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) = 18t - 3t²; 0 ≤ t ≤ 8

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 analyze the velocity function and its derivative.
2. Step 2: Find the velocity function v(t) by differentiating the position function f(t) with respect to time t. This gives v(t) = f'(t) = 18 - 6t.
3. Step 3: 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) = -6.
4. Step 4: Since the acceleration a(t) = -6 is constant and negative, the velocity is always decreasing. However, speed increases when velocity and acceleration have opposite signs.
5. Step 5: Analyze the intervals where velocity is negative (v(t) < 0) because speed increases when velocity is negative and decreasing. Solve 18 - 6t < 0 to find the interval where speed is increasing.

Key Concepts

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) = 18t - 3t² represents a quadratic function where the position is influenced by both linear and quadratic terms. 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.