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The amount of water in a tank is measured by the differentiable function f, where f(t) is measured in liters and t is measured in minutes. What are the units of int_(3)^(6)f^(')(t)dt ? liters minutes liters / minute minutes / liter liters / minute ^(2) minutes / liter ^(2)

The amount of water in a tank is measured by the differentiable function f f , where f(t) f(t) is measured in liters and t t is measured in minutes. What are the units of ∫36f′(t)dt \int_{3}^{6} f^{\prime}(t) d t ?\newlineliters\newlineminutes\newlineliters / minute\newlineminutes / liter\newlineliters / minute 2 { }^{2} \newlineminutes / liter 2 { }^{2}

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Q. The amount of water in a tank is measured by the differentiable function f f , where f(t) f(t) is measured in liters and t t is measured in minutes. What are the units of ∫36f′(t)dt \int_{3}^{6} f^{\prime}(t) d t ?\newlineliters\newlineminutes\newlineliters / minute\newlineminutes / liter\newlineliters / minute 2 { }^{2} \newlineminutes / liter 2 { }^{2}
  1. Rate of Change Function: The integral of a rate of change function gives the net change in the original function over the interval. In this case, f′(t)f'(t) represents the rate of change of the water volume in the tank with respect to time. The integral of f′(t)f'(t) from 33 to 66 will give the net change in the volume of water in the tank between t=3t=3 minutes and t=6t=6 minutes.
  2. Units of Measurement: Since f(t)f(t) is measured in liters and tt is measured in minutes, the rate of change f′(t)f'(t) would be measured in liters per minute. This is because derivatives represent the rate of change, and dividing the change in volume (liters) by the change in time (minutes) gives liters per minute.
  3. Integration Process: When we integrate f′(t)f'(t) over the interval from 33 to 66, we are essentially summing up all the little changes in volume over the time interval. The result of this integral will give us the total change in volume, which is measured in liters, since we are adding up liters per minute over a number of minutes.
  4. Total Change Calculation: Therefore, the units of the integral from 33 to 66 of f′(t)extdtf'(t) ext{ dt} are liters, because we are finding the total amount of water that has been added or removed from the tank over the time interval from 33 to 66 minutes.

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