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.
Suppose the speed of a car approaching a stop sign is given by v (t) = (t-5)², for 0 ≤ t ≤ 5, where t is measured in seconds and v(t) is measured in meters per second. Find v′(3).

Verified step by step guidance

1. First, identify the function given: v(t) = (t - 5)². This represents the speed of the car as a function of time.
2. To find the derivative v'(t), apply the power rule to the function v(t) = (t - 5)². The power rule states that if f(t) = tⁿ, then f'(t) = n * tⁿ⁻¹.
3. Using the power rule, differentiate v(t) = (t - 5)². The derivative v'(t) will be 2 * (t - 5) * 1, since the derivative of (t - 5) with respect to t is 1.
4. Substitute t = 3 into the derivative v'(t) to find v'(3). This will give you the rate of change of speed at t = 3 seconds.
5. Interpret the physical meaning of v'(3): This value represents the acceleration of the car at t = 3 seconds, measured in meters per second squared (m/s²). It indicates how quickly the car's speed is changing at that specific moment.

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes at a given point. In the context of motion, it represents the instantaneous rate of change of position with respect to time, which is the velocity. For the function v(t) = (t-5)², the derivative v'(t) will provide the speed of the car at any specific time t.

Interpretation of Derivatives in Physics

In physics, the derivative of a position function with respect to time gives the velocity of an object. This interpretation is crucial for understanding motion; for example, if v(t) represents the speed of a car, then v'(t) indicates how that speed is changing at a specific moment. This can inform us about acceleration or deceleration as the car approaches a stop sign.

Units of Measurement

When calculating derivatives in a physical context, it is essential to include units to convey meaningful information. In this case, time is measured in seconds and velocity in meters per second. When finding v'(3), the result should be expressed in appropriate units to reflect the physical quantity being measured, ensuring clarity in interpretation.