Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


f. When is it farthest from the axis origin?


s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

Verified step by step guidance

1. Step 1: Identify the position function s(t) = t³ - 6t² + 7t. This function describes the position of the object along the s-axis as a function of time t.
2. Step 2: Calculate the velocity function v(t) by finding the first derivative of the position function s(t) with respect to time t. This is v(t) = ds/dt = f'(t).
3. Step 3: Calculate the acceleration function a(t) by finding the second derivative of the position function s(t) with respect to time t. This is a(t) = d²s/dt² = f''(t).
4. Step 4: Graph the position function s(t), the velocity function v(t), and the acceleration function a(t) over the interval 0 ≤ t ≤ 4. Analyze the graphs to understand the object's motion.
5. Step 5: Determine when the object is farthest from the axis origin by finding the maximum value of the position function s(t) within the given interval. This involves finding critical points by setting the derivative v(t) = 0 and analyzing the behavior of s(t) at these points and the endpoints of the interval.

Key Concepts

Position Function

The position function, s = f(t), describes the location of an object along a specified axis as a function of time. In this context, s = t³ - 6t² + 7t represents the object's position on the s-axis over the interval 0 ≤ t ≤ 4. Understanding this function is crucial for determining the object's position at any given time and analyzing its motion.

Velocity Function

The velocity function, v(t) = ds/dt = f'(t), represents the rate of change of the object's position with respect to time, indicating how fast the object is moving along the s-axis. Calculating the derivative of the position function provides the velocity function, which is essential for understanding the object's speed and direction at any given moment.

Acceleration Function

The acceleration function, a(t) = d²s/dt² = f''(t), measures the rate of change of velocity with respect to time, indicating how the object's speed is increasing or decreasing. By taking the second derivative of the position function, we obtain the acceleration function, which helps in analyzing the object's dynamic behavior, such as speeding up or slowing down.