an increasing or decreasing function of time, t, where t represents the elapsed time since the patient went into shock? (d) If blood pressure is 3 mm Hg, explain how cardiac output changes as a function of time. In particular, does it change rapidly or slowly during the first two hours of shock? During hours 2 to 4? During the last hour of the study? Explain why this information is useful to a physician treating a patient for shock.

The cornea is the front surface of the eye. Corneal specialists use a TMS, or Topographical Modeling System, to produce a map of the curvature of the eyes surface. A computer analyzes light reflected off the eye and draws level curves joining points of constant curvature. The regions between these curves are colored different colors. The first two pictures in Figure 8.31 are crosssections of eyes with constant curvature, the smaller being about 38 units and the larger about 50 units. For contrast, the third eye has varying curvature. (a) Describe in words how the TMS map of an eye of constant curvature will look. (b) Draw the TMS map of an eye with the cross-section in Figure 8.32. Assume the eye is circular when viewed from the front, and the cross-section is the same in every direction. Put reasonable numeric labels on your level curves. Large curvature Small curvature Small (or zero) curvature Large curvature Figure 8.31: Pictures of eyes with different curvature Figure 8.32

The power P produced by a windmill is proportional to the square of the diameter d of the windmill and to the cube of the speed v of the wind.5 (a) Write a formula for P as a function of d and v. (b) A windmill generates 100 kW of power at a certain wind speed. If a second windmill is built having twice the diameter of the original, what fraction of the original wind speed is needed by the second windmill to produce 100 kW? (c) Sketch a contour diagram for P.

Antibiotics can be toxic in large doses. If repeated doses of an antibiotic are to be given, the rate at which the medicine is excreted through the kidneys should be monitored by a physician. One measure of kidney function is the glomerular filtration rate, or GFR, which measures the amount of material crossing the outer (or glomerular) membrane of the kidney, in milliliters per minute. A normal GFR is about 125 ml/min. Figure 8.33 gives a contour diagram of the percent, P, of a dose of mezlocillin (an antibiotic) excreted, as a function of the patients GFR and the time, t, in hours since the dose was administered. 6 (a) In a patient with a GFR of 50, approximately how long will it take for 30% of the dose to be excreted? (b) In a patient with a GFR of 60, approximately what percent of the dose has been excreted after 5 hours? (c) Explain how we can tell from the graph that, for a patient with a fixed GFR, the amount excreted changes very little after 12 hours. (d) Is the percent excreted an increasing or decreasing function of time? Explain why this makes sense. (e) Is the percent excreted an increasing or decreasing function of GFR? Explain what this means to a physician giving antibiotics to a patient with kidney disease. 4 8 12 16 20 24 10 20 30 40 50 60 70 80 90 100 P = 60 P = 50 P = 40 P = 30 P = 20 P = 10 t (hours) GFR(ml/min) Figure 8.33

Each contour diagram (a) and (b) shows satisfaction with quantities of two items X and Y combined. Match (a) and (b) with the items in (I) and (II). 50 40 30 20 10 X (a) Y 10 20 30 40 50 X (b) Y (I) X: Hours worked; Y : Time spent commuting (II) X: Hourly wage; Y : Time for leisure

Each contour diagram (a) and (b) shows satisfaction with quantities of two items X and Y combined. Match (a) and (b) with the items in (I) and (II). 10 20 30 40 50 X (a) Y 10 20 30 40 50 X (b) Y (I) X: Income; Y : Leisure time (II) X: Income; Y : Hours worked

Figure 8.34 shows a contour plot of job satisfaction as a function of the hourly wage and the safety of the workplace. Match the jobs at points P,Q, and R with the three descriptions. (a) The job is so unsafe that higher pay alone would not increase my satisfaction very much. (b) I could trade a little less safety for a littlemore pay. It would not matter to me. (c) The job pays so little that improving safety would not make me happier. 5 10 15 20 5 10 15 20 P Q R Hourly wage Safety level Figure 8.34

A shopper buys x units of item A and y units of item B, obtaining satisfaction s(x, y) from the purchase. (Satisfaction is called utility by economists.) The contours s(x, y) = xy = c are called indifference curves because they show pairs of purchases that give the shopper the same satisfaction. (a) A shopper buys 8 units of A and 2 units of B. What is the equation of the indifference curve showing the other purchases that give the shopper the same satisfaction? Sketch this curve. (b) After buying 4 units of item A, howmany units of B must the shopper buy to obtain the same satisfaction as obtained from buying 8 units of A and 2 units of B? (c) The shopper reduces the purchase of item A by k, a fixed number of units, while increasing the purchase of B to maintain satisfaction. In which of the following cases is the increase in B largest? Initial purchase of A is 6 units Initial purchase of A is 8 units