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

Verified step by step guidance

1. 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.
2. 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).
3. Step 3: Calculate the derivative: f'(t) = 2t - 4. This is the velocity function v(t).
4. 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).
5. 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.