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 miles, where p is measured in psi. What are the units of f^('')(p) ? psi miles psi / mile miles / psi psi // mile ^(2) miles //psi^(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 miles, where  p is measured in psi. What are the units of  f^('')(p) ? psi miles psi / mile miles / psi psi  // mile  ^(2) miles  //psi^(2)

Bytelearn · Accepted Answer

Define f'(p) units: The function f(p) measures the height above sea level in miles for a given pressure p in psi. To find the units of f''(p), we need to consider the units of the first derivative f'(p) first.Calculate f'(p) units: The first derivative f'(p) represents the rate of change of height with respect to pressure. Since f(p) is in miles and p is in psi, the units of f'(p) would be miles per psi (miles/psi).Determine f''(p) units: The second derivative f''(p) represents the rate of change of the first derivative f'(p) with respect to pressure. Therefore, we need to take the units of f'(p) and divide by the units of pressure again.Final units of f''(p): The units of f''(p) will be the units of f'(p) divided by the units of pressure, which is psi. So, the units of f''(p) will be (miles/psi) per psi, which simplifies to miles per psi squared (miles/psi^2).

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