Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.
On what intervals is the speed increasing?
f(t) = t² - 4t; 0 ≤ t ≤ 5

검증된 단계별 안내

Step 1: Understand that speed is the absolute value of velocity. To find when speed is increasing, we need to determine when the derivative of the speed function is positive.

Step 2: Find the velocity function by taking the derivative of the position function f(t). The velocity v(t) is given by v(t) = f'(t) = \(\frac{d}{dt}\)(t^2 - 4t).

Step 3: Calculate the derivative: f'(t) = 2t - 4. This is the velocity function v(t).

Step 4: Determine when the velocity is increasing by finding the derivative of the velocity function, which is the acceleration a(t). So, a(t) = v'(t) = \(\frac{d}{dt}\)(2t - 4).

Step 5: Calculate the acceleration: a(t) = 2. Since the acceleration is constant and positive, the velocity is increasing for all t in the interval [0, 5]. Therefore, the speed is increasing wherever the velocity is positive.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Position Function

The position function, denoted as s = f(t), describes the location of an object at any given time t. In this case, f(t) = t² - 4t represents a quadratic function that models the object's position in feet. Understanding this function is crucial for analyzing the object's motion and determining its speed and acceleration.

Velocity and Speed

Velocity is the rate of change of position with respect to time, calculated as the derivative of the position function, v(t) = f'(t). Speed, being the absolute value of velocity, indicates how fast the object is moving regardless of direction. To determine when speed is increasing, one must analyze the behavior of the velocity function over the specified interval.

Acceleration

Acceleration is the rate of change of velocity with respect to time, represented as the derivative of the velocity function, a(t) = v'(t). When acceleration is positive, the speed of the object is increasing. Thus, to find the intervals where speed is increasing, one must identify where the acceleration function is greater than zero within the given time frame.

Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.On what intervals is the speed increasing?f(t) = t2 - 4t; 0 ≤ t ≤ 5