The atmospheric pressure of the air changes with height above sea level. The height above sea level at a given pressure can be measured by the differentiable function f(p) in feet, where p is measured in psi. What are the units of f^('')(p)? feet psi feet / psi psi / foot feet //psi^(2) psi / foot ^(2)

Question

The atmospheric pressure of the air changes with height above sea level. The height above sea level at a given pressure can be measured by the differentiable function  f(p) in feet, where  p is measured in psi. What are the units of  f^('')(p)? feet psi feet / psi psi / foot feet  //psi^(2) psi / foot  ^(2)

Bytelearn · Accepted Answer

Function f(p): The function f(p) represents the height above sea level in feet as a function of pressure in psi. The first derivative f'(p) would represent the rate of change of height with respect to pressure. The units of f'(p) would therefore be the units of f divided by the units of p, which is feet per psi (feet/psi).First Derivative f'(p): The second derivative f''(p) represents the rate of change of the rate of change of height with respect to pressure. This means we are taking the derivative of f'(p) with respect to p again. To find the units of f''(p), we take the units of f'(p) and divide by the units of p again. Since f'(p) has units of feet/psi, when we take the derivative again with respect to p, we get units of (feet/psi) per psi, which simplifies to feet per psi squared (feet/psi^2).

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