106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (b) velocity equal to zero?
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Verified step by step guidance

1. To find when the velocity is zero, we need to identify the points where the derivative of the position function s=f(t) is zero. These points correspond to the local maxima and minima of the graph.
2. Examine the graph of s=f(t) and look for points where the tangent to the curve is horizontal. These are the points where the slope of the tangent line is zero, indicating that the velocity is zero.
3. From the graph, observe the peaks and troughs. The velocity is zero at these points because the object changes direction, which occurs at local maxima and minima.
4. Estimate the time values at these points by looking at the x-axis. For example, if a peak occurs at t=5 seconds, then the velocity is zero at t=5 seconds.
5. Repeat this process for each peak and trough in the graph to find all the times when the velocity is zero.

Key Concepts

Position Function

The position function, denoted as s = f(t), describes the location of an object along a coordinate line at any given time t. It is a continuous function that can be graphed to visualize the object's motion over time. Understanding this function is crucial for analyzing how the object's position changes, which directly relates to its velocity and acceleration.

Velocity

Velocity is the rate of change of the position function with respect to time, mathematically represented as v(t) = f'(t). It indicates how fast and in what direction the object is moving. When the velocity is equal to zero, it signifies that the object is momentarily at rest, which can be identified by finding the points where the tangent to the position graph is horizontal.

Critical Points

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. In the context of motion, these points are essential for determining when the velocity of the object is zero. Analyzing critical points helps in understanding the object's behavior, such as when it changes direction or comes to a stop.