To estimate the amount of blood that flows through a patient’s lung, cardiologists use the empirical formula

where P is a percentage of the total blood flow, x is the carbon dioxide output of the lung, y is the arteriovenous carbon dioxide difference in the lung, u is the carbon dioxide output of the lung, and v is the arteriovenous carbon dioxide difference in the other lung.

It is known that blood flows into the lungs to pick up oxygen and dump carbon dioxide, so the arteriovenous carbon dioxide difference measures the extent to which this exchange is accomplished. (The actual measurement is accomplished by a device called a cardiac shunt.) The carbon dioxide is then exhaled from the lungs so that oxygen-bearing air can be inhaled.

Compute the partial derivatives P_x, P_y, P_u, and P_v, and give a physiological interpretation of each derivative.

Transcribed Image Text:

P(x, y, u, v) 100xy xy + uv