Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.
When a faucet is turned ​​on to fill a bathtub, the volume of water in gallons in the tub after t minutes is V(t)=3t. Find V′(12).

Verified step by step guidance

1. First, identify the function given: V(t) = 3t, which represents the volume of water in the bathtub in gallons as a function of time in minutes.
2. To find the derivative V'(t), apply the basic rule of differentiation for a linear function. The derivative of V(t) = 3t with respect to t is V'(t) = 3.
3. Evaluate the derivative at the given point t = 12. Since V'(t) = 3, V'(12) = 3.
4. Interpret the physical meaning of the derivative V'(12) = 3. This means that the rate at which the volume of water in the bathtub is increasing is 3 gallons per minute.
5. Include units in your interpretation: The derivative V'(12) = 3 gallons per minute indicates that at t = 12 minutes, the water is being added to the bathtub at a constant rate of 3 gallons per minute.

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes with respect to changes in its input. In this context, it represents the instantaneous rate of change of the volume of water in the bathtub with respect to time. Mathematically, it is defined as the limit of the average rate of change as the interval approaches zero.

Physical Interpretation of Derivatives

In applied contexts, the derivative can be interpreted as a physical quantity. For the function V(t) = 3t, the derivative V′(t) indicates how quickly the volume of water is increasing at a specific time. This can be understood as the flow rate of water into the bathtub, typically measured in gallons per minute.

Units of Measurement

When calculating derivatives in real-world scenarios, it is essential to include appropriate units to convey the meaning of the result. In this case, since V(t) is measured in gallons and t in minutes, the derivative V′(t) will have units of gallons per minute, providing a clear understanding of the rate at which water is being added to the bathtub.