Interpretations of the Derivative

[AInerd72]

QUESTION TEXT: When you breath, a muscle (called the diaphragm) reduces the pressure around your lungs and they expand to fill with air. The table shows the volume of the lung as a function of the reduction in pressure from the diaphragm. Pulmonologists (lung doctors) define the "compliance" of the lung as the derivative of this function.

a) What are the units of compliance?

b) Estimate the maximum compliance of the lung.

c) Explain why the compliance gets small when the lung is nearly full (around 1 Liter)

TABLE
Pressure Reduction Volume
(cm of water) (liters)
0 0.20
5 0.29
10 0.49
15 0.70
20 0.86
25 0.95
30 1.00

MY (SEMI-CORRECT) SOLUTION: I got parts a and b correct. The units of compliance are liters per cm of water. The compliance gets small when the lung is nearly full because it's impossible for the lung to stretch much more. However, the answer given for part (b) is .042 L/cm. I have no idea how they even began to arrive at that answer so any advice on where to start would be GREATLY appreciated!!

 

[Ishuda]

QUESTION TEXT: When you breath, a muscle (called the diaphragm) reduces the pressure around your lungs and they expand to fill with air. The table shows the volume of the lung as a function of the reduction in pressure from the diaphragm. Pulmonologists (lung doctors) define the "compliance" of the lung as the derivative of this function.

a) What are the units of compliance?

b) Estimate the maximum compliance of the lung.

c) Explain why the compliance gets small when the lung is nearly full (around 1 Liter)

TABLE
Pressure Reduction Volume
(cm of water) (liters)
0 0.20
5 0.29
10 0.49
15 0.70
20 0.86
25 0.95
30 1.00

MY (SEMI-CORRECT) SOLUTION: I got parts a and b correct. The units of compliance are liters per cm of water. The compliance gets small when the lung is nearly full because it's impossible for the lung to stretch much more. However, the answer given for part (b) is .042 L/cm. I have no idea how they even began to arrive at that answer so any advice on where to start would be GREATLY appreciated!!

Go back to (one of) the definitions of the derivative of a function
f'(x₁)=$\displaystyle \lim_{x_2 \to x_1}\, \frac{f(x_1)\, -\, f(x_2)}{x_1\, -\, x_2}$
so an approximation to the derivative is
f'(x₁)~$\displaystyle \frac{f(x_1)\, -\, f(x_2)}{x_1\, -\, x_2}$
and do that for the whole table, i.e. x₁=0 and x₂=5 with corresponding f's, then x₁=5 and x₂=10 with corresponding f's, then ...
Then find the maximum of those numbers.