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) = 6t³ + 36t² - 54t; 0 ≤ t ≤ 4

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 derivative of the velocity function.
2. Step 2: First, find the velocity function v(t) by taking the derivative of the position function f(t). So, v(t) = f'(t) = \frac{d}{dt}(6t^3 + 36t^2 - 54t).
3. Step 3: Calculate the derivative: v(t) = 18t^2 + 72t - 54.
4. Step 4: To find when the speed is increasing, we need to find the derivative of the velocity function, which is the acceleration a(t). So, a(t) = v'(t) = \frac{d}{dt}(18t^2 + 72t - 54).
5. Step 5: Calculate the acceleration: a(t) = 36t + 72. Determine the intervals where a(t) > 0, as this indicates the velocity is increasing, and thus the speed is increasing.

Key Concepts

Velocity and Speed

Velocity is the rate of change of position with respect to time, represented mathematically as the derivative of the position function, f(t). Speed, being the absolute value of velocity, indicates how fast an object is moving regardless of direction. To determine when speed is increasing, one must analyze the behavior of the velocity function over the given interval.

Acceleration

Acceleration is the rate of change of velocity with respect to time, which can be found by taking the derivative of the velocity function. If the acceleration is positive, the speed of the object is increasing. Therefore, to find intervals where speed is increasing, we need to examine where the derivative of the velocity function is positive.

Critical Points and Intervals

Critical points occur where the first derivative of a function is zero or undefined, indicating potential maxima, minima, or points of inflection. By analyzing these points and the sign of the derivative in the intervals between them, one can determine where the function is increasing or decreasing. This is essential for identifying the intervals where speed is increasing in the context of the given position function.